Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a...
Transcript of Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a...
![Page 1: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/1.jpg)
Introduction
Linear Algebra. Session 1
Dr. Marco A Roque Sol
01/18/2018
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 2: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/2.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Table of contents
IntroductionSystems of linear equationsGaussian elimination, Gauss-Jordan reductionMatrices, matrix algebraDeterminantsVector spacesLinear independenceBasis and dimensionCoordinates, change of basisLinear transformationsOrthogonalityInner products and normsThe Gram-Schmidt orthogonalization processEigenvalues and eigenvectors. Singular Value DecompositionMatrix exponentials and Diagonalization
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 3: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/3.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Table of contents
Introduction
Systems of linear equationsGaussian elimination, Gauss-Jordan reductionMatrices, matrix algebraDeterminantsVector spacesLinear independenceBasis and dimensionCoordinates, change of basisLinear transformationsOrthogonalityInner products and normsThe Gram-Schmidt orthogonalization processEigenvalues and eigenvectors. Singular Value DecompositionMatrix exponentials and Diagonalization
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 4: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/4.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Table of contents
IntroductionSystems of linear equations
Gaussian elimination, Gauss-Jordan reductionMatrices, matrix algebraDeterminantsVector spacesLinear independenceBasis and dimensionCoordinates, change of basisLinear transformationsOrthogonalityInner products and normsThe Gram-Schmidt orthogonalization processEigenvalues and eigenvectors. Singular Value DecompositionMatrix exponentials and Diagonalization
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 5: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/5.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Table of contents
IntroductionSystems of linear equationsGaussian elimination, Gauss-Jordan reduction
Matrices, matrix algebraDeterminantsVector spacesLinear independenceBasis and dimensionCoordinates, change of basisLinear transformationsOrthogonalityInner products and normsThe Gram-Schmidt orthogonalization processEigenvalues and eigenvectors. Singular Value DecompositionMatrix exponentials and Diagonalization
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 6: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/6.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Table of contents
IntroductionSystems of linear equationsGaussian elimination, Gauss-Jordan reductionMatrices, matrix algebra
DeterminantsVector spacesLinear independenceBasis and dimensionCoordinates, change of basisLinear transformationsOrthogonalityInner products and normsThe Gram-Schmidt orthogonalization processEigenvalues and eigenvectors. Singular Value DecompositionMatrix exponentials and Diagonalization
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 7: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/7.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Table of contents
IntroductionSystems of linear equationsGaussian elimination, Gauss-Jordan reductionMatrices, matrix algebraDeterminants
Vector spacesLinear independenceBasis and dimensionCoordinates, change of basisLinear transformationsOrthogonalityInner products and normsThe Gram-Schmidt orthogonalization processEigenvalues and eigenvectors. Singular Value DecompositionMatrix exponentials and Diagonalization
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 8: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/8.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Table of contents
IntroductionSystems of linear equationsGaussian elimination, Gauss-Jordan reductionMatrices, matrix algebraDeterminantsVector spaces
Linear independenceBasis and dimensionCoordinates, change of basisLinear transformationsOrthogonalityInner products and normsThe Gram-Schmidt orthogonalization processEigenvalues and eigenvectors. Singular Value DecompositionMatrix exponentials and Diagonalization
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 9: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/9.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Table of contents
IntroductionSystems of linear equationsGaussian elimination, Gauss-Jordan reductionMatrices, matrix algebraDeterminantsVector spacesLinear independence
Basis and dimensionCoordinates, change of basisLinear transformationsOrthogonalityInner products and normsThe Gram-Schmidt orthogonalization processEigenvalues and eigenvectors. Singular Value DecompositionMatrix exponentials and Diagonalization
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 10: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/10.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Table of contents
IntroductionSystems of linear equationsGaussian elimination, Gauss-Jordan reductionMatrices, matrix algebraDeterminantsVector spacesLinear independenceBasis and dimension
Coordinates, change of basisLinear transformationsOrthogonalityInner products and normsThe Gram-Schmidt orthogonalization processEigenvalues and eigenvectors. Singular Value DecompositionMatrix exponentials and Diagonalization
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 11: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/11.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Table of contents
IntroductionSystems of linear equationsGaussian elimination, Gauss-Jordan reductionMatrices, matrix algebraDeterminantsVector spacesLinear independenceBasis and dimensionCoordinates, change of basis
Linear transformationsOrthogonalityInner products and normsThe Gram-Schmidt orthogonalization processEigenvalues and eigenvectors. Singular Value DecompositionMatrix exponentials and Diagonalization
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 12: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/12.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Table of contents
IntroductionSystems of linear equationsGaussian elimination, Gauss-Jordan reductionMatrices, matrix algebraDeterminantsVector spacesLinear independenceBasis and dimensionCoordinates, change of basisLinear transformations
OrthogonalityInner products and normsThe Gram-Schmidt orthogonalization processEigenvalues and eigenvectors. Singular Value DecompositionMatrix exponentials and Diagonalization
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 13: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/13.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Table of contents
IntroductionSystems of linear equationsGaussian elimination, Gauss-Jordan reductionMatrices, matrix algebraDeterminantsVector spacesLinear independenceBasis and dimensionCoordinates, change of basisLinear transformationsOrthogonality
Inner products and normsThe Gram-Schmidt orthogonalization processEigenvalues and eigenvectors. Singular Value DecompositionMatrix exponentials and Diagonalization
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 14: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/14.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Table of contents
IntroductionSystems of linear equationsGaussian elimination, Gauss-Jordan reductionMatrices, matrix algebraDeterminantsVector spacesLinear independenceBasis and dimensionCoordinates, change of basisLinear transformationsOrthogonalityInner products and norms
The Gram-Schmidt orthogonalization processEigenvalues and eigenvectors. Singular Value DecompositionMatrix exponentials and Diagonalization
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 15: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/15.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Table of contents
IntroductionSystems of linear equationsGaussian elimination, Gauss-Jordan reductionMatrices, matrix algebraDeterminantsVector spacesLinear independenceBasis and dimensionCoordinates, change of basisLinear transformationsOrthogonalityInner products and normsThe Gram-Schmidt orthogonalization process
Eigenvalues and eigenvectors. Singular Value DecompositionMatrix exponentials and Diagonalization
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 16: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/16.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Table of contents
IntroductionSystems of linear equationsGaussian elimination, Gauss-Jordan reductionMatrices, matrix algebraDeterminantsVector spacesLinear independenceBasis and dimensionCoordinates, change of basisLinear transformationsOrthogonalityInner products and normsThe Gram-Schmidt orthogonalization processEigenvalues and eigenvectors. Singular Value Decomposition
Matrix exponentials and Diagonalization
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 17: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/17.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Table of contents
IntroductionSystems of linear equationsGaussian elimination, Gauss-Jordan reductionMatrices, matrix algebraDeterminantsVector spacesLinear independenceBasis and dimensionCoordinates, change of basisLinear transformationsOrthogonalityInner products and normsThe Gram-Schmidt orthogonalization processEigenvalues and eigenvectors. Singular Value DecompositionMatrix exponentials and Diagonalization
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 18: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/18.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
A Survey
Linear Algebra
( Medical Imaging )
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 19: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/19.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
A Survey
Linear Algebra
( Medical Imaging )
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 20: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/20.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
A Survey
Linear Algebra
( Medical Imaging )
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 21: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/21.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
A Survey
Linear Algebra
( Medical Imaging )Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 22: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/22.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
A Survey
Linear algebra is a cornerstone in undergraduate mathematicaleducation. It develops a general language used by all scientists andis interdisciplinary in essence. It hence, evolves naturally towardsabstraction. For most students, it is a first contact with modernmathematics.
Here are some concrete areas where we can find applications ofLinear Algebra.
Abstract Thinking
Chemistry
Coding theory
Coupled oscillations
Cryptography
Economics
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 23: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/23.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
A Survey
Linear algebra
is a cornerstone in undergraduate mathematicaleducation. It develops a general language used by all scientists andis interdisciplinary in essence. It hence, evolves naturally towardsabstraction. For most students, it is a first contact with modernmathematics.
Here are some concrete areas where we can find applications ofLinear Algebra.
Abstract Thinking
Chemistry
Coding theory
Coupled oscillations
Cryptography
Economics
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 24: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/24.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
A Survey
Linear algebra is a cornerstone
in undergraduate mathematicaleducation. It develops a general language used by all scientists andis interdisciplinary in essence. It hence, evolves naturally towardsabstraction. For most students, it is a first contact with modernmathematics.
Here are some concrete areas where we can find applications ofLinear Algebra.
Abstract Thinking
Chemistry
Coding theory
Coupled oscillations
Cryptography
Economics
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 25: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/25.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
A Survey
Linear algebra is a cornerstone in undergraduate mathematicaleducation.
It develops a general language used by all scientists andis interdisciplinary in essence. It hence, evolves naturally towardsabstraction. For most students, it is a first contact with modernmathematics.
Here are some concrete areas where we can find applications ofLinear Algebra.
Abstract Thinking
Chemistry
Coding theory
Coupled oscillations
Cryptography
Economics
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 26: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/26.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
A Survey
Linear algebra is a cornerstone in undergraduate mathematicaleducation. It develops
a general language used by all scientists andis interdisciplinary in essence. It hence, evolves naturally towardsabstraction. For most students, it is a first contact with modernmathematics.
Here are some concrete areas where we can find applications ofLinear Algebra.
Abstract Thinking
Chemistry
Coding theory
Coupled oscillations
Cryptography
Economics
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 27: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/27.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
A Survey
Linear algebra is a cornerstone in undergraduate mathematicaleducation. It develops a general language used
by all scientists andis interdisciplinary in essence. It hence, evolves naturally towardsabstraction. For most students, it is a first contact with modernmathematics.
Here are some concrete areas where we can find applications ofLinear Algebra.
Abstract Thinking
Chemistry
Coding theory
Coupled oscillations
Cryptography
Economics
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 28: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/28.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
A Survey
Linear algebra is a cornerstone in undergraduate mathematicaleducation. It develops a general language used by all scientists and
is interdisciplinary in essence. It hence, evolves naturally towardsabstraction. For most students, it is a first contact with modernmathematics.
Here are some concrete areas where we can find applications ofLinear Algebra.
Abstract Thinking
Chemistry
Coding theory
Coupled oscillations
Cryptography
Economics
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 29: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/29.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
A Survey
Linear algebra is a cornerstone in undergraduate mathematicaleducation. It develops a general language used by all scientists andis interdisciplinary
in essence. It hence, evolves naturally towardsabstraction. For most students, it is a first contact with modernmathematics.
Here are some concrete areas where we can find applications ofLinear Algebra.
Abstract Thinking
Chemistry
Coding theory
Coupled oscillations
Cryptography
Economics
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 30: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/30.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
A Survey
Linear algebra is a cornerstone in undergraduate mathematicaleducation. It develops a general language used by all scientists andis interdisciplinary in essence.
It hence, evolves naturally towardsabstraction. For most students, it is a first contact with modernmathematics.
Here are some concrete areas where we can find applications ofLinear Algebra.
Abstract Thinking
Chemistry
Coding theory
Coupled oscillations
Cryptography
Economics
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 31: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/31.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
A Survey
Linear algebra is a cornerstone in undergraduate mathematicaleducation. It develops a general language used by all scientists andis interdisciplinary in essence. It hence,
evolves naturally towardsabstraction. For most students, it is a first contact with modernmathematics.
Here are some concrete areas where we can find applications ofLinear Algebra.
Abstract Thinking
Chemistry
Coding theory
Coupled oscillations
Cryptography
Economics
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 32: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/32.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
A Survey
Linear algebra is a cornerstone in undergraduate mathematicaleducation. It develops a general language used by all scientists andis interdisciplinary in essence. It hence, evolves naturally
towardsabstraction. For most students, it is a first contact with modernmathematics.
Here are some concrete areas where we can find applications ofLinear Algebra.
Abstract Thinking
Chemistry
Coding theory
Coupled oscillations
Cryptography
Economics
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 33: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/33.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
A Survey
Linear algebra is a cornerstone in undergraduate mathematicaleducation. It develops a general language used by all scientists andis interdisciplinary in essence. It hence, evolves naturally towardsabstraction.
For most students, it is a first contact with modernmathematics.
Here are some concrete areas where we can find applications ofLinear Algebra.
Abstract Thinking
Chemistry
Coding theory
Coupled oscillations
Cryptography
Economics
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 34: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/34.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
A Survey
Linear algebra is a cornerstone in undergraduate mathematicaleducation. It develops a general language used by all scientists andis interdisciplinary in essence. It hence, evolves naturally towardsabstraction. For most students,
it is a first contact with modernmathematics.
Here are some concrete areas where we can find applications ofLinear Algebra.
Abstract Thinking
Chemistry
Coding theory
Coupled oscillations
Cryptography
Economics
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 35: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/35.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
A Survey
Linear algebra is a cornerstone in undergraduate mathematicaleducation. It develops a general language used by all scientists andis interdisciplinary in essence. It hence, evolves naturally towardsabstraction. For most students, it is
a first contact with modernmathematics.
Here are some concrete areas where we can find applications ofLinear Algebra.
Abstract Thinking
Chemistry
Coding theory
Coupled oscillations
Cryptography
Economics
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 36: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/36.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
A Survey
Linear algebra is a cornerstone in undergraduate mathematicaleducation. It develops a general language used by all scientists andis interdisciplinary in essence. It hence, evolves naturally towardsabstraction. For most students, it is a first contact
with modernmathematics.
Here are some concrete areas where we can find applications ofLinear Algebra.
Abstract Thinking
Chemistry
Coding theory
Coupled oscillations
Cryptography
Economics
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 37: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/37.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
A Survey
Linear algebra is a cornerstone in undergraduate mathematicaleducation. It develops a general language used by all scientists andis interdisciplinary in essence. It hence, evolves naturally towardsabstraction. For most students, it is a first contact with modern
mathematics.
Here are some concrete areas where we can find applications ofLinear Algebra.
Abstract Thinking
Chemistry
Coding theory
Coupled oscillations
Cryptography
Economics
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 38: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/38.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
A Survey
Linear algebra is a cornerstone in undergraduate mathematicaleducation. It develops a general language used by all scientists andis interdisciplinary in essence. It hence, evolves naturally towardsabstraction. For most students, it is a first contact with modernmathematics.
Here are some concrete areas where we can find applications ofLinear Algebra.
Abstract Thinking
Chemistry
Coding theory
Coupled oscillations
Cryptography
Economics
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 39: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/39.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
A Survey
Linear algebra is a cornerstone in undergraduate mathematicaleducation. It develops a general language used by all scientists andis interdisciplinary in essence. It hence, evolves naturally towardsabstraction. For most students, it is a first contact with modernmathematics.
Here are
some concrete areas where we can find applications ofLinear Algebra.
Abstract Thinking
Chemistry
Coding theory
Coupled oscillations
Cryptography
Economics
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 40: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/40.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
A Survey
Linear algebra is a cornerstone in undergraduate mathematicaleducation. It develops a general language used by all scientists andis interdisciplinary in essence. It hence, evolves naturally towardsabstraction. For most students, it is a first contact with modernmathematics.
Here are some concrete areas
where we can find applications ofLinear Algebra.
Abstract Thinking
Chemistry
Coding theory
Coupled oscillations
Cryptography
Economics
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 41: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/41.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
A Survey
Linear algebra is a cornerstone in undergraduate mathematicaleducation. It develops a general language used by all scientists andis interdisciplinary in essence. It hence, evolves naturally towardsabstraction. For most students, it is a first contact with modernmathematics.
Here are some concrete areas where we can find
applications ofLinear Algebra.
Abstract Thinking
Chemistry
Coding theory
Coupled oscillations
Cryptography
Economics
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 42: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/42.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
A Survey
Linear algebra is a cornerstone in undergraduate mathematicaleducation. It develops a general language used by all scientists andis interdisciplinary in essence. It hence, evolves naturally towardsabstraction. For most students, it is a first contact with modernmathematics.
Here are some concrete areas where we can find applications of
Linear Algebra.
Abstract Thinking
Chemistry
Coding theory
Coupled oscillations
Cryptography
Economics
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 43: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/43.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
A Survey
Linear algebra is a cornerstone in undergraduate mathematicaleducation. It develops a general language used by all scientists andis interdisciplinary in essence. It hence, evolves naturally towardsabstraction. For most students, it is a first contact with modernmathematics.
Here are some concrete areas where we can find applications ofLinear Algebra.
Abstract Thinking
Chemistry
Coding theory
Coupled oscillations
Cryptography
Economics
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 44: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/44.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
A Survey
Linear algebra is a cornerstone in undergraduate mathematicaleducation. It develops a general language used by all scientists andis interdisciplinary in essence. It hence, evolves naturally towardsabstraction. For most students, it is a first contact with modernmathematics.
Here are some concrete areas where we can find applications ofLinear Algebra.
Abstract Thinking
Chemistry
Coding theory
Coupled oscillations
Cryptography
Economics
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 45: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/45.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
A Survey
Linear algebra is a cornerstone in undergraduate mathematicaleducation. It develops a general language used by all scientists andis interdisciplinary in essence. It hence, evolves naturally towardsabstraction. For most students, it is a first contact with modernmathematics.
Here are some concrete areas where we can find applications ofLinear Algebra.
Abstract Thinking
Chemistry
Coding theory
Coupled oscillations
Cryptography
Economics
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 46: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/46.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
A Survey
Linear algebra is a cornerstone in undergraduate mathematicaleducation. It develops a general language used by all scientists andis interdisciplinary in essence. It hence, evolves naturally towardsabstraction. For most students, it is a first contact with modernmathematics.
Here are some concrete areas where we can find applications ofLinear Algebra.
Abstract Thinking
Chemistry
Coding theory
Coupled oscillations
Cryptography
Economics
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 47: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/47.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
A Survey
Linear algebra is a cornerstone in undergraduate mathematicaleducation. It develops a general language used by all scientists andis interdisciplinary in essence. It hence, evolves naturally towardsabstraction. For most students, it is a first contact with modernmathematics.
Here are some concrete areas where we can find applications ofLinear Algebra.
Abstract Thinking
Chemistry
Coding theory
Coupled oscillations
Cryptography
Economics
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 48: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/48.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
A Survey
Linear algebra is a cornerstone in undergraduate mathematicaleducation. It develops a general language used by all scientists andis interdisciplinary in essence. It hence, evolves naturally towardsabstraction. For most students, it is a first contact with modernmathematics.
Here are some concrete areas where we can find applications ofLinear Algebra.
Abstract Thinking
Chemistry
Coding theory
Coupled oscillations
Cryptography
Economics
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 49: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/49.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
A Survey
Linear algebra is a cornerstone in undergraduate mathematicaleducation. It develops a general language used by all scientists andis interdisciplinary in essence. It hence, evolves naturally towardsabstraction. For most students, it is a first contact with modernmathematics.
Here are some concrete areas where we can find applications ofLinear Algebra.
Abstract Thinking
Chemistry
Coding theory
Coupled oscillations
Cryptography
Economics
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 50: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/50.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
IA Survey
Classical Electromagnetism.
Geophysics.
Elimination Theory.
Game Theory.
Genetics.
Geometry.
Graph theory.
Heat distribution.
Image compression.
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 51: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/51.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
IA Survey
Classical Electromagnetism.
Geophysics.
Elimination Theory.
Game Theory.
Genetics.
Geometry.
Graph theory.
Heat distribution.
Image compression.
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 52: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/52.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
IA Survey
Classical Electromagnetism.
Geophysics.
Elimination Theory.
Game Theory.
Genetics.
Geometry.
Graph theory.
Heat distribution.
Image compression.
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 53: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/53.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
IA Survey
Classical Electromagnetism.
Geophysics.
Elimination Theory.
Game Theory.
Genetics.
Geometry.
Graph theory.
Heat distribution.
Image compression.
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 54: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/54.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
IA Survey
Classical Electromagnetism.
Geophysics.
Elimination Theory.
Game Theory.
Genetics.
Geometry.
Graph theory.
Heat distribution.
Image compression.
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 55: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/55.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
IA Survey
Classical Electromagnetism.
Geophysics.
Elimination Theory.
Game Theory.
Genetics.
Geometry.
Graph theory.
Heat distribution.
Image compression.
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 56: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/56.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
IA Survey
Classical Electromagnetism.
Geophysics.
Elimination Theory.
Game Theory.
Genetics.
Geometry.
Graph theory.
Heat distribution.
Image compression.
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 57: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/57.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
IA Survey
Classical Electromagnetism.
Geophysics.
Elimination Theory.
Game Theory.
Genetics.
Geometry.
Graph theory.
Heat distribution.
Image compression.
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 58: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/58.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
IA Survey
Classical Electromagnetism.
Geophysics.
Elimination Theory.
Game Theory.
Genetics.
Geometry.
Graph theory.
Heat distribution.
Image compression.
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 59: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/59.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
IA Survey
Classical Electromagnetism.
Geophysics.
Elimination Theory.
Game Theory.
Genetics.
Geometry.
Graph theory.
Heat distribution.
Image compression.
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 60: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/60.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
IA Survey
Classical Electromagnetism.
Geophysics.
Elimination Theory.
Game Theory.
Genetics.
Geometry.
Graph theory.
Heat distribution.
Image compression.
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 61: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/61.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
A Survey
Linear Programming.
Markov Chains.
Networks.
Sociology
The Fibonacci numbers.
Eigenstates.
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 62: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/62.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
A Survey
Linear Programming.
Markov Chains.
Networks.
Sociology
The Fibonacci numbers.
Eigenstates.
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 63: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/63.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
A Survey
Linear Programming.
Markov Chains.
Networks.
Sociology
The Fibonacci numbers.
Eigenstates.
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 64: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/64.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
A Survey
Linear Programming.
Markov Chains.
Networks.
Sociology
The Fibonacci numbers.
Eigenstates.
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 65: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/65.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
A Survey
Linear Programming.
Markov Chains.
Networks.
Sociology
The Fibonacci numbers.
Eigenstates.
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 66: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/66.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
A Survey
Linear Programming.
Markov Chains.
Networks.
Sociology
The Fibonacci numbers.
Eigenstates.
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 67: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/67.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
A Survey
Linear Programming.
Markov Chains.
Networks.
Sociology
The Fibonacci numbers.
Eigenstates.
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 68: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/68.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
A Survey
Linear Programming.
Markov Chains.
Networks.
Sociology
The Fibonacci numbers.
Eigenstates.
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 69: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/69.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
A Survey
Definition (Linear Equation)An equation of the form ax + by = c (5x + 7y = 6) is called linearbecause its solution set is a straight line in R2
A solution of the equation is a pair of numbers (x0, y0) ∈ R2 suchthat ax0 + by0 = c (5x0 + 7y0 = 6).
For example, (1, 1/7) and (−1/5, 1), are solutions. In another way,we can write the first solution, as x = 1, y = 1/5 and the secondone, as x = −1/5, y = 1
And the graph of the lines is:
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 70: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/70.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
A Survey
Definition (Linear Equation)
An equation of the form ax + by = c (5x + 7y = 6) is called linearbecause its solution set is a straight line in R2
A solution of the equation is a pair of numbers (x0, y0) ∈ R2 suchthat ax0 + by0 = c (5x0 + 7y0 = 6).
For example, (1, 1/7) and (−1/5, 1), are solutions. In another way,we can write the first solution, as x = 1, y = 1/5 and the secondone, as x = −1/5, y = 1
And the graph of the lines is:
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 71: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/71.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
A Survey
Definition (Linear Equation)An equation
of the form ax + by = c (5x + 7y = 6) is called linearbecause its solution set is a straight line in R2
A solution of the equation is a pair of numbers (x0, y0) ∈ R2 suchthat ax0 + by0 = c (5x0 + 7y0 = 6).
For example, (1, 1/7) and (−1/5, 1), are solutions. In another way,we can write the first solution, as x = 1, y = 1/5 and the secondone, as x = −1/5, y = 1
And the graph of the lines is:
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 72: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/72.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
A Survey
Definition (Linear Equation)An equation of the form
ax + by = c (5x + 7y = 6) is called linearbecause its solution set is a straight line in R2
A solution of the equation is a pair of numbers (x0, y0) ∈ R2 suchthat ax0 + by0 = c (5x0 + 7y0 = 6).
For example, (1, 1/7) and (−1/5, 1), are solutions. In another way,we can write the first solution, as x = 1, y = 1/5 and the secondone, as x = −1/5, y = 1
And the graph of the lines is:
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 73: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/73.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
A Survey
Definition (Linear Equation)An equation of the form ax + by = c
(5x + 7y = 6) is called linearbecause its solution set is a straight line in R2
A solution of the equation is a pair of numbers (x0, y0) ∈ R2 suchthat ax0 + by0 = c (5x0 + 7y0 = 6).
For example, (1, 1/7) and (−1/5, 1), are solutions. In another way,we can write the first solution, as x = 1, y = 1/5 and the secondone, as x = −1/5, y = 1
And the graph of the lines is:
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 74: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/74.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
A Survey
Definition (Linear Equation)An equation of the form ax + by = c (5x + 7y = 6)
is called linearbecause its solution set is a straight line in R2
A solution of the equation is a pair of numbers (x0, y0) ∈ R2 suchthat ax0 + by0 = c (5x0 + 7y0 = 6).
For example, (1, 1/7) and (−1/5, 1), are solutions. In another way,we can write the first solution, as x = 1, y = 1/5 and the secondone, as x = −1/5, y = 1
And the graph of the lines is:
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 75: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/75.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
A Survey
Definition (Linear Equation)An equation of the form ax + by = c (5x + 7y = 6) is called linear
because its solution set is a straight line in R2
A solution of the equation is a pair of numbers (x0, y0) ∈ R2 suchthat ax0 + by0 = c (5x0 + 7y0 = 6).
For example, (1, 1/7) and (−1/5, 1), are solutions. In another way,we can write the first solution, as x = 1, y = 1/5 and the secondone, as x = −1/5, y = 1
And the graph of the lines is:
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 76: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/76.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
A Survey
Definition (Linear Equation)An equation of the form ax + by = c (5x + 7y = 6) is called linearbecause
its solution set is a straight line in R2
A solution of the equation is a pair of numbers (x0, y0) ∈ R2 suchthat ax0 + by0 = c (5x0 + 7y0 = 6).
For example, (1, 1/7) and (−1/5, 1), are solutions. In another way,we can write the first solution, as x = 1, y = 1/5 and the secondone, as x = −1/5, y = 1
And the graph of the lines is:
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 77: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/77.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
A Survey
Definition (Linear Equation)An equation of the form ax + by = c (5x + 7y = 6) is called linearbecause its solution set is
a straight line in R2
A solution of the equation is a pair of numbers (x0, y0) ∈ R2 suchthat ax0 + by0 = c (5x0 + 7y0 = 6).
For example, (1, 1/7) and (−1/5, 1), are solutions. In another way,we can write the first solution, as x = 1, y = 1/5 and the secondone, as x = −1/5, y = 1
And the graph of the lines is:
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 78: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/78.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
A Survey
Definition (Linear Equation)An equation of the form ax + by = c (5x + 7y = 6) is called linearbecause its solution set is a straight line in R2
A solution of the equation is a pair of numbers (x0, y0) ∈ R2 suchthat ax0 + by0 = c (5x0 + 7y0 = 6).
For example, (1, 1/7) and (−1/5, 1), are solutions. In another way,we can write the first solution, as x = 1, y = 1/5 and the secondone, as x = −1/5, y = 1
And the graph of the lines is:
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 79: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/79.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
A Survey
Definition (Linear Equation)An equation of the form ax + by = c (5x + 7y = 6) is called linearbecause its solution set is a straight line in R2
A solution
of the equation is a pair of numbers (x0, y0) ∈ R2 suchthat ax0 + by0 = c (5x0 + 7y0 = 6).
For example, (1, 1/7) and (−1/5, 1), are solutions. In another way,we can write the first solution, as x = 1, y = 1/5 and the secondone, as x = −1/5, y = 1
And the graph of the lines is:
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 80: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/80.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
A Survey
Definition (Linear Equation)An equation of the form ax + by = c (5x + 7y = 6) is called linearbecause its solution set is a straight line in R2
A solution of the equation
is a pair of numbers (x0, y0) ∈ R2 suchthat ax0 + by0 = c (5x0 + 7y0 = 6).
For example, (1, 1/7) and (−1/5, 1), are solutions. In another way,we can write the first solution, as x = 1, y = 1/5 and the secondone, as x = −1/5, y = 1
And the graph of the lines is:
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 81: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/81.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
A Survey
Definition (Linear Equation)An equation of the form ax + by = c (5x + 7y = 6) is called linearbecause its solution set is a straight line in R2
A solution of the equation is a pair of numbers
(x0, y0) ∈ R2 suchthat ax0 + by0 = c (5x0 + 7y0 = 6).
For example, (1, 1/7) and (−1/5, 1), are solutions. In another way,we can write the first solution, as x = 1, y = 1/5 and the secondone, as x = −1/5, y = 1
And the graph of the lines is:
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 82: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/82.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
A Survey
Definition (Linear Equation)An equation of the form ax + by = c (5x + 7y = 6) is called linearbecause its solution set is a straight line in R2
A solution of the equation is a pair of numbers (x0, y0) ∈ R2
suchthat ax0 + by0 = c (5x0 + 7y0 = 6).
For example, (1, 1/7) and (−1/5, 1), are solutions. In another way,we can write the first solution, as x = 1, y = 1/5 and the secondone, as x = −1/5, y = 1
And the graph of the lines is:
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 83: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/83.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
A Survey
Definition (Linear Equation)An equation of the form ax + by = c (5x + 7y = 6) is called linearbecause its solution set is a straight line in R2
A solution of the equation is a pair of numbers (x0, y0) ∈ R2 suchthat
ax0 + by0 = c (5x0 + 7y0 = 6).
For example, (1, 1/7) and (−1/5, 1), are solutions. In another way,we can write the first solution, as x = 1, y = 1/5 and the secondone, as x = −1/5, y = 1
And the graph of the lines is:
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 84: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/84.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
A Survey
Definition (Linear Equation)An equation of the form ax + by = c (5x + 7y = 6) is called linearbecause its solution set is a straight line in R2
A solution of the equation is a pair of numbers (x0, y0) ∈ R2 suchthat ax0 + by0 = c
(5x0 + 7y0 = 6).
For example, (1, 1/7) and (−1/5, 1), are solutions. In another way,we can write the first solution, as x = 1, y = 1/5 and the secondone, as x = −1/5, y = 1
And the graph of the lines is:
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 85: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/85.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
A Survey
Definition (Linear Equation)An equation of the form ax + by = c (5x + 7y = 6) is called linearbecause its solution set is a straight line in R2
A solution of the equation is a pair of numbers (x0, y0) ∈ R2 suchthat ax0 + by0 = c (5x0 + 7y0 = 6).
For example, (1, 1/7) and (−1/5, 1), are solutions. In another way,we can write the first solution, as x = 1, y = 1/5 and the secondone, as x = −1/5, y = 1
And the graph of the lines is:
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 86: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/86.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
A Survey
Definition (Linear Equation)An equation of the form ax + by = c (5x + 7y = 6) is called linearbecause its solution set is a straight line in R2
A solution of the equation is a pair of numbers (x0, y0) ∈ R2 suchthat ax0 + by0 = c (5x0 + 7y0 = 6).
For example,
(1, 1/7) and (−1/5, 1), are solutions. In another way,we can write the first solution, as x = 1, y = 1/5 and the secondone, as x = −1/5, y = 1
And the graph of the lines is:
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 87: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/87.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
A Survey
Definition (Linear Equation)An equation of the form ax + by = c (5x + 7y = 6) is called linearbecause its solution set is a straight line in R2
A solution of the equation is a pair of numbers (x0, y0) ∈ R2 suchthat ax0 + by0 = c (5x0 + 7y0 = 6).
For example, (1, 1/7) and
(−1/5, 1), are solutions. In another way,we can write the first solution, as x = 1, y = 1/5 and the secondone, as x = −1/5, y = 1
And the graph of the lines is:
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 88: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/88.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
A Survey
Definition (Linear Equation)An equation of the form ax + by = c (5x + 7y = 6) is called linearbecause its solution set is a straight line in R2
A solution of the equation is a pair of numbers (x0, y0) ∈ R2 suchthat ax0 + by0 = c (5x0 + 7y0 = 6).
For example, (1, 1/7) and (−1/5, 1),
are solutions. In another way,we can write the first solution, as x = 1, y = 1/5 and the secondone, as x = −1/5, y = 1
And the graph of the lines is:
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 89: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/89.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
A Survey
Definition (Linear Equation)An equation of the form ax + by = c (5x + 7y = 6) is called linearbecause its solution set is a straight line in R2
A solution of the equation is a pair of numbers (x0, y0) ∈ R2 suchthat ax0 + by0 = c (5x0 + 7y0 = 6).
For example, (1, 1/7) and (−1/5, 1), are solutions.
In another way,we can write the first solution, as x = 1, y = 1/5 and the secondone, as x = −1/5, y = 1
And the graph of the lines is:
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 90: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/90.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
A Survey
Definition (Linear Equation)An equation of the form ax + by = c (5x + 7y = 6) is called linearbecause its solution set is a straight line in R2
A solution of the equation is a pair of numbers (x0, y0) ∈ R2 suchthat ax0 + by0 = c (5x0 + 7y0 = 6).
For example, (1, 1/7) and (−1/5, 1), are solutions. In another way,
we can write the first solution, as x = 1, y = 1/5 and the secondone, as x = −1/5, y = 1
And the graph of the lines is:
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 91: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/91.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
A Survey
Definition (Linear Equation)An equation of the form ax + by = c (5x + 7y = 6) is called linearbecause its solution set is a straight line in R2
A solution of the equation is a pair of numbers (x0, y0) ∈ R2 suchthat ax0 + by0 = c (5x0 + 7y0 = 6).
For example, (1, 1/7) and (−1/5, 1), are solutions. In another way,we can write
the first solution, as x = 1, y = 1/5 and the secondone, as x = −1/5, y = 1
And the graph of the lines is:
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 92: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/92.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
A Survey
Definition (Linear Equation)An equation of the form ax + by = c (5x + 7y = 6) is called linearbecause its solution set is a straight line in R2
A solution of the equation is a pair of numbers (x0, y0) ∈ R2 suchthat ax0 + by0 = c (5x0 + 7y0 = 6).
For example, (1, 1/7) and (−1/5, 1), are solutions. In another way,we can write the first solution, as
x = 1, y = 1/5 and the secondone, as x = −1/5, y = 1
And the graph of the lines is:
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 93: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/93.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
A Survey
Definition (Linear Equation)An equation of the form ax + by = c (5x + 7y = 6) is called linearbecause its solution set is a straight line in R2
A solution of the equation is a pair of numbers (x0, y0) ∈ R2 suchthat ax0 + by0 = c (5x0 + 7y0 = 6).
For example, (1, 1/7) and (−1/5, 1), are solutions. In another way,we can write the first solution, as x = 1, y = 1/5
and the secondone, as x = −1/5, y = 1
And the graph of the lines is:
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 94: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/94.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
A Survey
Definition (Linear Equation)An equation of the form ax + by = c (5x + 7y = 6) is called linearbecause its solution set is a straight line in R2
A solution of the equation is a pair of numbers (x0, y0) ∈ R2 suchthat ax0 + by0 = c (5x0 + 7y0 = 6).
For example, (1, 1/7) and (−1/5, 1), are solutions. In another way,we can write the first solution, as x = 1, y = 1/5 and the secondone, as
x = −1/5, y = 1
And the graph of the lines is:
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 95: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/95.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
A Survey
Definition (Linear Equation)An equation of the form ax + by = c (5x + 7y = 6) is called linearbecause its solution set is a straight line in R2
A solution of the equation is a pair of numbers (x0, y0) ∈ R2 suchthat ax0 + by0 = c (5x0 + 7y0 = 6).
For example, (1, 1/7) and (−1/5, 1), are solutions. In another way,we can write the first solution, as x = 1, y = 1/5 and the secondone, as x = −1/5, y = 1
And the graph of the lines is:
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 96: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/96.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
A Survey
Definition (Linear Equation)An equation of the form ax + by = c (5x + 7y = 6) is called linearbecause its solution set is a straight line in R2
A solution of the equation is a pair of numbers (x0, y0) ∈ R2 suchthat ax0 + by0 = c (5x0 + 7y0 = 6).
For example, (1, 1/7) and (−1/5, 1), are solutions. In another way,we can write the first solution, as x = 1, y = 1/5 and the secondone, as x = −1/5, y = 1
And the graph of the lines is:
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 97: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/97.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
A Survey
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 98: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/98.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Sistems of linear equations
General equation of a line
The General equation of a line is given by
ax0 + by0 = c
where x , y are variables and a, b, and c are constants (except forthe case a = b = 0 )
DefinitionA linear equation in the variables x1, x2, · · · , xn, is an equation ofthe form:
a1x1 + a2x2 + · · · anxn = b
where a1, a2, · · · an and b are constants.
A solution of the equation is an array of numbersα1, α2, · · ·αn ∈ Rn such that
a1α1 + a2α2 + · · ·+ anαn = b
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 99: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/99.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Sistems of linear equations
General equation of a line
The General equation of a line is given by
ax0 + by0 = c
where x , y are variables and a, b, and c are constants (except forthe case a = b = 0 )
DefinitionA linear equation in the variables x1, x2, · · · , xn, is an equation ofthe form:
a1x1 + a2x2 + · · · anxn = b
where a1, a2, · · · an and b are constants.
A solution of the equation is an array of numbersα1, α2, · · ·αn ∈ Rn such that
a1α1 + a2α2 + · · ·+ anαn = b
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 100: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/100.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Sistems of linear equations
General equation of a line
The General equation
of a line is given by
ax0 + by0 = c
where x , y are variables and a, b, and c are constants (except forthe case a = b = 0 )
DefinitionA linear equation in the variables x1, x2, · · · , xn, is an equation ofthe form:
a1x1 + a2x2 + · · · anxn = b
where a1, a2, · · · an and b are constants.
A solution of the equation is an array of numbersα1, α2, · · ·αn ∈ Rn such that
a1α1 + a2α2 + · · ·+ anαn = b
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 101: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/101.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Sistems of linear equations
General equation of a line
The General equation of a line
is given by
ax0 + by0 = c
where x , y are variables and a, b, and c are constants (except forthe case a = b = 0 )
DefinitionA linear equation in the variables x1, x2, · · · , xn, is an equation ofthe form:
a1x1 + a2x2 + · · · anxn = b
where a1, a2, · · · an and b are constants.
A solution of the equation is an array of numbersα1, α2, · · ·αn ∈ Rn such that
a1α1 + a2α2 + · · ·+ anαn = b
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 102: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/102.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Sistems of linear equations
General equation of a line
The General equation of a line is given by
ax0 + by0 = c
where x , y are variables and a, b, and c are constants (except forthe case a = b = 0 )
DefinitionA linear equation in the variables x1, x2, · · · , xn, is an equation ofthe form:
a1x1 + a2x2 + · · · anxn = b
where a1, a2, · · · an and b are constants.
A solution of the equation is an array of numbersα1, α2, · · ·αn ∈ Rn such that
a1α1 + a2α2 + · · ·+ anαn = b
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 103: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/103.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Sistems of linear equations
General equation of a line
The General equation of a line is given by
ax0 + by0 = c
where x , y are variables and a, b, and c are constants (except forthe case a = b = 0 )
DefinitionA linear equation in the variables x1, x2, · · · , xn, is an equation ofthe form:
a1x1 + a2x2 + · · · anxn = b
where a1, a2, · · · an and b are constants.
A solution of the equation is an array of numbersα1, α2, · · ·αn ∈ Rn such that
a1α1 + a2α2 + · · ·+ anαn = b
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 104: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/104.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Sistems of linear equations
General equation of a line
The General equation of a line is given by
ax0 + by0 = c
where x , y
are variables and a, b, and c are constants (except forthe case a = b = 0 )
DefinitionA linear equation in the variables x1, x2, · · · , xn, is an equation ofthe form:
a1x1 + a2x2 + · · · anxn = b
where a1, a2, · · · an and b are constants.
A solution of the equation is an array of numbersα1, α2, · · ·αn ∈ Rn such that
a1α1 + a2α2 + · · ·+ anαn = b
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 105: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/105.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Sistems of linear equations
General equation of a line
The General equation of a line is given by
ax0 + by0 = c
where x , y are variables and
a, b, and c are constants (except forthe case a = b = 0 )
DefinitionA linear equation in the variables x1, x2, · · · , xn, is an equation ofthe form:
a1x1 + a2x2 + · · · anxn = b
where a1, a2, · · · an and b are constants.
A solution of the equation is an array of numbersα1, α2, · · ·αn ∈ Rn such that
a1α1 + a2α2 + · · ·+ anαn = b
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 106: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/106.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Sistems of linear equations
General equation of a line
The General equation of a line is given by
ax0 + by0 = c
where x , y are variables and a, b, and c
are constants (except forthe case a = b = 0 )
DefinitionA linear equation in the variables x1, x2, · · · , xn, is an equation ofthe form:
a1x1 + a2x2 + · · · anxn = b
where a1, a2, · · · an and b are constants.
A solution of the equation is an array of numbersα1, α2, · · ·αn ∈ Rn such that
a1α1 + a2α2 + · · ·+ anαn = b
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 107: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/107.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Sistems of linear equations
General equation of a line
The General equation of a line is given by
ax0 + by0 = c
where x , y are variables and a, b, and c are constants (except forthe case a = b = 0 )
DefinitionA linear equation in the variables x1, x2, · · · , xn, is an equation ofthe form:
a1x1 + a2x2 + · · · anxn = b
where a1, a2, · · · an and b are constants.
A solution of the equation is an array of numbersα1, α2, · · ·αn ∈ Rn such that
a1α1 + a2α2 + · · ·+ anαn = b
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 108: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/108.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Sistems of linear equations
General equation of a line
The General equation of a line is given by
ax0 + by0 = c
where x , y are variables and a, b, and c are constants (except forthe case a = b = 0 )
Definition
A linear equation in the variables x1, x2, · · · , xn, is an equation ofthe form:
a1x1 + a2x2 + · · · anxn = b
where a1, a2, · · · an and b are constants.
A solution of the equation is an array of numbersα1, α2, · · ·αn ∈ Rn such that
a1α1 + a2α2 + · · ·+ anαn = b
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 109: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/109.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Sistems of linear equations
General equation of a line
The General equation of a line is given by
ax0 + by0 = c
where x , y are variables and a, b, and c are constants (except forthe case a = b = 0 )
DefinitionA linear equation
in the variables x1, x2, · · · , xn, is an equation ofthe form:
a1x1 + a2x2 + · · · anxn = b
where a1, a2, · · · an and b are constants.
A solution of the equation is an array of numbersα1, α2, · · ·αn ∈ Rn such that
a1α1 + a2α2 + · · ·+ anαn = b
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 110: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/110.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Sistems of linear equations
General equation of a line
The General equation of a line is given by
ax0 + by0 = c
where x , y are variables and a, b, and c are constants (except forthe case a = b = 0 )
DefinitionA linear equation in the variables
x1, x2, · · · , xn, is an equation ofthe form:
a1x1 + a2x2 + · · · anxn = b
where a1, a2, · · · an and b are constants.
A solution of the equation is an array of numbersα1, α2, · · ·αn ∈ Rn such that
a1α1 + a2α2 + · · ·+ anαn = b
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 111: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/111.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Sistems of linear equations
General equation of a line
The General equation of a line is given by
ax0 + by0 = c
where x , y are variables and a, b, and c are constants (except forthe case a = b = 0 )
DefinitionA linear equation in the variables x1, x2, · · · , xn, is an equation ofthe form:
a1x1 + a2x2 + · · · anxn = b
where a1, a2, · · · an and b are constants.
A solution of the equation is an array of numbersα1, α2, · · ·αn ∈ Rn such that
a1α1 + a2α2 + · · ·+ anαn = b
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 112: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/112.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Sistems of linear equations
General equation of a line
The General equation of a line is given by
ax0 + by0 = c
where x , y are variables and a, b, and c are constants (except forthe case a = b = 0 )
DefinitionA linear equation in the variables x1, x2, · · · , xn, is an equation ofthe form:
a1x1 + a2x2 + · · · anxn = b
where a1, a2, · · · an and b are constants.
A solution of the equation is an array of numbersα1, α2, · · ·αn ∈ Rn such that
a1α1 + a2α2 + · · ·+ anαn = b
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 113: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/113.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Sistems of linear equations
General equation of a line
The General equation of a line is given by
ax0 + by0 = c
where x , y are variables and a, b, and c are constants (except forthe case a = b = 0 )
DefinitionA linear equation in the variables x1, x2, · · · , xn, is an equation ofthe form:
a1x1 + a2x2 + · · · anxn = b
where a1, a2, · · · an and b are constants.
A solution of the equation is an array of numbersα1, α2, · · ·αn ∈ Rn such that
a1α1 + a2α2 + · · ·+ anαn = b
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 114: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/114.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Sistems of linear equations
General equation of a line
The General equation of a line is given by
ax0 + by0 = c
where x , y are variables and a, b, and c are constants (except forthe case a = b = 0 )
DefinitionA linear equation in the variables x1, x2, · · · , xn, is an equation ofthe form:
a1x1 + a2x2 + · · · anxn = b
where a1, a2, · · · an and b are constants.
A solution
of the equation is an array of numbersα1, α2, · · ·αn ∈ Rn such that
a1α1 + a2α2 + · · ·+ anαn = b
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 115: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/115.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Sistems of linear equations
General equation of a line
The General equation of a line is given by
ax0 + by0 = c
where x , y are variables and a, b, and c are constants (except forthe case a = b = 0 )
DefinitionA linear equation in the variables x1, x2, · · · , xn, is an equation ofthe form:
a1x1 + a2x2 + · · · anxn = b
where a1, a2, · · · an and b are constants.
A solution of the equation
is an array of numbersα1, α2, · · ·αn ∈ Rn such that
a1α1 + a2α2 + · · ·+ anαn = b
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 116: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/116.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Sistems of linear equations
General equation of a line
The General equation of a line is given by
ax0 + by0 = c
where x , y are variables and a, b, and c are constants (except forthe case a = b = 0 )
DefinitionA linear equation in the variables x1, x2, · · · , xn, is an equation ofthe form:
a1x1 + a2x2 + · · · anxn = b
where a1, a2, · · · an and b are constants.
A solution of the equation is an array of numbersα1, α2, · · ·αn ∈ Rn
such that
a1α1 + a2α2 + · · ·+ anαn = b
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 117: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/117.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Sistems of linear equations
General equation of a line
The General equation of a line is given by
ax0 + by0 = c
where x , y are variables and a, b, and c are constants (except forthe case a = b = 0 )
DefinitionA linear equation in the variables x1, x2, · · · , xn, is an equation ofthe form:
a1x1 + a2x2 + · · · anxn = b
where a1, a2, · · · an and b are constants.
A solution of the equation is an array of numbersα1, α2, · · ·αn ∈ Rn such that
a1α1 + a2α2 + · · ·+ anαn = b
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 118: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/118.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Sistems of linear equations
General equation of a line
The General equation of a line is given by
ax0 + by0 = c
where x , y are variables and a, b, and c are constants (except forthe case a = b = 0 )
DefinitionA linear equation in the variables x1, x2, · · · , xn, is an equation ofthe form:
a1x1 + a2x2 + · · · anxn = b
where a1, a2, · · · an and b are constants.
A solution of the equation is an array of numbersα1, α2, · · ·αn ∈ Rn such that
a1α1 + a2α2 + · · ·+ anαn = bDr. Marco A Roque Sol Linear Algebra. Session 1
![Page 119: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/119.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
System of linear equations
System of linear equations
A System of linear equations is an expression of the form:a11x1 + a12x2 + · · ·+ a1nxn = b1
a21x1 + a22x2 + · · ·+ a2nxn = b2...
am1x1 + am2x2 + · · ·+ amnxn = bm
Here x1, x2, · · · xn are variables and aij , bj are constants.
In many applications,we will find such a systems of equations andit is here, where all the study of Linear Algebra Starts.
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 120: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/120.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
System of linear equations
System of linear equations
A System of linear equations is an expression of the form:a11x1 + a12x2 + · · ·+ a1nxn = b1
a21x1 + a22x2 + · · ·+ a2nxn = b2...
am1x1 + am2x2 + · · ·+ amnxn = bm
Here x1, x2, · · · xn are variables and aij , bj are constants.
In many applications,we will find such a systems of equations andit is here, where all the study of Linear Algebra Starts.
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 121: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/121.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
System of linear equations
System of linear equations
A System of linear equations
is an expression of the form:a11x1 + a12x2 + · · ·+ a1nxn = b1
a21x1 + a22x2 + · · ·+ a2nxn = b2...
am1x1 + am2x2 + · · ·+ amnxn = bm
Here x1, x2, · · · xn are variables and aij , bj are constants.
In many applications,we will find such a systems of equations andit is here, where all the study of Linear Algebra Starts.
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 122: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/122.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
System of linear equations
System of linear equations
A System of linear equations is an expression
of the form:a11x1 + a12x2 + · · ·+ a1nxn = b1
a21x1 + a22x2 + · · ·+ a2nxn = b2...
am1x1 + am2x2 + · · ·+ amnxn = bm
Here x1, x2, · · · xn are variables and aij , bj are constants.
In many applications,we will find such a systems of equations andit is here, where all the study of Linear Algebra Starts.
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 123: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/123.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
System of linear equations
System of linear equations
A System of linear equations is an expression of the form:
a11x1 + a12x2 + · · ·+ a1nxn = b1
a21x1 + a22x2 + · · ·+ a2nxn = b2...
am1x1 + am2x2 + · · ·+ amnxn = bm
Here x1, x2, · · · xn are variables and aij , bj are constants.
In many applications,we will find such a systems of equations andit is here, where all the study of Linear Algebra Starts.
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 124: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/124.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
System of linear equations
System of linear equations
A System of linear equations is an expression of the form:a11x1 + a12x2 + · · ·+ a1nxn = b1
a21x1 + a22x2 + · · ·+ a2nxn = b2...
am1x1 + am2x2 + · · ·+ amnxn = bm
Here x1, x2, · · · xn are variables and aij , bj are constants.
In many applications,we will find such a systems of equations andit is here, where all the study of Linear Algebra Starts.
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 125: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/125.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
System of linear equations
System of linear equations
A System of linear equations is an expression of the form:a11x1 + a12x2 + · · ·+ a1nxn = b1
a21x1 + a22x2 + · · ·+ a2nxn = b2
...am1x1 + am2x2 + · · ·+ amnxn = bm
Here x1, x2, · · · xn are variables and aij , bj are constants.
In many applications,we will find such a systems of equations andit is here, where all the study of Linear Algebra Starts.
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 126: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/126.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
System of linear equations
System of linear equations
A System of linear equations is an expression of the form:a11x1 + a12x2 + · · ·+ a1nxn = b1
a21x1 + a22x2 + · · ·+ a2nxn = b2...
am1x1 + am2x2 + · · ·+ amnxn = bm
Here x1, x2, · · · xn are variables and aij , bj are constants.
In many applications,we will find such a systems of equations andit is here, where all the study of Linear Algebra Starts.
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 127: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/127.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
System of linear equations
System of linear equations
A System of linear equations is an expression of the form:a11x1 + a12x2 + · · ·+ a1nxn = b1
a21x1 + a22x2 + · · ·+ a2nxn = b2...
am1x1 + am2x2 + · · ·+ amnxn = bm
Here x1, x2, · · · xn are variables and aij , bj are constants.
In many applications,we will find such a systems of equations andit is here, where all the study of Linear Algebra Starts.
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 128: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/128.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
System of linear equations
System of linear equations
A System of linear equations is an expression of the form:a11x1 + a12x2 + · · ·+ a1nxn = b1
a21x1 + a22x2 + · · ·+ a2nxn = b2...
am1x1 + am2x2 + · · ·+ amnxn = bm
Here
x1, x2, · · · xn are variables and aij , bj are constants.
In many applications,we will find such a systems of equations andit is here, where all the study of Linear Algebra Starts.
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 129: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/129.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
System of linear equations
System of linear equations
A System of linear equations is an expression of the form:a11x1 + a12x2 + · · ·+ a1nxn = b1
a21x1 + a22x2 + · · ·+ a2nxn = b2...
am1x1 + am2x2 + · · ·+ amnxn = bm
Here x1, x2, · · · xn
are variables and aij , bj are constants.
In many applications,we will find such a systems of equations andit is here, where all the study of Linear Algebra Starts.
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 130: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/130.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
System of linear equations
System of linear equations
A System of linear equations is an expression of the form:a11x1 + a12x2 + · · ·+ a1nxn = b1
a21x1 + a22x2 + · · ·+ a2nxn = b2...
am1x1 + am2x2 + · · ·+ amnxn = bm
Here x1, x2, · · · xn are variables and
aij , bj are constants.
In many applications,we will find such a systems of equations andit is here, where all the study of Linear Algebra Starts.
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 131: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/131.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
System of linear equations
System of linear equations
A System of linear equations is an expression of the form:a11x1 + a12x2 + · · ·+ a1nxn = b1
a21x1 + a22x2 + · · ·+ a2nxn = b2...
am1x1 + am2x2 + · · ·+ amnxn = bm
Here x1, x2, · · · xn are variables and aij , bj
are constants.
In many applications,we will find such a systems of equations andit is here, where all the study of Linear Algebra Starts.
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 132: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/132.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
System of linear equations
System of linear equations
A System of linear equations is an expression of the form:a11x1 + a12x2 + · · ·+ a1nxn = b1
a21x1 + a22x2 + · · ·+ a2nxn = b2...
am1x1 + am2x2 + · · ·+ amnxn = bm
Here x1, x2, · · · xn are variables and aij , bj are constants.
In many applications,we will find such a systems of equations andit is here, where all the study of Linear Algebra Starts.
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 133: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/133.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
System of linear equations
System of linear equations
A System of linear equations is an expression of the form:a11x1 + a12x2 + · · ·+ a1nxn = b1
a21x1 + a22x2 + · · ·+ a2nxn = b2...
am1x1 + am2x2 + · · ·+ amnxn = bm
Here x1, x2, · · · xn are variables and aij , bj are constants.
In many applications,
we will find such a systems of equations andit is here, where all the study of Linear Algebra Starts.
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 134: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/134.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
System of linear equations
System of linear equations
A System of linear equations is an expression of the form:a11x1 + a12x2 + · · ·+ a1nxn = b1
a21x1 + a22x2 + · · ·+ a2nxn = b2...
am1x1 + am2x2 + · · ·+ amnxn = bm
Here x1, x2, · · · xn are variables and aij , bj are constants.
In many applications,we will find such a systems of equations
andit is here, where all the study of Linear Algebra Starts.
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 135: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/135.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
System of linear equations
System of linear equations
A System of linear equations is an expression of the form:a11x1 + a12x2 + · · ·+ a1nxn = b1
a21x1 + a22x2 + · · ·+ a2nxn = b2...
am1x1 + am2x2 + · · ·+ amnxn = bm
Here x1, x2, · · · xn are variables and aij , bj are constants.
In many applications,we will find such a systems of equations andit is here,
where all the study of Linear Algebra Starts.
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 136: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/136.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
System of linear equations
System of linear equations
A System of linear equations is an expression of the form:a11x1 + a12x2 + · · ·+ a1nxn = b1
a21x1 + a22x2 + · · ·+ a2nxn = b2...
am1x1 + am2x2 + · · ·+ amnxn = bm
Here x1, x2, · · · xn are variables and aij , bj are constants.
In many applications,we will find such a systems of equations andit is here, where all the study
of Linear Algebra Starts.
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 137: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/137.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
System of linear equations
System of linear equations
A System of linear equations is an expression of the form:a11x1 + a12x2 + · · ·+ a1nxn = b1
a21x1 + a22x2 + · · ·+ a2nxn = b2...
am1x1 + am2x2 + · · ·+ amnxn = bm
Here x1, x2, · · · xn are variables and aij , bj are constants.
In many applications,we will find such a systems of equations andit is here, where all the study of Linear Algebra
Starts.
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 138: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/138.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
System of linear equations
System of linear equations
A System of linear equations is an expression of the form:a11x1 + a12x2 + · · ·+ a1nxn = b1
a21x1 + a22x2 + · · ·+ a2nxn = b2...
am1x1 + am2x2 + · · ·+ amnxn = bm
Here x1, x2, · · · xn are variables and aij , bj are constants.
In many applications,we will find such a systems of equations andit is here, where all the study of Linear Algebra Starts.
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 139: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/139.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Systems of linear equations
Example 1.1
Find the point of intersection of the lines x − y = −2 and2x + 3y = in R2. You can find the solution of this problem bysetting up and solving the system of two equations in twounknowns {
x − y = −22x + 3y = 6
⇔ ( Equivalent System){x = y −2
2x + 3y = 6
⇔ {x = y − 2
2(y − 2) + 3y = 6
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 140: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/140.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Systems of linear equations
Example 1.1
Find the point of intersection of the lines x − y = −2 and2x + 3y = in R2.
You can find the solution of this problem bysetting up and solving the system of two equations in twounknowns {
x − y = −22x + 3y = 6
⇔ ( Equivalent System){x = y −2
2x + 3y = 6
⇔ {x = y − 2
2(y − 2) + 3y = 6
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 141: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/141.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Systems of linear equations
Example 1.1
Find the point of intersection of the lines x − y = −2 and2x + 3y = in R2. You can find
the solution of this problem bysetting up and solving the system of two equations in twounknowns {
x − y = −22x + 3y = 6
⇔ ( Equivalent System){x = y −2
2x + 3y = 6
⇔ {x = y − 2
2(y − 2) + 3y = 6
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 142: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/142.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Systems of linear equations
Example 1.1
Find the point of intersection of the lines x − y = −2 and2x + 3y = in R2. You can find the solution
of this problem bysetting up and solving the system of two equations in twounknowns {
x − y = −22x + 3y = 6
⇔ ( Equivalent System){x = y −2
2x + 3y = 6
⇔ {x = y − 2
2(y − 2) + 3y = 6
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 143: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/143.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Systems of linear equations
Example 1.1
Find the point of intersection of the lines x − y = −2 and2x + 3y = in R2. You can find the solution of this problem
bysetting up and solving the system of two equations in twounknowns {
x − y = −22x + 3y = 6
⇔ ( Equivalent System){x = y −2
2x + 3y = 6
⇔ {x = y − 2
2(y − 2) + 3y = 6
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 144: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/144.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Systems of linear equations
Example 1.1
Find the point of intersection of the lines x − y = −2 and2x + 3y = in R2. You can find the solution of this problem bysetting up and
solving the system of two equations in twounknowns {
x − y = −22x + 3y = 6
⇔ ( Equivalent System){x = y −2
2x + 3y = 6
⇔ {x = y − 2
2(y − 2) + 3y = 6
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 145: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/145.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Systems of linear equations
Example 1.1
Find the point of intersection of the lines x − y = −2 and2x + 3y = in R2. You can find the solution of this problem bysetting up and solving the system
of two equations in twounknowns {
x − y = −22x + 3y = 6
⇔ ( Equivalent System){x = y −2
2x + 3y = 6
⇔ {x = y − 2
2(y − 2) + 3y = 6
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 146: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/146.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Systems of linear equations
Example 1.1
Find the point of intersection of the lines x − y = −2 and2x + 3y = in R2. You can find the solution of this problem bysetting up and solving the system of two equations
in twounknowns {
x − y = −22x + 3y = 6
⇔ ( Equivalent System){x = y −2
2x + 3y = 6
⇔ {x = y − 2
2(y − 2) + 3y = 6
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 147: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/147.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Systems of linear equations
Example 1.1
Find the point of intersection of the lines x − y = −2 and2x + 3y = in R2. You can find the solution of this problem bysetting up and solving the system of two equations in twounknowns
{x − y = −2
2x + 3y = 6
⇔ ( Equivalent System){x = y −2
2x + 3y = 6
⇔ {x = y − 2
2(y − 2) + 3y = 6
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 148: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/148.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Systems of linear equations
Example 1.1
Find the point of intersection of the lines x − y = −2 and2x + 3y = in R2. You can find the solution of this problem bysetting up and solving the system of two equations in twounknowns {
x − y = −22x + 3y = 6
⇔ ( Equivalent System){x = y −2
2x + 3y = 6
⇔ {x = y − 2
2(y − 2) + 3y = 6
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 149: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/149.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Systems of linear equations
Example 1.1
Find the point of intersection of the lines x − y = −2 and2x + 3y = in R2. You can find the solution of this problem bysetting up and solving the system of two equations in twounknowns {
x − y = −22x + 3y = 6
⇔ ( Equivalent System)
{x = y −2
2x + 3y = 6
⇔ {x = y − 2
2(y − 2) + 3y = 6
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 150: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/150.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Systems of linear equations
Example 1.1
Find the point of intersection of the lines x − y = −2 and2x + 3y = in R2. You can find the solution of this problem bysetting up and solving the system of two equations in twounknowns {
x − y = −22x + 3y = 6
⇔ ( Equivalent System){x = y −2
2x + 3y = 6
⇔
{x = y − 2
2(y − 2) + 3y = 6
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 151: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/151.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Systems of linear equations
Example 1.1
Find the point of intersection of the lines x − y = −2 and2x + 3y = in R2. You can find the solution of this problem bysetting up and solving the system of two equations in twounknowns {
x − y = −22x + 3y = 6
⇔ ( Equivalent System){x = y −2
2x + 3y = 6
⇔ {x = y − 2
2(y − 2) + 3y = 6
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 152: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/152.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Systems of linear equations
⇔ {x = y − 2
5y = 10
⇔ {x = y − 2y = 2
⇔ {x = 0y = 2
Thus, the solution is given by the point (0, 2)
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 153: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/153.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Systems of linear equations
⇔
{x = y − 2
5y = 10
⇔ {x = y − 2y = 2
⇔ {x = 0y = 2
Thus, the solution is given by the point (0, 2)
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 154: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/154.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Systems of linear equations
⇔ {x = y − 2
5y = 10
⇔ {x = y − 2y = 2
⇔ {x = 0y = 2
Thus, the solution is given by the point (0, 2)
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 155: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/155.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Systems of linear equations
⇔ {x = y − 2
5y = 10
⇔
{x = y − 2y = 2
⇔ {x = 0y = 2
Thus, the solution is given by the point (0, 2)
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 156: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/156.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Systems of linear equations
⇔ {x = y − 2
5y = 10
⇔ {x = y − 2y = 2
⇔
{x = 0y = 2
Thus, the solution is given by the point (0, 2)
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 157: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/157.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Systems of linear equations
⇔ {x = y − 2
5y = 10
⇔ {x = y − 2y = 2
⇔ {x = 0y = 2
Thus, the solution is given by the point (0, 2)
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 158: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/158.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Systems of linear equations
⇔ {x = y − 2
5y = 10
⇔ {x = y − 2y = 2
⇔ {x = 0y = 2
Thus,
the solution is given by the point (0, 2)
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 159: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/159.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Systems of linear equations
⇔ {x = y − 2
5y = 10
⇔ {x = y − 2y = 2
⇔ {x = 0y = 2
Thus, the solution
is given by the point (0, 2)
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 160: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/160.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Systems of linear equations
⇔ {x = y − 2
5y = 10
⇔ {x = y − 2y = 2
⇔ {x = 0y = 2
Thus, the solution is given by
the point (0, 2)
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 161: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/161.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Systems of linear equations
⇔ {x = y − 2
5y = 10
⇔ {x = y − 2y = 2
⇔ {x = 0y = 2
Thus, the solution is given by the point (0, 2)
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 162: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/162.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Systems of linear equations
{x − y = −2
2x + 3y = 6x = 0; y = 2
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 163: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/163.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Systems of linear equations
{x − y = −2
2x + 3y = 6
x = 0; y = 2
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 164: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/164.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Systems of linear equations
{x − y = −2
2x + 3y = 6x = 0; y = 2
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 165: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/165.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Systems of linear equations
In a similar way, we have
{2x + 3y = 22x + 3y = 6
inconsistent system (no solution)
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 166: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/166.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Systems of linear equations
In a similar way, we have
{2x + 3y = 22x + 3y = 6
inconsistent system (no solution)
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 167: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/167.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Systems of linear equations
In a similar way, we have
{2x + 3y = 22x + 3y = 6
inconsistent system (no solution)
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 168: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/168.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Systems of linear equations
In a similar way, we have
{2x + 3y = 22x + 3y = 6
inconsistent system (no solution)
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 169: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/169.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Systems of linear equations
In a similar way, we have
{2x + 3y = 22x + 3y = 6
inconsistent system
(no solution)
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 170: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/170.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Systems of linear equations
In a similar way, we have
{2x + 3y = 22x + 3y = 6
inconsistent system (no solution)
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 171: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/171.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Systems of linear equations
{4x + 6y = 122x + 3y = 6
2x+3y = 6 (infinitely many solutions)
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 172: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/172.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Systems of linear equations
{4x + 6y = 122x + 3y = 6
2x+3y = 6 (infinitely many solutions)
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 173: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/173.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Systems of linear equations
{4x + 6y = 122x + 3y = 6
2x+3y = 6 (infinitely many solutions)
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 174: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/174.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Systems of linear equations
{4x + 6y = 122x + 3y = 6
2x+3y = 6
(infinitely many solutions)
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 175: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/175.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Systems of linear equations
{4x + 6y = 122x + 3y = 6
2x+3y = 6 (infinitely many solutions)
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 176: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/176.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Systems of linear equations
Solving systems of linear equations
Elimination Method
Algorithm:(1) pick a variable, solve one of the equations for it, and eliminateit from the other equations;
(2) put aside the equation used in the elimination, and return tostep (1).
The algorithm reduces the number of variables (as well as thenumber of equations), hence it stops after a finite number of steps.After the algorithm stops, the system is simplified so that it shouldbe clear how to complete solution
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 177: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/177.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Systems of linear equations
Solving systems of linear equations
Elimination Method
Algorithm:(1) pick a variable, solve one of the equations for it, and eliminateit from the other equations;
(2) put aside the equation used in the elimination, and return tostep (1).
The algorithm reduces the number of variables (as well as thenumber of equations), hence it stops after a finite number of steps.After the algorithm stops, the system is simplified so that it shouldbe clear how to complete solution
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 178: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/178.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Systems of linear equations
Solving systems of linear equations
Elimination Method
Algorithm:(1) pick a variable, solve one of the equations for it, and eliminateit from the other equations;
(2) put aside the equation used in the elimination, and return tostep (1).
The algorithm reduces the number of variables (as well as thenumber of equations), hence it stops after a finite number of steps.After the algorithm stops, the system is simplified so that it shouldbe clear how to complete solution
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 179: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/179.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Systems of linear equations
Solving systems of linear equations
Elimination Method
Algorithm:
(1) pick a variable, solve one of the equations for it, and eliminateit from the other equations;
(2) put aside the equation used in the elimination, and return tostep (1).
The algorithm reduces the number of variables (as well as thenumber of equations), hence it stops after a finite number of steps.After the algorithm stops, the system is simplified so that it shouldbe clear how to complete solution
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 180: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/180.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Systems of linear equations
Solving systems of linear equations
Elimination Method
Algorithm:(1) pick a variable,
solve one of the equations for it, and eliminateit from the other equations;
(2) put aside the equation used in the elimination, and return tostep (1).
The algorithm reduces the number of variables (as well as thenumber of equations), hence it stops after a finite number of steps.After the algorithm stops, the system is simplified so that it shouldbe clear how to complete solution
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 181: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/181.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Systems of linear equations
Solving systems of linear equations
Elimination Method
Algorithm:(1) pick a variable, solve one of the equations for it, and
eliminateit from the other equations;
(2) put aside the equation used in the elimination, and return tostep (1).
The algorithm reduces the number of variables (as well as thenumber of equations), hence it stops after a finite number of steps.After the algorithm stops, the system is simplified so that it shouldbe clear how to complete solution
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 182: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/182.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Systems of linear equations
Solving systems of linear equations
Elimination Method
Algorithm:(1) pick a variable, solve one of the equations for it, and eliminateit
from the other equations;
(2) put aside the equation used in the elimination, and return tostep (1).
The algorithm reduces the number of variables (as well as thenumber of equations), hence it stops after a finite number of steps.After the algorithm stops, the system is simplified so that it shouldbe clear how to complete solution
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 183: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/183.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Systems of linear equations
Solving systems of linear equations
Elimination Method
Algorithm:(1) pick a variable, solve one of the equations for it, and eliminateit from the other equations;
(2) put aside the equation used in the elimination, and return tostep (1).
The algorithm reduces the number of variables (as well as thenumber of equations), hence it stops after a finite number of steps.After the algorithm stops, the system is simplified so that it shouldbe clear how to complete solution
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 184: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/184.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Systems of linear equations
Solving systems of linear equations
Elimination Method
Algorithm:(1) pick a variable, solve one of the equations for it, and eliminateit from the other equations;
(2) put aside the equation
used in the elimination, and return tostep (1).
The algorithm reduces the number of variables (as well as thenumber of equations), hence it stops after a finite number of steps.After the algorithm stops, the system is simplified so that it shouldbe clear how to complete solution
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 185: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/185.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Systems of linear equations
Solving systems of linear equations
Elimination Method
Algorithm:(1) pick a variable, solve one of the equations for it, and eliminateit from the other equations;
(2) put aside the equation used in the elimination, and
return tostep (1).
The algorithm reduces the number of variables (as well as thenumber of equations), hence it stops after a finite number of steps.After the algorithm stops, the system is simplified so that it shouldbe clear how to complete solution
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 186: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/186.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Systems of linear equations
Solving systems of linear equations
Elimination Method
Algorithm:(1) pick a variable, solve one of the equations for it, and eliminateit from the other equations;
(2) put aside the equation used in the elimination, and return tostep (1).
The algorithm reduces the number of variables (as well as thenumber of equations), hence it stops after a finite number of steps.After the algorithm stops, the system is simplified so that it shouldbe clear how to complete solution
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 187: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/187.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Systems of linear equations
Solving systems of linear equations
Elimination Method
Algorithm:(1) pick a variable, solve one of the equations for it, and eliminateit from the other equations;
(2) put aside the equation used in the elimination, and return tostep (1).
The algorithm
reduces the number of variables (as well as thenumber of equations), hence it stops after a finite number of steps.After the algorithm stops, the system is simplified so that it shouldbe clear how to complete solution
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 188: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/188.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Systems of linear equations
Solving systems of linear equations
Elimination Method
Algorithm:(1) pick a variable, solve one of the equations for it, and eliminateit from the other equations;
(2) put aside the equation used in the elimination, and return tostep (1).
The algorithm reduces the number of variables
(as well as thenumber of equations), hence it stops after a finite number of steps.After the algorithm stops, the system is simplified so that it shouldbe clear how to complete solution
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 189: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/189.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Systems of linear equations
Solving systems of linear equations
Elimination Method
Algorithm:(1) pick a variable, solve one of the equations for it, and eliminateit from the other equations;
(2) put aside the equation used in the elimination, and return tostep (1).
The algorithm reduces the number of variables (as well as thenumber of equations),
hence it stops after a finite number of steps.After the algorithm stops, the system is simplified so that it shouldbe clear how to complete solution
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 190: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/190.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Systems of linear equations
Solving systems of linear equations
Elimination Method
Algorithm:(1) pick a variable, solve one of the equations for it, and eliminateit from the other equations;
(2) put aside the equation used in the elimination, and return tostep (1).
The algorithm reduces the number of variables (as well as thenumber of equations), hence it stops after
a finite number of steps.After the algorithm stops, the system is simplified so that it shouldbe clear how to complete solution
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 191: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/191.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Systems of linear equations
Solving systems of linear equations
Elimination Method
Algorithm:(1) pick a variable, solve one of the equations for it, and eliminateit from the other equations;
(2) put aside the equation used in the elimination, and return tostep (1).
The algorithm reduces the number of variables (as well as thenumber of equations), hence it stops after a finite number of steps.After the algorithm stops,
the system is simplified so that it shouldbe clear how to complete solution
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 192: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/192.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Systems of linear equations
Solving systems of linear equations
Elimination Method
Algorithm:(1) pick a variable, solve one of the equations for it, and eliminateit from the other equations;
(2) put aside the equation used in the elimination, and return tostep (1).
The algorithm reduces the number of variables (as well as thenumber of equations), hence it stops after a finite number of steps.After the algorithm stops, the system is simplified so that
it shouldbe clear how to complete solution
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 193: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/193.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Systems of linear equations
Solving systems of linear equations
Elimination Method
Algorithm:(1) pick a variable, solve one of the equations for it, and eliminateit from the other equations;
(2) put aside the equation used in the elimination, and return tostep (1).
The algorithm reduces the number of variables (as well as thenumber of equations), hence it stops after a finite number of steps.After the algorithm stops, the system is simplified so that it shouldbe clear
how to complete solution
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 194: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/194.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Systems of linear equations
Solving systems of linear equations
Elimination Method
Algorithm:(1) pick a variable, solve one of the equations for it, and eliminateit from the other equations;
(2) put aside the equation used in the elimination, and return tostep (1).
The algorithm reduces the number of variables (as well as thenumber of equations), hence it stops after a finite number of steps.After the algorithm stops, the system is simplified so that it shouldbe clear how to complete solution
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 195: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/195.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Systems of linear equations
Example 1.2 x − y = 2
2x − y− z = 3x + y+ z = 6
Solve the 1st equation for x :x = y + 2
2x − y− z = 3x + y+ z = 6
⇔Eliminate x from the 2nd and 3rd equations:
x = y + 22(y + 2) − y− z = 3(y + 2) + y+ z = 6
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 196: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/196.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Systems of linear equations
Example 1.2
x − y = 2
2x − y− z = 3x + y+ z = 6
Solve the 1st equation for x :x = y + 2
2x − y− z = 3x + y+ z = 6
⇔Eliminate x from the 2nd and 3rd equations:
x = y + 22(y + 2) − y− z = 3(y + 2) + y+ z = 6
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 197: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/197.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Systems of linear equations
Example 1.2 x − y = 2
2x − y− z = 3x + y+ z = 6
Solve the 1st equation for x :x = y + 2
2x − y− z = 3x + y+ z = 6
⇔Eliminate x from the 2nd and 3rd equations:
x = y + 22(y + 2) − y− z = 3(y + 2) + y+ z = 6
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 198: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/198.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Systems of linear equations
Example 1.2 x − y = 2
2x − y− z = 3x + y+ z = 6
Solve the 1st equation for x :
x = y + 2
2x − y− z = 3x + y+ z = 6
⇔Eliminate x from the 2nd and 3rd equations:
x = y + 22(y + 2) − y− z = 3(y + 2) + y+ z = 6
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 199: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/199.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Systems of linear equations
Example 1.2 x − y = 2
2x − y− z = 3x + y+ z = 6
Solve the 1st equation for x :x = y + 2
2x − y− z = 3x + y+ z = 6
⇔
Eliminate x from the 2nd and 3rd equations:x = y + 2
2(y + 2) − y− z = 3(y + 2) + y+ z = 6
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 200: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/200.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Systems of linear equations
Example 1.2 x − y = 2
2x − y− z = 3x + y+ z = 6
Solve the 1st equation for x :x = y + 2
2x − y− z = 3x + y+ z = 6
⇔Eliminate x from the 2nd and 3rd equations:
x = y + 2
2(y + 2) − y− z = 3(y + 2) + y+ z = 6
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 201: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/201.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Systems of linear equations
Example 1.2 x − y = 2
2x − y− z = 3x + y+ z = 6
Solve the 1st equation for x :x = y + 2
2x − y− z = 3x + y+ z = 6
⇔Eliminate x from the 2nd and 3rd equations:
x = y + 22(y + 2) − y− z = 3(y + 2) + y+ z = 6
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 202: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/202.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Systems of linear equations
⇔
Simplify: x = y+ 2y− z = −12y + z = 4
In this way we have that, the whole system has been reduced to asystem (2nd and 3rd equations) of two linear equations in twovariables.
Solve the 2nd equation for y :x = y+ 2y = z− 12y + z = 4
⇔
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 203: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/203.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Systems of linear equations
⇔Simplify:
x = y+ 2y− z = −12y + z = 4
In this way we have that, the whole system has been reduced to asystem (2nd and 3rd equations) of two linear equations in twovariables.
Solve the 2nd equation for y :x = y+ 2y = z− 12y + z = 4
⇔
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 204: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/204.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Systems of linear equations
⇔Simplify:
x = y+ 2y− z = −12y + z = 4
In this way we have that, the whole system has been reduced to asystem (2nd and 3rd equations) of two linear equations in twovariables.
Solve the 2nd equation for y :x = y+ 2y = z− 12y + z = 4
⇔
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 205: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/205.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Systems of linear equations
⇔Simplify:
x = y+ 2y− z = −12y + z = 4
In this way
we have that, the whole system has been reduced to asystem (2nd and 3rd equations) of two linear equations in twovariables.
Solve the 2nd equation for y :x = y+ 2y = z− 12y + z = 4
⇔
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 206: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/206.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Systems of linear equations
⇔Simplify:
x = y+ 2y− z = −12y + z = 4
In this way we have that,
the whole system has been reduced to asystem (2nd and 3rd equations) of two linear equations in twovariables.
Solve the 2nd equation for y :x = y+ 2y = z− 12y + z = 4
⇔
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 207: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/207.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Systems of linear equations
⇔Simplify:
x = y+ 2y− z = −12y + z = 4
In this way we have that, the whole system has been reduced
to asystem (2nd and 3rd equations) of two linear equations in twovariables.
Solve the 2nd equation for y :x = y+ 2y = z− 12y + z = 4
⇔
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 208: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/208.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Systems of linear equations
⇔Simplify:
x = y+ 2y− z = −12y + z = 4
In this way we have that, the whole system has been reduced to asystem (2nd and 3rd equations)
of two linear equations in twovariables.
Solve the 2nd equation for y :x = y+ 2y = z− 12y + z = 4
⇔
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 209: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/209.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Systems of linear equations
⇔Simplify:
x = y+ 2y− z = −12y + z = 4
In this way we have that, the whole system has been reduced to asystem (2nd and 3rd equations) of two linear equations
in twovariables.
Solve the 2nd equation for y :x = y+ 2y = z− 12y + z = 4
⇔
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 210: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/210.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Systems of linear equations
⇔Simplify:
x = y+ 2y− z = −12y + z = 4
In this way we have that, the whole system has been reduced to asystem (2nd and 3rd equations) of two linear equations in twovariables.
Solve the 2nd equation for y :x = y+ 2y = z− 12y + z = 4
⇔
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 211: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/211.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Systems of linear equations
⇔Simplify:
x = y+ 2y− z = −12y + z = 4
In this way we have that, the whole system has been reduced to asystem (2nd and 3rd equations) of two linear equations in twovariables.
Solve the 2nd equation for y :
x = y+ 2y = z− 12y + z = 4
⇔
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 212: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/212.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Systems of linear equations
⇔Simplify:
x = y+ 2y− z = −12y + z = 4
In this way we have that, the whole system has been reduced to asystem (2nd and 3rd equations) of two linear equations in twovariables.
Solve the 2nd equation for y :x = y+ 2y = z− 12y + z = 4
⇔Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 213: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/213.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Systems of linear equations
Eliminate y from the 3rd equation:x = y+ 2y = z+ 1
2(z − 1) + z = 4⇔
Simplify x = y+ 2y = z+ 13z = 6
⇔
Thus,the elimination process, has been completed.Now, the systemis easily solved by back substitution
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 214: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/214.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Systems of linear equations
Eliminate y from the 3rd equation:
x = y+ 2y = z+ 1
2(z − 1) + z = 4⇔
Simplify x = y+ 2y = z+ 13z = 6
⇔
Thus,the elimination process, has been completed.Now, the systemis easily solved by back substitution
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 215: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/215.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Systems of linear equations
Eliminate y from the 3rd equation:x = y+ 2y = z+ 1
2(z − 1) + z = 4⇔
Simplify x = y+ 2y = z+ 13z = 6
⇔
Thus,the elimination process, has been completed.Now, the systemis easily solved by back substitution
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 216: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/216.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Systems of linear equations
Eliminate y from the 3rd equation:x = y+ 2y = z+ 1
2(z − 1) + z = 4⇔
Simplify
x = y+ 2y = z+ 13z = 6
⇔
Thus,the elimination process, has been completed.Now, the systemis easily solved by back substitution
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 217: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/217.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Systems of linear equations
Eliminate y from the 3rd equation:x = y+ 2y = z+ 1
2(z − 1) + z = 4⇔
Simplify x = y+ 2y = z+ 13z = 6
⇔
Thus,the elimination process, has been completed.Now, the systemis easily solved by back substitution
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 218: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/218.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Systems of linear equations
Eliminate y from the 3rd equation:x = y+ 2y = z+ 1
2(z − 1) + z = 4⇔
Simplify x = y+ 2y = z+ 13z = 6
⇔
Thus,
the elimination process, has been completed.Now, the systemis easily solved by back substitution
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 219: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/219.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Systems of linear equations
Eliminate y from the 3rd equation:x = y+ 2y = z+ 1
2(z − 1) + z = 4⇔
Simplify x = y+ 2y = z+ 13z = 6
⇔
Thus,the elimination process,
has been completed.Now, the systemis easily solved by back substitution
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 220: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/220.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Systems of linear equations
Eliminate y from the 3rd equation:x = y+ 2y = z+ 1
2(z − 1) + z = 4⇔
Simplify x = y+ 2y = z+ 13z = 6
⇔
Thus,the elimination process, has been completed.
Now, the systemis easily solved by back substitution
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 221: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/221.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Systems of linear equations
Eliminate y from the 3rd equation:x = y+ 2y = z+ 1
2(z − 1) + z = 4⇔
Simplify x = y+ 2y = z+ 13z = 6
⇔
Thus,the elimination process, has been completed.Now,
the systemis easily solved by back substitution
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 222: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/222.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Systems of linear equations
Eliminate y from the 3rd equation:x = y+ 2y = z+ 1
2(z − 1) + z = 4⇔
Simplify x = y+ 2y = z+ 13z = 6
⇔
Thus,the elimination process, has been completed.Now, the system
is easily solved by back substitution
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 223: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/223.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Systems of linear equations
Eliminate y from the 3rd equation:x = y+ 2y = z+ 1
2(z − 1) + z = 4⇔
Simplify x = y+ 2y = z+ 13z = 6
⇔
Thus,the elimination process, has been completed.Now, the systemis easily solved by
back substitution
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 224: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/224.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Systems of linear equations
Eliminate y from the 3rd equation:x = y+ 2y = z+ 1
2(z − 1) + z = 4⇔
Simplify x = y+ 2y = z+ 13z = 6
⇔
Thus,the elimination process, has been completed.Now, the systemis easily solved by back substitution
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 225: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/225.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Systems of linear equations
That is, we find z from the 3rd equation, then substitute it in the2nd equation and find y , then substitute y and z in the 1stequation and find x .
x = y + 2y = z + 1
z = 2
x = y + 2y = 1z = 2
x = 3y = 1z = 2
Finally, the System of linear equations:x − y = 2
2x − y− z = 3x + y+ z = 6
has as a solution, the point (x , y , z) = (3, 1, 2) .
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 226: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/226.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Systems of linear equations
That is,
we find z from the 3rd equation, then substitute it in the2nd equation and find y , then substitute y and z in the 1stequation and find x .
x = y + 2y = z + 1
z = 2
x = y + 2y = 1z = 2
x = 3y = 1z = 2
Finally, the System of linear equations:x − y = 2
2x − y− z = 3x + y+ z = 6
has as a solution, the point (x , y , z) = (3, 1, 2) .
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 227: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/227.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Systems of linear equations
That is, we find z
from the 3rd equation, then substitute it in the2nd equation and find y , then substitute y and z in the 1stequation and find x .
x = y + 2y = z + 1
z = 2
x = y + 2y = 1z = 2
x = 3y = 1z = 2
Finally, the System of linear equations:x − y = 2
2x − y− z = 3x + y+ z = 6
has as a solution, the point (x , y , z) = (3, 1, 2) .
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 228: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/228.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Systems of linear equations
That is, we find z from the 3rd equation,
then substitute it in the2nd equation and find y , then substitute y and z in the 1stequation and find x .
x = y + 2y = z + 1
z = 2
x = y + 2y = 1z = 2
x = 3y = 1z = 2
Finally, the System of linear equations:x − y = 2
2x − y− z = 3x + y+ z = 6
has as a solution, the point (x , y , z) = (3, 1, 2) .
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 229: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/229.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Systems of linear equations
That is, we find z from the 3rd equation, then substitute it
in the2nd equation and find y , then substitute y and z in the 1stequation and find x .
x = y + 2y = z + 1
z = 2
x = y + 2y = 1z = 2
x = 3y = 1z = 2
Finally, the System of linear equations:x − y = 2
2x − y− z = 3x + y+ z = 6
has as a solution, the point (x , y , z) = (3, 1, 2) .
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 230: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/230.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Systems of linear equations
That is, we find z from the 3rd equation, then substitute it in the2nd equation and
find y , then substitute y and z in the 1stequation and find x .
x = y + 2y = z + 1
z = 2
x = y + 2y = 1z = 2
x = 3y = 1z = 2
Finally, the System of linear equations:x − y = 2
2x − y− z = 3x + y+ z = 6
has as a solution, the point (x , y , z) = (3, 1, 2) .
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 231: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/231.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Systems of linear equations
That is, we find z from the 3rd equation, then substitute it in the2nd equation and find y ,
then substitute y and z in the 1stequation and find x .
x = y + 2y = z + 1
z = 2
x = y + 2y = 1z = 2
x = 3y = 1z = 2
Finally, the System of linear equations:x − y = 2
2x − y− z = 3x + y+ z = 6
has as a solution, the point (x , y , z) = (3, 1, 2) .
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 232: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/232.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Systems of linear equations
That is, we find z from the 3rd equation, then substitute it in the2nd equation and find y , then substitute y and z
in the 1stequation and find x .
x = y + 2y = z + 1
z = 2
x = y + 2y = 1z = 2
x = 3y = 1z = 2
Finally, the System of linear equations:x − y = 2
2x − y− z = 3x + y+ z = 6
has as a solution, the point (x , y , z) = (3, 1, 2) .
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 233: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/233.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Systems of linear equations
That is, we find z from the 3rd equation, then substitute it in the2nd equation and find y , then substitute y and z in the 1stequation and
find x .
x = y + 2y = z + 1
z = 2
x = y + 2y = 1z = 2
x = 3y = 1z = 2
Finally, the System of linear equations:x − y = 2
2x − y− z = 3x + y+ z = 6
has as a solution, the point (x , y , z) = (3, 1, 2) .
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 234: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/234.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Systems of linear equations
That is, we find z from the 3rd equation, then substitute it in the2nd equation and find y , then substitute y and z in the 1stequation and find x .
x = y + 2y = z + 1
z = 2
x = y + 2y = 1z = 2
x = 3y = 1z = 2
Finally, the System of linear equations:x − y = 2
2x − y− z = 3x + y+ z = 6
has as a solution, the point (x , y , z) = (3, 1, 2) .
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 235: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/235.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Systems of linear equations
That is, we find z from the 3rd equation, then substitute it in the2nd equation and find y , then substitute y and z in the 1stequation and find x .
x = y + 2y = z + 1
z = 2
x = y + 2y = 1z = 2
x = 3y = 1z = 2
Finally, the System of linear equations:x − y = 2
2x − y− z = 3x + y+ z = 6
has as a solution, the point (x , y , z) = (3, 1, 2) .
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 236: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/236.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Systems of linear equations
That is, we find z from the 3rd equation, then substitute it in the2nd equation and find y , then substitute y and z in the 1stequation and find x .
x = y + 2y = z + 1
z = 2
x = y + 2y = 1z = 2
x = 3y = 1z = 2
Finally,
the System of linear equations:x − y = 2
2x − y− z = 3x + y+ z = 6
has as a solution, the point (x , y , z) = (3, 1, 2) .
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 237: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/237.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Systems of linear equations
That is, we find z from the 3rd equation, then substitute it in the2nd equation and find y , then substitute y and z in the 1stequation and find x .
x = y + 2y = z + 1
z = 2
x = y + 2y = 1z = 2
x = 3y = 1z = 2
Finally, the System of linear equations:
x − y = 2
2x − y− z = 3x + y+ z = 6
has as a solution, the point (x , y , z) = (3, 1, 2) .
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 238: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/238.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Systems of linear equations
That is, we find z from the 3rd equation, then substitute it in the2nd equation and find y , then substitute y and z in the 1stequation and find x .
x = y + 2y = z + 1
z = 2
x = y + 2y = 1z = 2
x = 3y = 1z = 2
Finally, the System of linear equations:x − y = 2
2x − y− z = 3x + y+ z = 6
has as a solution, the point (x , y , z) = (3, 1, 2) .
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 239: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/239.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Systems of linear equations
That is, we find z from the 3rd equation, then substitute it in the2nd equation and find y , then substitute y and z in the 1stequation and find x .
x = y + 2y = z + 1
z = 2
x = y + 2y = 1z = 2
x = 3y = 1z = 2
Finally, the System of linear equations:x − y = 2
2x − y− z = 3x + y+ z = 6
has as a solution,
the point (x , y , z) = (3, 1, 2) .
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 240: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/240.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Systems of linear equations
That is, we find z from the 3rd equation, then substitute it in the2nd equation and find y , then substitute y and z in the 1stequation and find x .
x = y + 2y = z + 1
z = 2
x = y + 2y = 1z = 2
x = 3y = 1z = 2
Finally, the System of linear equations:x − y = 2
2x − y− z = 3x + y+ z = 6
has as a solution, the point
(x , y , z) = (3, 1, 2) .
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 241: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/241.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Systems of linear equations
That is, we find z from the 3rd equation, then substitute it in the2nd equation and find y , then substitute y and z in the 1stequation and find x .
x = y + 2y = z + 1
z = 2
x = y + 2y = 1z = 2
x = 3y = 1z = 2
Finally, the System of linear equations:x − y = 2
2x − y− z = 3x + y+ z = 6
has as a solution, the point (x , y , z) = (3, 1, 2) .Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 242: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/242.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Gaussian Elimination. Gauss-Jordan Reduction
Gaussian elimination
System of linear equations
Remember that a System of linear equations is an expression ofthe form:
a11x1 + a12x2 + · · ·+ a1nxn = b1
a21x1 + a22x2 + · · ·+ a2nxn = b2...
am1x1 + am2x2 + · · ·+ amnxn = bm
Here x1, x2, · · · xn are variables and aij , bj are constants.
A solution of the system is a common solution of all equationspresent in the system.
A system of linear equations can have one solution, infinitely manysolutions, or no solution at all.
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 243: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/243.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Gaussian Elimination. Gauss-Jordan Reduction
Gaussian elimination
System of linear equations
Remember that a System of linear equations is an expression ofthe form:
a11x1 + a12x2 + · · ·+ a1nxn = b1
a21x1 + a22x2 + · · ·+ a2nxn = b2...
am1x1 + am2x2 + · · ·+ amnxn = bm
Here x1, x2, · · · xn are variables and aij , bj are constants.
A solution of the system is a common solution of all equationspresent in the system.
A system of linear equations can have one solution, infinitely manysolutions, or no solution at all.
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 244: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/244.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Gaussian Elimination. Gauss-Jordan Reduction
Gaussian elimination
System of linear equations
Remember that a System of linear equations is an expression ofthe form:
a11x1 + a12x2 + · · ·+ a1nxn = b1
a21x1 + a22x2 + · · ·+ a2nxn = b2...
am1x1 + am2x2 + · · ·+ amnxn = bm
Here x1, x2, · · · xn are variables and aij , bj are constants.
A solution of the system is a common solution of all equationspresent in the system.
A system of linear equations can have one solution, infinitely manysolutions, or no solution at all.
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 245: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/245.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Gaussian Elimination. Gauss-Jordan Reduction
Gaussian elimination
System of linear equations
Remember
that a System of linear equations is an expression ofthe form:
a11x1 + a12x2 + · · ·+ a1nxn = b1
a21x1 + a22x2 + · · ·+ a2nxn = b2...
am1x1 + am2x2 + · · ·+ amnxn = bm
Here x1, x2, · · · xn are variables and aij , bj are constants.
A solution of the system is a common solution of all equationspresent in the system.
A system of linear equations can have one solution, infinitely manysolutions, or no solution at all.
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 246: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/246.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Gaussian Elimination. Gauss-Jordan Reduction
Gaussian elimination
System of linear equations
Remember that a System of linear equations
is an expression ofthe form:
a11x1 + a12x2 + · · ·+ a1nxn = b1
a21x1 + a22x2 + · · ·+ a2nxn = b2...
am1x1 + am2x2 + · · ·+ amnxn = bm
Here x1, x2, · · · xn are variables and aij , bj are constants.
A solution of the system is a common solution of all equationspresent in the system.
A system of linear equations can have one solution, infinitely manysolutions, or no solution at all.
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 247: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/247.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Gaussian Elimination. Gauss-Jordan Reduction
Gaussian elimination
System of linear equations
Remember that a System of linear equations is an expression
ofthe form:
a11x1 + a12x2 + · · ·+ a1nxn = b1
a21x1 + a22x2 + · · ·+ a2nxn = b2...
am1x1 + am2x2 + · · ·+ amnxn = bm
Here x1, x2, · · · xn are variables and aij , bj are constants.
A solution of the system is a common solution of all equationspresent in the system.
A system of linear equations can have one solution, infinitely manysolutions, or no solution at all.
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 248: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/248.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Gaussian Elimination. Gauss-Jordan Reduction
Gaussian elimination
System of linear equations
Remember that a System of linear equations is an expression ofthe form:
a11x1 + a12x2 + · · ·+ a1nxn = b1
a21x1 + a22x2 + · · ·+ a2nxn = b2...
am1x1 + am2x2 + · · ·+ amnxn = bm
Here x1, x2, · · · xn are variables and aij , bj are constants.
A solution of the system is a common solution of all equationspresent in the system.
A system of linear equations can have one solution, infinitely manysolutions, or no solution at all.
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 249: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/249.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Gaussian Elimination. Gauss-Jordan Reduction
Gaussian elimination
System of linear equations
Remember that a System of linear equations is an expression ofthe form:
a11x1 + a12x2 + · · ·+ a1nxn = b1
a21x1 + a22x2 + · · ·+ a2nxn = b2...
am1x1 + am2x2 + · · ·+ amnxn = bm
Here x1, x2, · · · xn are variables and aij , bj are constants.
A solution of the system is a common solution of all equationspresent in the system.
A system of linear equations can have one solution, infinitely manysolutions, or no solution at all.
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 250: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/250.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Gaussian Elimination. Gauss-Jordan Reduction
Gaussian elimination
System of linear equations
Remember that a System of linear equations is an expression ofthe form:
a11x1 + a12x2 + · · ·+ a1nxn = b1
a21x1 + a22x2 + · · ·+ a2nxn = b2...
am1x1 + am2x2 + · · ·+ amnxn = bm
Here x1, x2, · · · xn
are variables and aij , bj are constants.
A solution of the system is a common solution of all equationspresent in the system.
A system of linear equations can have one solution, infinitely manysolutions, or no solution at all.
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 251: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/251.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Gaussian Elimination. Gauss-Jordan Reduction
Gaussian elimination
System of linear equations
Remember that a System of linear equations is an expression ofthe form:
a11x1 + a12x2 + · · ·+ a1nxn = b1
a21x1 + a22x2 + · · ·+ a2nxn = b2...
am1x1 + am2x2 + · · ·+ amnxn = bm
Here x1, x2, · · · xn are variables and
aij , bj are constants.
A solution of the system is a common solution of all equationspresent in the system.
A system of linear equations can have one solution, infinitely manysolutions, or no solution at all.
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 252: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/252.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Gaussian Elimination. Gauss-Jordan Reduction
Gaussian elimination
System of linear equations
Remember that a System of linear equations is an expression ofthe form:
a11x1 + a12x2 + · · ·+ a1nxn = b1
a21x1 + a22x2 + · · ·+ a2nxn = b2...
am1x1 + am2x2 + · · ·+ amnxn = bm
Here x1, x2, · · · xn are variables and aij , bj are constants.
A solution of the system is a common solution of all equationspresent in the system.
A system of linear equations can have one solution, infinitely manysolutions, or no solution at all.
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 253: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/253.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Gaussian Elimination. Gauss-Jordan Reduction
Gaussian elimination
System of linear equations
Remember that a System of linear equations is an expression ofthe form:
a11x1 + a12x2 + · · ·+ a1nxn = b1
a21x1 + a22x2 + · · ·+ a2nxn = b2...
am1x1 + am2x2 + · · ·+ amnxn = bm
Here x1, x2, · · · xn are variables and aij , bj are constants.
A solution of the system
is a common solution of all equationspresent in the system.
A system of linear equations can have one solution, infinitely manysolutions, or no solution at all.
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 254: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/254.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Gaussian Elimination. Gauss-Jordan Reduction
Gaussian elimination
System of linear equations
Remember that a System of linear equations is an expression ofthe form:
a11x1 + a12x2 + · · ·+ a1nxn = b1
a21x1 + a22x2 + · · ·+ a2nxn = b2...
am1x1 + am2x2 + · · ·+ amnxn = bm
Here x1, x2, · · · xn are variables and aij , bj are constants.
A solution of the system is a common solution of all equations
present in the system.
A system of linear equations can have one solution, infinitely manysolutions, or no solution at all.
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 255: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/255.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Gaussian Elimination. Gauss-Jordan Reduction
Gaussian elimination
System of linear equations
Remember that a System of linear equations is an expression ofthe form:
a11x1 + a12x2 + · · ·+ a1nxn = b1
a21x1 + a22x2 + · · ·+ a2nxn = b2...
am1x1 + am2x2 + · · ·+ amnxn = bm
Here x1, x2, · · · xn are variables and aij , bj are constants.
A solution of the system is a common solution of all equationspresent in the system.
A system of linear equations can have one solution, infinitely manysolutions, or no solution at all.
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 256: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/256.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Gaussian Elimination. Gauss-Jordan Reduction
Gaussian elimination
System of linear equations
Remember that a System of linear equations is an expression ofthe form:
a11x1 + a12x2 + · · ·+ a1nxn = b1
a21x1 + a22x2 + · · ·+ a2nxn = b2...
am1x1 + am2x2 + · · ·+ amnxn = bm
Here x1, x2, · · · xn are variables and aij , bj are constants.
A solution of the system is a common solution of all equationspresent in the system.
A system
of linear equations can have one solution, infinitely manysolutions, or no solution at all.
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 257: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/257.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Gaussian Elimination. Gauss-Jordan Reduction
Gaussian elimination
System of linear equations
Remember that a System of linear equations is an expression ofthe form:
a11x1 + a12x2 + · · ·+ a1nxn = b1
a21x1 + a22x2 + · · ·+ a2nxn = b2...
am1x1 + am2x2 + · · ·+ amnxn = bm
Here x1, x2, · · · xn are variables and aij , bj are constants.
A solution of the system is a common solution of all equationspresent in the system.
A system of linear equations
can have one solution, infinitely manysolutions, or no solution at all.
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 258: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/258.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Gaussian Elimination. Gauss-Jordan Reduction
Gaussian elimination
System of linear equations
Remember that a System of linear equations is an expression ofthe form:
a11x1 + a12x2 + · · ·+ a1nxn = b1
a21x1 + a22x2 + · · ·+ a2nxn = b2...
am1x1 + am2x2 + · · ·+ amnxn = bm
Here x1, x2, · · · xn are variables and aij , bj are constants.
A solution of the system is a common solution of all equationspresent in the system.
A system of linear equations can have
one solution, infinitely manysolutions, or no solution at all.
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 259: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/259.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Gaussian Elimination. Gauss-Jordan Reduction
Gaussian elimination
System of linear equations
Remember that a System of linear equations is an expression ofthe form:
a11x1 + a12x2 + · · ·+ a1nxn = b1
a21x1 + a22x2 + · · ·+ a2nxn = b2...
am1x1 + am2x2 + · · ·+ amnxn = bm
Here x1, x2, · · · xn are variables and aij , bj are constants.
A solution of the system is a common solution of all equationspresent in the system.
A system of linear equations can have one solution,
infinitely manysolutions, or no solution at all.
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 260: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/260.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Gaussian Elimination. Gauss-Jordan Reduction
Gaussian elimination
System of linear equations
Remember that a System of linear equations is an expression ofthe form:
a11x1 + a12x2 + · · ·+ a1nxn = b1
a21x1 + a22x2 + · · ·+ a2nxn = b2...
am1x1 + am2x2 + · · ·+ amnxn = bm
Here x1, x2, · · · xn are variables and aij , bj are constants.
A solution of the system is a common solution of all equationspresent in the system.
A system of linear equations can have one solution, infinitely manysolutions, or
no solution at all.
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 261: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/261.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Gaussian Elimination. Gauss-Jordan Reduction
Gaussian elimination
System of linear equations
Remember that a System of linear equations is an expression ofthe form:
a11x1 + a12x2 + · · ·+ a1nxn = b1
a21x1 + a22x2 + · · ·+ a2nxn = b2...
am1x1 + am2x2 + · · ·+ amnxn = bm
Here x1, x2, · · · xn are variables and aij , bj are constants.
A solution of the system is a common solution of all equationspresent in the system.
A system of linear equations can have one solution, infinitely manysolutions, or no solution at all.
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 262: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/262.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Gaussian Elimination. Gauss-Jordan Reduction
Using the Elimination method we had already shown the first caseand using it again we can illustrate the other two extra cases
Example 1.3 x + y− 2z = 1
y− z = 3−x + 4y− 3z = 14
Solve the 1st equation for x :⇔ ( Equivalent Systems)
x = −y+ 2z+ 1y − z = 3
−x + 4y− 3z = 14
Eliminate x from the 3rd equation :
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 263: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/263.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Gaussian Elimination. Gauss-Jordan Reduction
Using the Elimination method
we had already shown the first caseand using it again we can illustrate the other two extra cases
Example 1.3 x + y− 2z = 1
y− z = 3−x + 4y− 3z = 14
Solve the 1st equation for x :⇔ ( Equivalent Systems)
x = −y+ 2z+ 1y − z = 3
−x + 4y− 3z = 14
Eliminate x from the 3rd equation :
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 264: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/264.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Gaussian Elimination. Gauss-Jordan Reduction
Using the Elimination method we had already shown
the first caseand using it again we can illustrate the other two extra cases
Example 1.3 x + y− 2z = 1
y− z = 3−x + 4y− 3z = 14
Solve the 1st equation for x :⇔ ( Equivalent Systems)
x = −y+ 2z+ 1y − z = 3
−x + 4y− 3z = 14
Eliminate x from the 3rd equation :
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 265: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/265.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Gaussian Elimination. Gauss-Jordan Reduction
Using the Elimination method we had already shown the first caseand
using it again we can illustrate the other two extra cases
Example 1.3 x + y− 2z = 1
y− z = 3−x + 4y− 3z = 14
Solve the 1st equation for x :⇔ ( Equivalent Systems)
x = −y+ 2z+ 1y − z = 3
−x + 4y− 3z = 14
Eliminate x from the 3rd equation :
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 266: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/266.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Gaussian Elimination. Gauss-Jordan Reduction
Using the Elimination method we had already shown the first caseand using it again
we can illustrate the other two extra cases
Example 1.3 x + y− 2z = 1
y− z = 3−x + 4y− 3z = 14
Solve the 1st equation for x :⇔ ( Equivalent Systems)
x = −y+ 2z+ 1y − z = 3
−x + 4y− 3z = 14
Eliminate x from the 3rd equation :
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 267: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/267.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Gaussian Elimination. Gauss-Jordan Reduction
Using the Elimination method we had already shown the first caseand using it again we can illustrate
the other two extra cases
Example 1.3 x + y− 2z = 1
y− z = 3−x + 4y− 3z = 14
Solve the 1st equation for x :⇔ ( Equivalent Systems)
x = −y+ 2z+ 1y − z = 3
−x + 4y− 3z = 14
Eliminate x from the 3rd equation :
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 268: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/268.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Gaussian Elimination. Gauss-Jordan Reduction
Using the Elimination method we had already shown the first caseand using it again we can illustrate the other
two extra cases
Example 1.3 x + y− 2z = 1
y− z = 3−x + 4y− 3z = 14
Solve the 1st equation for x :⇔ ( Equivalent Systems)
x = −y+ 2z+ 1y − z = 3
−x + 4y− 3z = 14
Eliminate x from the 3rd equation :
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 269: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/269.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Gaussian Elimination. Gauss-Jordan Reduction
Using the Elimination method we had already shown the first caseand using it again we can illustrate the other two extra cases
Example 1.3 x + y− 2z = 1
y− z = 3−x + 4y− 3z = 14
Solve the 1st equation for x :⇔ ( Equivalent Systems)
x = −y+ 2z+ 1y − z = 3
−x + 4y− 3z = 14
Eliminate x from the 3rd equation :
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 270: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/270.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Gaussian Elimination. Gauss-Jordan Reduction
Using the Elimination method we had already shown the first caseand using it again we can illustrate the other two extra cases
Example 1.3
x + y− 2z = 1
y− z = 3−x + 4y− 3z = 14
Solve the 1st equation for x :⇔ ( Equivalent Systems)
x = −y+ 2z+ 1y − z = 3
−x + 4y− 3z = 14
Eliminate x from the 3rd equation :
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 271: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/271.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Gaussian Elimination. Gauss-Jordan Reduction
Using the Elimination method we had already shown the first caseand using it again we can illustrate the other two extra cases
Example 1.3 x + y− 2z = 1
y− z = 3−x + 4y− 3z = 14
Solve the 1st equation for x :⇔ ( Equivalent Systems)
x = −y+ 2z+ 1y − z = 3
−x + 4y− 3z = 14
Eliminate x from the 3rd equation :
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 272: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/272.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Gaussian Elimination. Gauss-Jordan Reduction
Using the Elimination method we had already shown the first caseand using it again we can illustrate the other two extra cases
Example 1.3 x + y− 2z = 1
y− z = 3−x + 4y− 3z = 14
Solve the 1st equation for x :
⇔ ( Equivalent Systems)x = −y+ 2z+ 1
y − z = 3−x + 4y− 3z = 14
Eliminate x from the 3rd equation :
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 273: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/273.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Gaussian Elimination. Gauss-Jordan Reduction
Using the Elimination method we had already shown the first caseand using it again we can illustrate the other two extra cases
Example 1.3 x + y− 2z = 1
y− z = 3−x + 4y− 3z = 14
Solve the 1st equation for x :⇔ ( Equivalent Systems)
x = −y+ 2z+ 1
y − z = 3−x + 4y− 3z = 14
Eliminate x from the 3rd equation :
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 274: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/274.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Gaussian Elimination. Gauss-Jordan Reduction
Using the Elimination method we had already shown the first caseand using it again we can illustrate the other two extra cases
Example 1.3 x + y− 2z = 1
y− z = 3−x + 4y− 3z = 14
Solve the 1st equation for x :⇔ ( Equivalent Systems)
x = −y+ 2z+ 1y − z = 3
−x + 4y− 3z = 14
Eliminate x from the 3rd equation :
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 275: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/275.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Gaussian Elimination. Gauss-Jordan Reduction
Using the Elimination method we had already shown the first caseand using it again we can illustrate the other two extra cases
Example 1.3 x + y− 2z = 1
y− z = 3−x + 4y− 3z = 14
Solve the 1st equation for x :⇔ ( Equivalent Systems)
x = −y+ 2z+ 1y − z = 3
−x + 4y− 3z = 14
Eliminate x from the 3rd equation :Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 276: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/276.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
⇔
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 277: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/277.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Gaussian Elimination. Gauss-Jordan Reduction
⇔ x = −y + 2z + 1
y − z = 3−(−y + 2z + 1) + 4y − 3z = 14
Simplify :⇔ x = −y + 2z + 1
y − z = 35y − 5z = 15⇔Solve the 2nd equation for y
x = −y+ 2z+ 1y = z+ 35y − 5z = 15
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 278: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/278.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Gaussian Elimination. Gauss-Jordan Reduction
⇔
x = −y + 2z + 1
y − z = 3−(−y + 2z + 1) + 4y − 3z = 14
Simplify :⇔ x = −y + 2z + 1
y − z = 35y − 5z = 15⇔Solve the 2nd equation for y
x = −y+ 2z+ 1y = z+ 35y − 5z = 15
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 279: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/279.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Gaussian Elimination. Gauss-Jordan Reduction
⇔ x = −y + 2z + 1
y − z = 3−(−y + 2z + 1) + 4y − 3z = 14
Simplify :⇔ x = −y + 2z + 1
y − z = 35y − 5z = 15⇔Solve the 2nd equation for y
x = −y+ 2z+ 1y = z+ 35y − 5z = 15
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 280: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/280.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Gaussian Elimination. Gauss-Jordan Reduction
⇔ x = −y + 2z + 1
y − z = 3−(−y + 2z + 1) + 4y − 3z = 14
Simplify :
⇔ x = −y + 2z + 1
y − z = 35y − 5z = 15⇔Solve the 2nd equation for y
x = −y+ 2z+ 1y = z+ 35y − 5z = 15
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 281: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/281.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Gaussian Elimination. Gauss-Jordan Reduction
⇔ x = −y + 2z + 1
y − z = 3−(−y + 2z + 1) + 4y − 3z = 14
Simplify :⇔
x = −y + 2z + 1
y − z = 35y − 5z = 15⇔Solve the 2nd equation for y
x = −y+ 2z+ 1y = z+ 35y − 5z = 15
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 282: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/282.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Gaussian Elimination. Gauss-Jordan Reduction
⇔ x = −y + 2z + 1
y − z = 3−(−y + 2z + 1) + 4y − 3z = 14
Simplify :⇔ x = −y + 2z + 1
y − z = 35y − 5z = 15⇔
Solve the 2nd equation for yx = −y+ 2z+ 1y = z+ 35y − 5z = 15
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 283: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/283.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Gaussian Elimination. Gauss-Jordan Reduction
⇔ x = −y + 2z + 1
y − z = 3−(−y + 2z + 1) + 4y − 3z = 14
Simplify :⇔ x = −y + 2z + 1
y − z = 35y − 5z = 15⇔Solve the 2nd equation for y
x = −y+ 2z+ 1y = z+ 35y − 5z = 15
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 284: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/284.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Gaussian Elimination. Gauss-Jordan Reduction
⇔ x = −y + 2z + 1
y − z = 3−(−y + 2z + 1) + 4y − 3z = 14
Simplify :⇔ x = −y + 2z + 1
y − z = 35y − 5z = 15⇔Solve the 2nd equation for y
x = −y+ 2z+ 1y = z+ 35y − 5z = 15
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 285: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/285.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Gaussian Elimination. Gauss-Jordan Reduction
Eliminate y from the 3rd equation:x = −y+ 2z+ 1y = z+ 3
5(z + 3)− 5z = 15
⇔Simplify :
x = −y+ 2z+ 1y = z + 3
0 = 0
Now, the elimination process is completed. The last equation isactually 0z = 0. Hence z , is a free variable, that is, it can beassigned an arbitrary value. Then, x and y are found by backsubstitution .
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 286: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/286.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Gaussian Elimination. Gauss-Jordan Reduction
Eliminate y from the 3rd equation:
x = −y+ 2z+ 1y = z+ 3
5(z + 3)− 5z = 15
⇔Simplify :
x = −y+ 2z+ 1y = z + 3
0 = 0
Now, the elimination process is completed. The last equation isactually 0z = 0. Hence z , is a free variable, that is, it can beassigned an arbitrary value. Then, x and y are found by backsubstitution .
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 287: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/287.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Gaussian Elimination. Gauss-Jordan Reduction
Eliminate y from the 3rd equation:x = −y+ 2z+ 1y = z+ 3
5(z + 3)− 5z = 15
⇔Simplify :
x = −y+ 2z+ 1y = z + 3
0 = 0
Now, the elimination process is completed. The last equation isactually 0z = 0. Hence z , is a free variable, that is, it can beassigned an arbitrary value. Then, x and y are found by backsubstitution .
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 288: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/288.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Gaussian Elimination. Gauss-Jordan Reduction
Eliminate y from the 3rd equation:x = −y+ 2z+ 1y = z+ 3
5(z + 3)− 5z = 15
⇔
Simplify : x = −y+ 2z+ 1y = z + 3
0 = 0
Now, the elimination process is completed. The last equation isactually 0z = 0. Hence z , is a free variable, that is, it can beassigned an arbitrary value. Then, x and y are found by backsubstitution .
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 289: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/289.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Gaussian Elimination. Gauss-Jordan Reduction
Eliminate y from the 3rd equation:x = −y+ 2z+ 1y = z+ 3
5(z + 3)− 5z = 15
⇔Simplify :
x = −y+ 2z+ 1y = z + 3
0 = 0
Now, the elimination process is completed. The last equation isactually 0z = 0. Hence z , is a free variable, that is, it can beassigned an arbitrary value. Then, x and y are found by backsubstitution .
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 290: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/290.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Gaussian Elimination. Gauss-Jordan Reduction
Eliminate y from the 3rd equation:x = −y+ 2z+ 1y = z+ 3
5(z + 3)− 5z = 15
⇔Simplify :
x = −y+ 2z+ 1y = z + 3
0 = 0
Now, the elimination process is completed. The last equation isactually 0z = 0. Hence z , is a free variable, that is, it can beassigned an arbitrary value. Then, x and y are found by backsubstitution .
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 291: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/291.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Gaussian Elimination. Gauss-Jordan Reduction
Eliminate y from the 3rd equation:x = −y+ 2z+ 1y = z+ 3
5(z + 3)− 5z = 15
⇔Simplify :
x = −y+ 2z+ 1y = z + 3
0 = 0
Now,
the elimination process is completed. The last equation isactually 0z = 0. Hence z , is a free variable, that is, it can beassigned an arbitrary value. Then, x and y are found by backsubstitution .
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 292: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/292.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Gaussian Elimination. Gauss-Jordan Reduction
Eliminate y from the 3rd equation:x = −y+ 2z+ 1y = z+ 3
5(z + 3)− 5z = 15
⇔Simplify :
x = −y+ 2z+ 1y = z + 3
0 = 0
Now, the elimination process is completed.
The last equation isactually 0z = 0. Hence z , is a free variable, that is, it can beassigned an arbitrary value. Then, x and y are found by backsubstitution .
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 293: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/293.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Gaussian Elimination. Gauss-Jordan Reduction
Eliminate y from the 3rd equation:x = −y+ 2z+ 1y = z+ 3
5(z + 3)− 5z = 15
⇔Simplify :
x = −y+ 2z+ 1y = z + 3
0 = 0
Now, the elimination process is completed. The last equation
isactually 0z = 0. Hence z , is a free variable, that is, it can beassigned an arbitrary value. Then, x and y are found by backsubstitution .
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 294: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/294.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Gaussian Elimination. Gauss-Jordan Reduction
Eliminate y from the 3rd equation:x = −y+ 2z+ 1y = z+ 3
5(z + 3)− 5z = 15
⇔Simplify :
x = −y+ 2z+ 1y = z + 3
0 = 0
Now, the elimination process is completed. The last equation isactually 0z = 0.
Hence z , is a free variable, that is, it can beassigned an arbitrary value. Then, x and y are found by backsubstitution .
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 295: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/295.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Gaussian Elimination. Gauss-Jordan Reduction
Eliminate y from the 3rd equation:x = −y+ 2z+ 1y = z+ 3
5(z + 3)− 5z = 15
⇔Simplify :
x = −y+ 2z+ 1y = z + 3
0 = 0
Now, the elimination process is completed. The last equation isactually 0z = 0. Hence z ,
is a free variable, that is, it can beassigned an arbitrary value. Then, x and y are found by backsubstitution .
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 296: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/296.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Gaussian Elimination. Gauss-Jordan Reduction
Eliminate y from the 3rd equation:x = −y+ 2z+ 1y = z+ 3
5(z + 3)− 5z = 15
⇔Simplify :
x = −y+ 2z+ 1y = z + 3
0 = 0
Now, the elimination process is completed. The last equation isactually 0z = 0. Hence z , is a free variable, that is,
it can beassigned an arbitrary value. Then, x and y are found by backsubstitution .
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 297: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/297.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Gaussian Elimination. Gauss-Jordan Reduction
Eliminate y from the 3rd equation:x = −y+ 2z+ 1y = z+ 3
5(z + 3)− 5z = 15
⇔Simplify :
x = −y+ 2z+ 1y = z + 3
0 = 0
Now, the elimination process is completed. The last equation isactually 0z = 0. Hence z , is a free variable, that is, it can beassigned
an arbitrary value. Then, x and y are found by backsubstitution .
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 298: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/298.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Gaussian Elimination. Gauss-Jordan Reduction
Eliminate y from the 3rd equation:x = −y+ 2z+ 1y = z+ 3
5(z + 3)− 5z = 15
⇔Simplify :
x = −y+ 2z+ 1y = z + 3
0 = 0
Now, the elimination process is completed. The last equation isactually 0z = 0. Hence z , is a free variable, that is, it can beassigned an arbitrary value.
Then, x and y are found by backsubstitution .
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 299: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/299.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Gaussian Elimination. Gauss-Jordan Reduction
Eliminate y from the 3rd equation:x = −y+ 2z+ 1y = z+ 3
5(z + 3)− 5z = 15
⇔Simplify :
x = −y+ 2z+ 1y = z + 3
0 = 0
Now, the elimination process is completed. The last equation isactually 0z = 0. Hence z , is a free variable, that is, it can beassigned an arbitrary value. Then, x and y
are found by backsubstitution .
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 300: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/300.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Gaussian Elimination. Gauss-Jordan Reduction
Eliminate y from the 3rd equation:x = −y+ 2z+ 1y = z+ 3
5(z + 3)− 5z = 15
⇔Simplify :
x = −y+ 2z+ 1y = z + 3
0 = 0
Now, the elimination process is completed. The last equation isactually 0z = 0. Hence z , is a free variable, that is, it can beassigned an arbitrary value. Then, x and y are found by
backsubstitution .
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 301: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/301.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Gaussian Elimination. Gauss-Jordan Reduction
Eliminate y from the 3rd equation:x = −y+ 2z+ 1y = z+ 3
5(z + 3)− 5z = 15
⇔Simplify :
x = −y+ 2z+ 1y = z + 3
0 = 0
Now, the elimination process is completed. The last equation isactually 0z = 0. Hence z , is a free variable, that is, it can beassigned an arbitrary value. Then, x and y are found by backsubstitution .
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 302: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/302.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Gaussian Elimination. Gauss-Jordan Reduction
z = t (a parameter)
y = z + 3x = −y + 2z + 1
⇔ z = t
y = t + 3x = t − 2
Thus, the system x + y− 2z = 1
y − z = 3−x + 4y− 3z = 14
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 303: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/303.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Gaussian Elimination. Gauss-Jordan Reduction
z = t (a parameter)
y = z + 3x = −y + 2z + 1
⇔
z = t
y = t + 3x = t − 2
Thus, the system x + y− 2z = 1
y − z = 3−x + 4y− 3z = 14
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 304: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/304.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Gaussian Elimination. Gauss-Jordan Reduction
z = t (a parameter)
y = z + 3x = −y + 2z + 1
⇔ z = t
y = t + 3x = t − 2
Thus, the system x + y− 2z = 1
y − z = 3−x + 4y− 3z = 14
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 305: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/305.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Gaussian Elimination. Gauss-Jordan Reduction
z = t (a parameter)
y = z + 3x = −y + 2z + 1
⇔ z = t
y = t + 3x = t − 2
Thus, the system
x + y− 2z = 1
y − z = 3−x + 4y− 3z = 14
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 306: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/306.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Gaussian Elimination. Gauss-Jordan Reduction
z = t (a parameter)
y = z + 3x = −y + 2z + 1
⇔ z = t
y = t + 3x = t − 2
Thus, the system x + y− 2z = 1
y − z = 3−x + 4y− 3z = 14
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 307: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/307.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Gaussian Elimination. Gauss-Jordan Reduction
has the General Solution
(x , y , z) = (t − 2, t + 3, t)
or in vector form
(x , y , z) = (−2, 3, 0) + t(1, 1, 1)
The set of all solutions is a straight line in R3 passing through thepoint P = (−2, 3, 0) in the direction v =< 1, 1, 1 > .
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 308: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/308.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Gaussian Elimination. Gauss-Jordan Reduction
has the General Solution
(x , y , z) = (t − 2, t + 3, t)
or in vector form
(x , y , z) = (−2, 3, 0) + t(1, 1, 1)
The set of all solutions is a straight line in R3 passing through thepoint P = (−2, 3, 0) in the direction v =< 1, 1, 1 > .
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 309: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/309.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Gaussian Elimination. Gauss-Jordan Reduction
has the General Solution
(x , y , z) = (t − 2, t + 3, t)
or in vector form
(x , y , z) = (−2, 3, 0) + t(1, 1, 1)
The set of all solutions is a straight line in R3 passing through thepoint P = (−2, 3, 0) in the direction v =< 1, 1, 1 > .
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 310: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/310.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Gaussian Elimination. Gauss-Jordan Reduction
has the General Solution
(x , y , z) = (t − 2, t + 3, t)
or in vector form
(x , y , z) = (−2, 3, 0) + t(1, 1, 1)
The set of all solutions is a straight line in R3 passing through thepoint P = (−2, 3, 0) in the direction v =< 1, 1, 1 > .
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 311: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/311.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Gaussian Elimination. Gauss-Jordan Reduction
has the General Solution
(x , y , z) = (t − 2, t + 3, t)
or in vector form
(x , y , z) = (−2, 3, 0) + t(1, 1, 1)
The set of all solutions is a straight line in R3 passing through thepoint P = (−2, 3, 0) in the direction v =< 1, 1, 1 > .
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 312: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/312.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Gaussian Elimination. Gauss-Jordan Reduction
has the General Solution
(x , y , z) = (t − 2, t + 3, t)
or in vector form
(x , y , z) = (−2, 3, 0) + t(1, 1, 1)
The set of all solutions
is a straight line in R3 passing through thepoint P = (−2, 3, 0) in the direction v =< 1, 1, 1 > .
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 313: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/313.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Gaussian Elimination. Gauss-Jordan Reduction
has the General Solution
(x , y , z) = (t − 2, t + 3, t)
or in vector form
(x , y , z) = (−2, 3, 0) + t(1, 1, 1)
The set of all solutions is a straight line
in R3 passing through thepoint P = (−2, 3, 0) in the direction v =< 1, 1, 1 > .
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 314: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/314.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Gaussian Elimination. Gauss-Jordan Reduction
has the General Solution
(x , y , z) = (t − 2, t + 3, t)
or in vector form
(x , y , z) = (−2, 3, 0) + t(1, 1, 1)
The set of all solutions is a straight line in R3
passing through thepoint P = (−2, 3, 0) in the direction v =< 1, 1, 1 > .
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 315: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/315.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Gaussian Elimination. Gauss-Jordan Reduction
has the General Solution
(x , y , z) = (t − 2, t + 3, t)
or in vector form
(x , y , z) = (−2, 3, 0) + t(1, 1, 1)
The set of all solutions is a straight line in R3 passing through thepoint
P = (−2, 3, 0) in the direction v =< 1, 1, 1 > .
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 316: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/316.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Gaussian Elimination. Gauss-Jordan Reduction
has the General Solution
(x , y , z) = (t − 2, t + 3, t)
or in vector form
(x , y , z) = (−2, 3, 0) + t(1, 1, 1)
The set of all solutions is a straight line in R3 passing through thepoint P = (−2, 3, 0)
in the direction v =< 1, 1, 1 > .
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 317: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/317.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Gaussian Elimination. Gauss-Jordan Reduction
has the General Solution
(x , y , z) = (t − 2, t + 3, t)
or in vector form
(x , y , z) = (−2, 3, 0) + t(1, 1, 1)
The set of all solutions is a straight line in R3 passing through thepoint P = (−2, 3, 0) in the direction
v =< 1, 1, 1 > .
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 318: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/318.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Gaussian Elimination. Gauss-Jordan Reduction
has the General Solution
(x , y , z) = (t − 2, t + 3, t)
or in vector form
(x , y , z) = (−2, 3, 0) + t(1, 1, 1)
The set of all solutions is a straight line in R3 passing through thepoint P = (−2, 3, 0) in the direction v =< 1, 1, 1 > .
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 319: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/319.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Gaussian Elimination. Gauss-Jordan Reduction
Example 1.4
x + y− 2z = 1
y− z = 3−x + 4y− 3z = 1
Solve the 1st equation for x :
⇔ (Equivalent System )x = −y + 2z + 1
y − z = 3−x + 4y − 3z = 1
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 320: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/320.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Gaussian Elimination. Gauss-Jordan Reduction
Example 1.4
x + y− 2z = 1
y− z = 3−x + 4y− 3z = 1
Solve the 1st equation for x :
⇔ (Equivalent System )x = −y + 2z + 1
y − z = 3−x + 4y − 3z = 1
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 321: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/321.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Gaussian Elimination. Gauss-Jordan Reduction
Example 1.4
x + y− 2z = 1
y− z = 3−x + 4y− 3z = 1
Solve the 1st equation for x :
⇔ (Equivalent System )x = −y + 2z + 1
y − z = 3−x + 4y − 3z = 1
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 322: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/322.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Gaussian Elimination. Gauss-Jordan Reduction
Example 1.4
x + y− 2z = 1
y− z = 3−x + 4y− 3z = 1
Solve the 1st equation for x :
⇔ (Equivalent System )x = −y + 2z + 1
y − z = 3−x + 4y − 3z = 1
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 323: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/323.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Gaussian Elimination. Gauss-Jordan Reduction
Example 1.4
x + y− 2z = 1
y− z = 3−x + 4y− 3z = 1
Solve the 1st equation for x :
⇔ (Equivalent System )
x = −y + 2z + 1
y − z = 3−x + 4y − 3z = 1
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 324: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/324.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Gaussian Elimination. Gauss-Jordan Reduction
Example 1.4
x + y− 2z = 1
y− z = 3−x + 4y− 3z = 1
Solve the 1st equation for x :
⇔ (Equivalent System )x = −y + 2z + 1
y − z = 3−x + 4y − 3z = 1
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 325: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/325.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Gaussian Elimination. Gauss-Jordan Reduction
⇔Eliminate x from the 3rd equation:
x = −y + 2z + 1y − z = 3
−(−y + 2z + 1) + 4y − 3z = 1⇔Simplify:
x = −y + 2z + 1y − z = 3
5y − 5z = 2⇔Solve the second equation for y :
x = −y + 2z + 1y = z + 3
5y − 5z = 2
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 326: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/326.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Gaussian Elimination. Gauss-Jordan Reduction
⇔
Eliminate x from the 3rd equation:x = −y + 2z + 1
y − z = 3−(−y + 2z + 1) + 4y − 3z = 1
⇔Simplify:
x = −y + 2z + 1y − z = 3
5y − 5z = 2⇔Solve the second equation for y :
x = −y + 2z + 1y = z + 3
5y − 5z = 2
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 327: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/327.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Gaussian Elimination. Gauss-Jordan Reduction
⇔Eliminate x from the 3rd equation:
x = −y + 2z + 1
y − z = 3−(−y + 2z + 1) + 4y − 3z = 1
⇔Simplify:
x = −y + 2z + 1y − z = 3
5y − 5z = 2⇔Solve the second equation for y :
x = −y + 2z + 1y = z + 3
5y − 5z = 2
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 328: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/328.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Gaussian Elimination. Gauss-Jordan Reduction
⇔Eliminate x from the 3rd equation:
x = −y + 2z + 1y − z = 3
−(−y + 2z + 1) + 4y − 3z = 1⇔
Simplify: x = −y + 2z + 1
y − z = 35y − 5z = 2
⇔Solve the second equation for y :
x = −y + 2z + 1y = z + 3
5y − 5z = 2
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 329: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/329.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Gaussian Elimination. Gauss-Jordan Reduction
⇔Eliminate x from the 3rd equation:
x = −y + 2z + 1y − z = 3
−(−y + 2z + 1) + 4y − 3z = 1⇔Simplify:
x = −y + 2z + 1
y − z = 35y − 5z = 2
⇔Solve the second equation for y :
x = −y + 2z + 1y = z + 3
5y − 5z = 2
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 330: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/330.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Gaussian Elimination. Gauss-Jordan Reduction
⇔Eliminate x from the 3rd equation:
x = −y + 2z + 1y − z = 3
−(−y + 2z + 1) + 4y − 3z = 1⇔Simplify:
x = −y + 2z + 1y − z = 3
5y − 5z = 2
⇔Solve the second equation for y :
x = −y + 2z + 1y = z + 3
5y − 5z = 2
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 331: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/331.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Gaussian Elimination. Gauss-Jordan Reduction
⇔Eliminate x from the 3rd equation:
x = −y + 2z + 1y − z = 3
−(−y + 2z + 1) + 4y − 3z = 1⇔Simplify:
x = −y + 2z + 1y − z = 3
5y − 5z = 2⇔
Solve the second equation for y :x = −y + 2z + 1
y = z + 35y − 5z = 2
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 332: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/332.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Gaussian Elimination. Gauss-Jordan Reduction
⇔Eliminate x from the 3rd equation:
x = −y + 2z + 1y − z = 3
−(−y + 2z + 1) + 4y − 3z = 1⇔Simplify:
x = −y + 2z + 1y − z = 3
5y − 5z = 2⇔Solve the second equation for y :
x = −y + 2z + 1
y = z + 35y − 5z = 2
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 333: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/333.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Gaussian Elimination. Gauss-Jordan Reduction
⇔Eliminate x from the 3rd equation:
x = −y + 2z + 1y − z = 3
−(−y + 2z + 1) + 4y − 3z = 1⇔Simplify:
x = −y + 2z + 1y − z = 3
5y − 5z = 2⇔Solve the second equation for y :
x = −y + 2z + 1y = z + 3
5y − 5z = 2Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 334: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/334.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Gaussian Elimination. Gauss-Jordan Reduction
⇔Eliminate y from the 3rd equation:
x = −y + 2z + 1y = z + 3
5(z + 3)− 5z = 2⇔Simplify:
x = −y + 2z + 1y = z + 3
15 = 2
Now, the elimination process is completed. The last equationactually is giving us a contradiction. Hence, there is no solution forthis system.
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 335: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/335.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Gaussian Elimination. Gauss-Jordan Reduction
⇔
Eliminate y from the 3rd equation:x = −y + 2z + 1
y = z + 35(z + 3)− 5z = 2
⇔Simplify:
x = −y + 2z + 1y = z + 3
15 = 2
Now, the elimination process is completed. The last equationactually is giving us a contradiction. Hence, there is no solution forthis system.
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 336: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/336.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Gaussian Elimination. Gauss-Jordan Reduction
⇔Eliminate y from the 3rd equation:
x = −y + 2z + 1
y = z + 35(z + 3)− 5z = 2
⇔Simplify:
x = −y + 2z + 1y = z + 3
15 = 2
Now, the elimination process is completed. The last equationactually is giving us a contradiction. Hence, there is no solution forthis system.
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 337: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/337.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Gaussian Elimination. Gauss-Jordan Reduction
⇔Eliminate y from the 3rd equation:
x = −y + 2z + 1y = z + 3
5(z + 3)− 5z = 2
⇔Simplify:
x = −y + 2z + 1y = z + 3
15 = 2
Now, the elimination process is completed. The last equationactually is giving us a contradiction. Hence, there is no solution forthis system.
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 338: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/338.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Gaussian Elimination. Gauss-Jordan Reduction
⇔Eliminate y from the 3rd equation:
x = −y + 2z + 1y = z + 3
5(z + 3)− 5z = 2⇔
Simplify: x = −y + 2z + 1
y = z + 315 = 2
Now, the elimination process is completed. The last equationactually is giving us a contradiction. Hence, there is no solution forthis system.
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 339: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/339.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Gaussian Elimination. Gauss-Jordan Reduction
⇔Eliminate y from the 3rd equation:
x = −y + 2z + 1y = z + 3
5(z + 3)− 5z = 2⇔Simplify:
x = −y + 2z + 1
y = z + 315 = 2
Now, the elimination process is completed. The last equationactually is giving us a contradiction. Hence, there is no solution forthis system.
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 340: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/340.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Gaussian Elimination. Gauss-Jordan Reduction
⇔Eliminate y from the 3rd equation:
x = −y + 2z + 1y = z + 3
5(z + 3)− 5z = 2⇔Simplify:
x = −y + 2z + 1y = z + 3
15 = 2
Now, the elimination process is completed. The last equationactually is giving us a contradiction. Hence, there is no solution forthis system.
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 341: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/341.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Gaussian Elimination. Gauss-Jordan Reduction
⇔Eliminate y from the 3rd equation:
x = −y + 2z + 1y = z + 3
5(z + 3)− 5z = 2⇔Simplify:
x = −y + 2z + 1y = z + 3
15 = 2
Now,
the elimination process is completed. The last equationactually is giving us a contradiction. Hence, there is no solution forthis system.
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 342: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/342.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Gaussian Elimination. Gauss-Jordan Reduction
⇔Eliminate y from the 3rd equation:
x = −y + 2z + 1y = z + 3
5(z + 3)− 5z = 2⇔Simplify:
x = −y + 2z + 1y = z + 3
15 = 2
Now, the elimination process is completed.
The last equationactually is giving us a contradiction. Hence, there is no solution forthis system.
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 343: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/343.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Gaussian Elimination. Gauss-Jordan Reduction
⇔Eliminate y from the 3rd equation:
x = −y + 2z + 1y = z + 3
5(z + 3)− 5z = 2⇔Simplify:
x = −y + 2z + 1y = z + 3
15 = 2
Now, the elimination process is completed. The last equation
actually is giving us a contradiction. Hence, there is no solution forthis system.
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 344: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/344.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Gaussian Elimination. Gauss-Jordan Reduction
⇔Eliminate y from the 3rd equation:
x = −y + 2z + 1y = z + 3
5(z + 3)− 5z = 2⇔Simplify:
x = −y + 2z + 1y = z + 3
15 = 2
Now, the elimination process is completed. The last equationactually is giving us
a contradiction. Hence, there is no solution forthis system.
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 345: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/345.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Gaussian Elimination. Gauss-Jordan Reduction
⇔Eliminate y from the 3rd equation:
x = −y + 2z + 1y = z + 3
5(z + 3)− 5z = 2⇔Simplify:
x = −y + 2z + 1y = z + 3
15 = 2
Now, the elimination process is completed. The last equationactually is giving us a contradiction. Hence, there is no solution forthis system.
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 346: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/346.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Gaussian Elimination. Gauss-Jordan Reduction
Thus, the system x + y− 2z = 1
y− z = 3−x + 4y− 3z = 1
has the General Solution
φ = empty set
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 347: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/347.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Gaussian Elimination. Gauss-Jordan Reduction
Thus, the system
x + y− 2z = 1
y− z = 3−x + 4y− 3z = 1
has the General Solution
φ = empty set
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 348: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/348.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Gaussian Elimination. Gauss-Jordan Reduction
Thus, the system x + y− 2z = 1
y− z = 3−x + 4y− 3z = 1
has the General Solution
φ = empty set
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 349: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/349.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Gaussian Elimination. Gauss-Jordan Reduction
Thus, the system x + y− 2z = 1
y− z = 3−x + 4y− 3z = 1
has the General Solution
φ = empty set
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 350: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/350.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Gaussian Elimination. Gauss-Jordan Reduction
Thus, the system x + y− 2z = 1
y− z = 3−x + 4y− 3z = 1
has the General Solution
φ = empty set
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 351: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/351.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Gaussian Elimination. Gauss-Jordan Reduction
Gaussian elimination
Gaussian elimination is a modification of the elimination methodthat allows only so-called elementary operations .
Elementary operations for systems of linear equations:
(1) to multiply an equation by a nonzero scalar;
(2) to add an equation multiplied by a scalar to another equation;
(3) to interchange any two equations.
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 352: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/352.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Gaussian Elimination. Gauss-Jordan Reduction
Gaussian elimination
Gaussian elimination is a modification of the elimination methodthat allows only so-called elementary operations .
Elementary operations for systems of linear equations:
(1) to multiply an equation by a nonzero scalar;
(2) to add an equation multiplied by a scalar to another equation;
(3) to interchange any two equations.
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 353: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/353.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Gaussian Elimination. Gauss-Jordan Reduction
Gaussian elimination
Gaussian elimination
is a modification of the elimination methodthat allows only so-called elementary operations .
Elementary operations for systems of linear equations:
(1) to multiply an equation by a nonzero scalar;
(2) to add an equation multiplied by a scalar to another equation;
(3) to interchange any two equations.
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 354: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/354.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Gaussian Elimination. Gauss-Jordan Reduction
Gaussian elimination
Gaussian elimination is a modification of
the elimination methodthat allows only so-called elementary operations .
Elementary operations for systems of linear equations:
(1) to multiply an equation by a nonzero scalar;
(2) to add an equation multiplied by a scalar to another equation;
(3) to interchange any two equations.
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 355: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/355.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Gaussian Elimination. Gauss-Jordan Reduction
Gaussian elimination
Gaussian elimination is a modification of the elimination method
that allows only so-called elementary operations .
Elementary operations for systems of linear equations:
(1) to multiply an equation by a nonzero scalar;
(2) to add an equation multiplied by a scalar to another equation;
(3) to interchange any two equations.
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 356: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/356.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Gaussian Elimination. Gauss-Jordan Reduction
Gaussian elimination
Gaussian elimination is a modification of the elimination methodthat allows only
so-called elementary operations .
Elementary operations for systems of linear equations:
(1) to multiply an equation by a nonzero scalar;
(2) to add an equation multiplied by a scalar to another equation;
(3) to interchange any two equations.
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 357: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/357.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Gaussian Elimination. Gauss-Jordan Reduction
Gaussian elimination
Gaussian elimination is a modification of the elimination methodthat allows only so-called elementary operations .
Elementary operations for systems of linear equations:
(1) to multiply an equation by a nonzero scalar;
(2) to add an equation multiplied by a scalar to another equation;
(3) to interchange any two equations.
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 358: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/358.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Gaussian Elimination. Gauss-Jordan Reduction
Gaussian elimination
Gaussian elimination is a modification of the elimination methodthat allows only so-called elementary operations .
Elementary operations
for systems of linear equations:
(1) to multiply an equation by a nonzero scalar;
(2) to add an equation multiplied by a scalar to another equation;
(3) to interchange any two equations.
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 359: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/359.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Gaussian Elimination. Gauss-Jordan Reduction
Gaussian elimination
Gaussian elimination is a modification of the elimination methodthat allows only so-called elementary operations .
Elementary operations for systems of
linear equations:
(1) to multiply an equation by a nonzero scalar;
(2) to add an equation multiplied by a scalar to another equation;
(3) to interchange any two equations.
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 360: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/360.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Gaussian Elimination. Gauss-Jordan Reduction
Gaussian elimination
Gaussian elimination is a modification of the elimination methodthat allows only so-called elementary operations .
Elementary operations for systems of linear equations:
(1) to multiply an equation by a nonzero scalar;
(2) to add an equation multiplied by a scalar to another equation;
(3) to interchange any two equations.
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 361: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/361.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Gaussian Elimination. Gauss-Jordan Reduction
Gaussian elimination
Gaussian elimination is a modification of the elimination methodthat allows only so-called elementary operations .
Elementary operations for systems of linear equations:
(1) to multiply
an equation by a nonzero scalar;
(2) to add an equation multiplied by a scalar to another equation;
(3) to interchange any two equations.
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 362: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/362.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Gaussian Elimination. Gauss-Jordan Reduction
Gaussian elimination
Gaussian elimination is a modification of the elimination methodthat allows only so-called elementary operations .
Elementary operations for systems of linear equations:
(1) to multiply an equation
by a nonzero scalar;
(2) to add an equation multiplied by a scalar to another equation;
(3) to interchange any two equations.
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 363: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/363.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Gaussian Elimination. Gauss-Jordan Reduction
Gaussian elimination
Gaussian elimination is a modification of the elimination methodthat allows only so-called elementary operations .
Elementary operations for systems of linear equations:
(1) to multiply an equation by a nonzero scalar;
(2) to add an equation multiplied by a scalar to another equation;
(3) to interchange any two equations.
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 364: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/364.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Gaussian Elimination. Gauss-Jordan Reduction
Gaussian elimination
Gaussian elimination is a modification of the elimination methodthat allows only so-called elementary operations .
Elementary operations for systems of linear equations:
(1) to multiply an equation by a nonzero scalar;
(2) to add
an equation multiplied by a scalar to another equation;
(3) to interchange any two equations.
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 365: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/365.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Gaussian Elimination. Gauss-Jordan Reduction
Gaussian elimination
Gaussian elimination is a modification of the elimination methodthat allows only so-called elementary operations .
Elementary operations for systems of linear equations:
(1) to multiply an equation by a nonzero scalar;
(2) to add an equation
multiplied by a scalar to another equation;
(3) to interchange any two equations.
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 366: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/366.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Gaussian Elimination. Gauss-Jordan Reduction
Gaussian elimination
Gaussian elimination is a modification of the elimination methodthat allows only so-called elementary operations .
Elementary operations for systems of linear equations:
(1) to multiply an equation by a nonzero scalar;
(2) to add an equation multiplied by a scalar
to another equation;
(3) to interchange any two equations.
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 367: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/367.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Gaussian Elimination. Gauss-Jordan Reduction
Gaussian elimination
Gaussian elimination is a modification of the elimination methodthat allows only so-called elementary operations .
Elementary operations for systems of linear equations:
(1) to multiply an equation by a nonzero scalar;
(2) to add an equation multiplied by a scalar to another equation;
(3) to interchange any two equations.
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 368: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/368.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Gaussian Elimination. Gauss-Jordan Reduction
Gaussian elimination
Gaussian elimination is a modification of the elimination methodthat allows only so-called elementary operations .
Elementary operations for systems of linear equations:
(1) to multiply an equation by a nonzero scalar;
(2) to add an equation multiplied by a scalar to another equation;
(3) to interchange
any two equations.
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 369: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/369.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Gaussian Elimination. Gauss-Jordan Reduction
Gaussian elimination
Gaussian elimination is a modification of the elimination methodthat allows only so-called elementary operations .
Elementary operations for systems of linear equations:
(1) to multiply an equation by a nonzero scalar;
(2) to add an equation multiplied by a scalar to another equation;
(3) to interchange any two equations.
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 370: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/370.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Gaussian Elimination. Gauss-Jordan Reduction
The next result, is very important, because it ensures that thisoperations will not modify the solution set of the original system.
Theorem
(i) Applying elementary operations to a system of linear equationsdoes not change the solution set of the system.
(ii) Any elementary operation can be undone by anotherelementary operation.
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 371: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/371.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Gaussian Elimination. Gauss-Jordan Reduction
The next result,
is very important, because it ensures that thisoperations will not modify the solution set of the original system.
Theorem
(i) Applying elementary operations to a system of linear equationsdoes not change the solution set of the system.
(ii) Any elementary operation can be undone by anotherelementary operation.
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 372: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/372.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Gaussian Elimination. Gauss-Jordan Reduction
The next result, is very important,
because it ensures that thisoperations will not modify the solution set of the original system.
Theorem
(i) Applying elementary operations to a system of linear equationsdoes not change the solution set of the system.
(ii) Any elementary operation can be undone by anotherelementary operation.
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 373: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/373.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Gaussian Elimination. Gauss-Jordan Reduction
The next result, is very important, because it ensures
that thisoperations will not modify the solution set of the original system.
Theorem
(i) Applying elementary operations to a system of linear equationsdoes not change the solution set of the system.
(ii) Any elementary operation can be undone by anotherelementary operation.
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 374: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/374.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Gaussian Elimination. Gauss-Jordan Reduction
The next result, is very important, because it ensures that thisoperations
will not modify the solution set of the original system.
Theorem
(i) Applying elementary operations to a system of linear equationsdoes not change the solution set of the system.
(ii) Any elementary operation can be undone by anotherelementary operation.
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 375: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/375.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Gaussian Elimination. Gauss-Jordan Reduction
The next result, is very important, because it ensures that thisoperations will not
modify the solution set of the original system.
Theorem
(i) Applying elementary operations to a system of linear equationsdoes not change the solution set of the system.
(ii) Any elementary operation can be undone by anotherelementary operation.
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 376: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/376.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Gaussian Elimination. Gauss-Jordan Reduction
The next result, is very important, because it ensures that thisoperations will not modify the solution set
of the original system.
Theorem
(i) Applying elementary operations to a system of linear equationsdoes not change the solution set of the system.
(ii) Any elementary operation can be undone by anotherelementary operation.
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 377: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/377.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Gaussian Elimination. Gauss-Jordan Reduction
The next result, is very important, because it ensures that thisoperations will not modify the solution set of the original system.
Theorem
(i) Applying elementary operations to a system of linear equationsdoes not change the solution set of the system.
(ii) Any elementary operation can be undone by anotherelementary operation.
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 378: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/378.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Gaussian Elimination. Gauss-Jordan Reduction
The next result, is very important, because it ensures that thisoperations will not modify the solution set of the original system.
Theorem
(i) Applying
elementary operations to a system of linear equationsdoes not change the solution set of the system.
(ii) Any elementary operation can be undone by anotherelementary operation.
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 379: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/379.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Gaussian Elimination. Gauss-Jordan Reduction
The next result, is very important, because it ensures that thisoperations will not modify the solution set of the original system.
Theorem
(i) Applying elementary operations
to a system of linear equationsdoes not change the solution set of the system.
(ii) Any elementary operation can be undone by anotherelementary operation.
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 380: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/380.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Gaussian Elimination. Gauss-Jordan Reduction
The next result, is very important, because it ensures that thisoperations will not modify the solution set of the original system.
Theorem
(i) Applying elementary operations to a system
of linear equationsdoes not change the solution set of the system.
(ii) Any elementary operation can be undone by anotherelementary operation.
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 381: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/381.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Gaussian Elimination. Gauss-Jordan Reduction
The next result, is very important, because it ensures that thisoperations will not modify the solution set of the original system.
Theorem
(i) Applying elementary operations to a system of linear equations
does not change the solution set of the system.
(ii) Any elementary operation can be undone by anotherelementary operation.
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 382: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/382.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Gaussian Elimination. Gauss-Jordan Reduction
The next result, is very important, because it ensures that thisoperations will not modify the solution set of the original system.
Theorem
(i) Applying elementary operations to a system of linear equationsdoes not change
the solution set of the system.
(ii) Any elementary operation can be undone by anotherelementary operation.
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 383: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/383.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Gaussian Elimination. Gauss-Jordan Reduction
The next result, is very important, because it ensures that thisoperations will not modify the solution set of the original system.
Theorem
(i) Applying elementary operations to a system of linear equationsdoes not change the solution set
of the system.
(ii) Any elementary operation can be undone by anotherelementary operation.
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 384: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/384.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Gaussian Elimination. Gauss-Jordan Reduction
The next result, is very important, because it ensures that thisoperations will not modify the solution set of the original system.
Theorem
(i) Applying elementary operations to a system of linear equationsdoes not change the solution set of the system.
(ii) Any elementary operation can be undone by anotherelementary operation.
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 385: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/385.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Gaussian Elimination. Gauss-Jordan Reduction
The next result, is very important, because it ensures that thisoperations will not modify the solution set of the original system.
Theorem
(i) Applying elementary operations to a system of linear equationsdoes not change the solution set of the system.
(ii) Any
elementary operation can be undone by anotherelementary operation.
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 386: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/386.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Gaussian Elimination. Gauss-Jordan Reduction
The next result, is very important, because it ensures that thisoperations will not modify the solution set of the original system.
Theorem
(i) Applying elementary operations to a system of linear equationsdoes not change the solution set of the system.
(ii) Any elementary operation
can be undone by anotherelementary operation.
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 387: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/387.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Gaussian Elimination. Gauss-Jordan Reduction
The next result, is very important, because it ensures that thisoperations will not modify the solution set of the original system.
Theorem
(i) Applying elementary operations to a system of linear equationsdoes not change the solution set of the system.
(ii) Any elementary operation can be undone
by anotherelementary operation.
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 388: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/388.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Gaussian Elimination. Gauss-Jordan Reduction
The next result, is very important, because it ensures that thisoperations will not modify the solution set of the original system.
Theorem
(i) Applying elementary operations to a system of linear equationsdoes not change the solution set of the system.
(ii) Any elementary operation can be undone by another
elementary operation.
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 389: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/389.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Gaussian Elimination. Gauss-Jordan Reduction
The next result, is very important, because it ensures that thisoperations will not modify the solution set of the original system.
Theorem
(i) Applying elementary operations to a system of linear equationsdoes not change the solution set of the system.
(ii) Any elementary operation can be undone by anotherelementary operation.
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 390: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/390.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Gaussian Elimination. Gauss-Jordan Reduction
Operation 1: Multiply the ith equation by r 6= 0
a11x1 + a12x2 + · · ·+ a1nxn = b1...
ai1x1 + ai2x2 + · · ·+ ainxn = bi...
am1x1 + am2x2 + · · ·+ amnxn = bm
⇒
a11x1 + a12x2 + · · ·+ a1nxn = b1...
(rai1)x1 + (rai2)x2 + · · ·+ (rain)xn = rbi...
am1x1 + am2x2 + · · ·+ amnxn = bm
To undo the operation, multiply the ith equation by r−1
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 391: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/391.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Gaussian Elimination. Gauss-Jordan Reduction
Operation 1: Multiply
the ith equation by r 6= 0
a11x1 + a12x2 + · · ·+ a1nxn = b1...
ai1x1 + ai2x2 + · · ·+ ainxn = bi...
am1x1 + am2x2 + · · ·+ amnxn = bm
⇒
a11x1 + a12x2 + · · ·+ a1nxn = b1...
(rai1)x1 + (rai2)x2 + · · ·+ (rain)xn = rbi...
am1x1 + am2x2 + · · ·+ amnxn = bm
To undo the operation, multiply the ith equation by r−1
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 392: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/392.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Gaussian Elimination. Gauss-Jordan Reduction
Operation 1: Multiply the ith equation
by r 6= 0
a11x1 + a12x2 + · · ·+ a1nxn = b1...
ai1x1 + ai2x2 + · · ·+ ainxn = bi...
am1x1 + am2x2 + · · ·+ amnxn = bm
⇒
a11x1 + a12x2 + · · ·+ a1nxn = b1...
(rai1)x1 + (rai2)x2 + · · ·+ (rain)xn = rbi...
am1x1 + am2x2 + · · ·+ amnxn = bm
To undo the operation, multiply the ith equation by r−1
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 393: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/393.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Gaussian Elimination. Gauss-Jordan Reduction
Operation 1: Multiply the ith equation by
r 6= 0
a11x1 + a12x2 + · · ·+ a1nxn = b1...
ai1x1 + ai2x2 + · · ·+ ainxn = bi...
am1x1 + am2x2 + · · ·+ amnxn = bm
⇒
a11x1 + a12x2 + · · ·+ a1nxn = b1...
(rai1)x1 + (rai2)x2 + · · ·+ (rain)xn = rbi...
am1x1 + am2x2 + · · ·+ amnxn = bm
To undo the operation, multiply the ith equation by r−1
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 394: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/394.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Gaussian Elimination. Gauss-Jordan Reduction
Operation 1: Multiply the ith equation by r 6= 0
a11x1 + a12x2 + · · ·+ a1nxn = b1...
ai1x1 + ai2x2 + · · ·+ ainxn = bi...
am1x1 + am2x2 + · · ·+ amnxn = bm
⇒
a11x1 + a12x2 + · · ·+ a1nxn = b1...
(rai1)x1 + (rai2)x2 + · · ·+ (rain)xn = rbi...
am1x1 + am2x2 + · · ·+ amnxn = bm
To undo the operation, multiply the ith equation by r−1
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 395: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/395.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Gaussian Elimination. Gauss-Jordan Reduction
Operation 1: Multiply the ith equation by r 6= 0
a11x1 + a12x2 + · · ·+ a1nxn = b1...
ai1x1 + ai2x2 + · · ·+ ainxn = bi...
am1x1 + am2x2 + · · ·+ amnxn = bm
⇒
a11x1 + a12x2 + · · ·+ a1nxn = b1...
(rai1)x1 + (rai2)x2 + · · ·+ (rain)xn = rbi...
am1x1 + am2x2 + · · ·+ amnxn = bm
To undo the operation, multiply the ith equation by r−1
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 396: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/396.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Gaussian Elimination. Gauss-Jordan Reduction
Operation 1: Multiply the ith equation by r 6= 0
a11x1 + a12x2 + · · ·+ a1nxn = b1...
ai1x1 + ai2x2 + · · ·+ ainxn = bi...
am1x1 + am2x2 + · · ·+ amnxn = bm
⇒
a11x1 + a12x2 + · · ·+ a1nxn = b1...
(rai1)x1 + (rai2)x2 + · · ·+ (rain)xn = rbi...
am1x1 + am2x2 + · · ·+ amnxn = bm
To undo the operation, multiply the ith equation by r−1
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 397: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/397.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Gaussian Elimination. Gauss-Jordan Reduction
Operation 1: Multiply the ith equation by r 6= 0
a11x1 + a12x2 + · · ·+ a1nxn = b1...
ai1x1 + ai2x2 + · · ·+ ainxn = bi...
am1x1 + am2x2 + · · ·+ amnxn = bm
⇒
a11x1 + a12x2 + · · ·+ a1nxn = b1...
(rai1)x1 + (rai2)x2 + · · ·+ (rain)xn = rbi...
am1x1 + am2x2 + · · ·+ amnxn = bm
To undo
the operation, multiply the ith equation by r−1
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 398: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/398.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Gaussian Elimination. Gauss-Jordan Reduction
Operation 1: Multiply the ith equation by r 6= 0
a11x1 + a12x2 + · · ·+ a1nxn = b1...
ai1x1 + ai2x2 + · · ·+ ainxn = bi...
am1x1 + am2x2 + · · ·+ amnxn = bm
⇒
a11x1 + a12x2 + · · ·+ a1nxn = b1...
(rai1)x1 + (rai2)x2 + · · ·+ (rain)xn = rbi...
am1x1 + am2x2 + · · ·+ amnxn = bm
To undo the operation,
multiply the ith equation by r−1
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 399: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/399.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Gaussian Elimination. Gauss-Jordan Reduction
Operation 1: Multiply the ith equation by r 6= 0
a11x1 + a12x2 + · · ·+ a1nxn = b1...
ai1x1 + ai2x2 + · · ·+ ainxn = bi...
am1x1 + am2x2 + · · ·+ amnxn = bm
⇒
a11x1 + a12x2 + · · ·+ a1nxn = b1...
(rai1)x1 + (rai2)x2 + · · ·+ (rain)xn = rbi...
am1x1 + am2x2 + · · ·+ amnxn = bm
To undo the operation, multiply the ith equation
by r−1
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 400: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/400.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Gaussian Elimination. Gauss-Jordan Reduction
Operation 1: Multiply the ith equation by r 6= 0
a11x1 + a12x2 + · · ·+ a1nxn = b1...
ai1x1 + ai2x2 + · · ·+ ainxn = bi...
am1x1 + am2x2 + · · ·+ amnxn = bm
⇒
a11x1 + a12x2 + · · ·+ a1nxn = b1...
(rai1)x1 + (rai2)x2 + · · ·+ (rain)xn = rbi...
am1x1 + am2x2 + · · ·+ amnxn = bm
To undo the operation, multiply the ith equation by
r−1
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 401: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/401.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Gaussian Elimination. Gauss-Jordan Reduction
Operation 1: Multiply the ith equation by r 6= 0
a11x1 + a12x2 + · · ·+ a1nxn = b1...
ai1x1 + ai2x2 + · · ·+ ainxn = bi...
am1x1 + am2x2 + · · ·+ amnxn = bm
⇒
a11x1 + a12x2 + · · ·+ a1nxn = b1...
(rai1)x1 + (rai2)x2 + · · ·+ (rain)xn = rbi...
am1x1 + am2x2 + · · ·+ amnxn = bm
To undo the operation, multiply the ith equation by r−1Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 402: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/402.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Gaussian Elimination. Gauss-Jordan Reduction
Operation 2: Add r times the ith equation to the jth equation
...ai1x1 + ai2x2 + · · ·+ ainxn = bi
...aj1x1 + aj2x2 + · · ·+ ajnxn = bj
...
⇒
...ai1x1 + ai2x2 + · · ·+ ainxn = bi
...(aj1 + rai1)x1 + (aj2 + rai2)x2 + · · ·+ (ajn + rain)xn = bj + rbi
...To undo the operation, add −r times the ith equation to the jthequation .
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 403: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/403.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Gaussian Elimination. Gauss-Jordan Reduction
Operation 2: Add
r times the ith equation to the jth equation
...ai1x1 + ai2x2 + · · ·+ ainxn = bi
...aj1x1 + aj2x2 + · · ·+ ajnxn = bj
...
⇒
...ai1x1 + ai2x2 + · · ·+ ainxn = bi
...(aj1 + rai1)x1 + (aj2 + rai2)x2 + · · ·+ (ajn + rain)xn = bj + rbi
...To undo the operation, add −r times the ith equation to the jthequation .
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 404: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/404.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Gaussian Elimination. Gauss-Jordan Reduction
Operation 2: Add r times
the ith equation to the jth equation
...ai1x1 + ai2x2 + · · ·+ ainxn = bi
...aj1x1 + aj2x2 + · · ·+ ajnxn = bj
...
⇒
...ai1x1 + ai2x2 + · · ·+ ainxn = bi
...(aj1 + rai1)x1 + (aj2 + rai2)x2 + · · ·+ (ajn + rain)xn = bj + rbi
...To undo the operation, add −r times the ith equation to the jthequation .
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 405: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/405.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Gaussian Elimination. Gauss-Jordan Reduction
Operation 2: Add r times the ith equation
to the jth equation
...ai1x1 + ai2x2 + · · ·+ ainxn = bi
...aj1x1 + aj2x2 + · · ·+ ajnxn = bj
...
⇒
...ai1x1 + ai2x2 + · · ·+ ainxn = bi
...(aj1 + rai1)x1 + (aj2 + rai2)x2 + · · ·+ (ajn + rain)xn = bj + rbi
...To undo the operation, add −r times the ith equation to the jthequation .
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 406: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/406.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Gaussian Elimination. Gauss-Jordan Reduction
Operation 2: Add r times the ith equation to the jth equation
...ai1x1 + ai2x2 + · · ·+ ainxn = bi
...aj1x1 + aj2x2 + · · ·+ ajnxn = bj
...
⇒
...ai1x1 + ai2x2 + · · ·+ ainxn = bi
...(aj1 + rai1)x1 + (aj2 + rai2)x2 + · · ·+ (ajn + rain)xn = bj + rbi
...To undo the operation, add −r times the ith equation to the jthequation .
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 407: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/407.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Gaussian Elimination. Gauss-Jordan Reduction
Operation 2: Add r times the ith equation to the jth equation
...ai1x1 + ai2x2 + · · ·+ ainxn = bi
...aj1x1 + aj2x2 + · · ·+ ajnxn = bj
...
⇒
...ai1x1 + ai2x2 + · · ·+ ainxn = bi
...(aj1 + rai1)x1 + (aj2 + rai2)x2 + · · ·+ (ajn + rain)xn = bj + rbi
...To undo the operation, add −r times the ith equation to the jthequation .
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 408: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/408.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Gaussian Elimination. Gauss-Jordan Reduction
Operation 2: Add r times the ith equation to the jth equation
...ai1x1 + ai2x2 + · · ·+ ainxn = bi
...aj1x1 + aj2x2 + · · ·+ ajnxn = bj
...
⇒
...ai1x1 + ai2x2 + · · ·+ ainxn = bi
...(aj1 + rai1)x1 + (aj2 + rai2)x2 + · · ·+ (ajn + rain)xn = bj + rbi
...To undo the operation, add −r times the ith equation to the jthequation .
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 409: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/409.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Gaussian Elimination. Gauss-Jordan Reduction
Operation 2: Add r times the ith equation to the jth equation
...ai1x1 + ai2x2 + · · ·+ ainxn = bi
...aj1x1 + aj2x2 + · · ·+ ajnxn = bj
...
⇒
...ai1x1 + ai2x2 + · · ·+ ainxn = bi
...(aj1 + rai1)x1 + (aj2 + rai2)x2 + · · ·+ (ajn + rain)xn = bj + rbi
...
To undo the operation, add −r times the ith equation to the jthequation .
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 410: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/410.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Gaussian Elimination. Gauss-Jordan Reduction
Operation 2: Add r times the ith equation to the jth equation
...ai1x1 + ai2x2 + · · ·+ ainxn = bi
...aj1x1 + aj2x2 + · · ·+ ajnxn = bj
...
⇒
...ai1x1 + ai2x2 + · · ·+ ainxn = bi
...(aj1 + rai1)x1 + (aj2 + rai2)x2 + · · ·+ (ajn + rain)xn = bj + rbi
...To undo
the operation, add −r times the ith equation to the jthequation .
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 411: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/411.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Gaussian Elimination. Gauss-Jordan Reduction
Operation 2: Add r times the ith equation to the jth equation
...ai1x1 + ai2x2 + · · ·+ ainxn = bi
...aj1x1 + aj2x2 + · · ·+ ajnxn = bj
...
⇒
...ai1x1 + ai2x2 + · · ·+ ainxn = bi
...(aj1 + rai1)x1 + (aj2 + rai2)x2 + · · ·+ (ajn + rain)xn = bj + rbi
...To undo the operation,
add −r times the ith equation to the jthequation .
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 412: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/412.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Gaussian Elimination. Gauss-Jordan Reduction
Operation 2: Add r times the ith equation to the jth equation
...ai1x1 + ai2x2 + · · ·+ ainxn = bi
...aj1x1 + aj2x2 + · · ·+ ajnxn = bj
...
⇒
...ai1x1 + ai2x2 + · · ·+ ainxn = bi
...(aj1 + rai1)x1 + (aj2 + rai2)x2 + · · ·+ (ajn + rain)xn = bj + rbi
...To undo the operation, add −r times
the ith equation to the jthequation .
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 413: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/413.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Gaussian Elimination. Gauss-Jordan Reduction
Operation 2: Add r times the ith equation to the jth equation
...ai1x1 + ai2x2 + · · ·+ ainxn = bi
...aj1x1 + aj2x2 + · · ·+ ajnxn = bj
...
⇒
...ai1x1 + ai2x2 + · · ·+ ainxn = bi
...(aj1 + rai1)x1 + (aj2 + rai2)x2 + · · ·+ (ajn + rain)xn = bj + rbi
...To undo the operation, add −r times the ith equation
to the jthequation .
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 414: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/414.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Gaussian Elimination. Gauss-Jordan Reduction
Operation 2: Add r times the ith equation to the jth equation
...ai1x1 + ai2x2 + · · ·+ ainxn = bi
...aj1x1 + aj2x2 + · · ·+ ajnxn = bj
...
⇒
...ai1x1 + ai2x2 + · · ·+ ainxn = bi
...(aj1 + rai1)x1 + (aj2 + rai2)x2 + · · ·+ (ajn + rain)xn = bj + rbi
...To undo the operation, add −r times the ith equation to the jthequation . Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 415: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/415.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Gaussian Elimination. Gauss-Jordan Reduction
Operation 3: interchange the ith and jth equations.
...ai1x1 + ai2x2 + · · ·+ ainxn = bi
...aj1x1 + aj2x2 + · · ·+ ajnxn = bj
...
⇒
...aj1x1 + aj2x2 + · · ·+ ajnxn = bj
...ai1x1 + ai2x2 + · · ·+ ainxn = bi
...
To undo the operation, apply it once more .
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 416: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/416.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Gaussian Elimination. Gauss-Jordan Reduction
Operation 3: interchange
the ith and jth equations.
...ai1x1 + ai2x2 + · · ·+ ainxn = bi
...aj1x1 + aj2x2 + · · ·+ ajnxn = bj
...
⇒
...aj1x1 + aj2x2 + · · ·+ ajnxn = bj
...ai1x1 + ai2x2 + · · ·+ ainxn = bi
...
To undo the operation, apply it once more .
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 417: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/417.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Gaussian Elimination. Gauss-Jordan Reduction
Operation 3: interchange the ith and
jth equations.
...ai1x1 + ai2x2 + · · ·+ ainxn = bi
...aj1x1 + aj2x2 + · · ·+ ajnxn = bj
...
⇒
...aj1x1 + aj2x2 + · · ·+ ajnxn = bj
...ai1x1 + ai2x2 + · · ·+ ainxn = bi
...
To undo the operation, apply it once more .
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 418: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/418.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Gaussian Elimination. Gauss-Jordan Reduction
Operation 3: interchange the ith and jth equations.
...ai1x1 + ai2x2 + · · ·+ ainxn = bi
...aj1x1 + aj2x2 + · · ·+ ajnxn = bj
...
⇒
...aj1x1 + aj2x2 + · · ·+ ajnxn = bj
...ai1x1 + ai2x2 + · · ·+ ainxn = bi
...
To undo the operation, apply it once more .
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 419: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/419.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Gaussian Elimination. Gauss-Jordan Reduction
Operation 3: interchange the ith and jth equations.
...ai1x1 + ai2x2 + · · ·+ ainxn = bi
...aj1x1 + aj2x2 + · · ·+ ajnxn = bj
...
⇒
...aj1x1 + aj2x2 + · · ·+ ajnxn = bj
...ai1x1 + ai2x2 + · · ·+ ainxn = bi
...
To undo the operation, apply it once more .
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 420: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/420.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Gaussian Elimination. Gauss-Jordan Reduction
Operation 3: interchange the ith and jth equations.
...ai1x1 + ai2x2 + · · ·+ ainxn = bi
...aj1x1 + aj2x2 + · · ·+ ajnxn = bj
...
⇒
...aj1x1 + aj2x2 + · · ·+ ajnxn = bj
...ai1x1 + ai2x2 + · · ·+ ainxn = bi
...
To undo the operation, apply it once more .
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 421: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/421.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Gaussian Elimination. Gauss-Jordan Reduction
Operation 3: interchange the ith and jth equations.
...ai1x1 + ai2x2 + · · ·+ ainxn = bi
...aj1x1 + aj2x2 + · · ·+ ajnxn = bj
...
⇒
...aj1x1 + aj2x2 + · · ·+ ajnxn = bj
...ai1x1 + ai2x2 + · · ·+ ainxn = bi
...
To undo the operation, apply it once more .
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 422: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/422.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Gaussian Elimination. Gauss-Jordan Reduction
Operation 3: interchange the ith and jth equations.
...ai1x1 + ai2x2 + · · ·+ ainxn = bi
...aj1x1 + aj2x2 + · · ·+ ajnxn = bj
...
⇒
...aj1x1 + aj2x2 + · · ·+ ajnxn = bj
...ai1x1 + ai2x2 + · · ·+ ainxn = bi
...
To undo
the operation, apply it once more .
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 423: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/423.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Gaussian Elimination. Gauss-Jordan Reduction
Operation 3: interchange the ith and jth equations.
...ai1x1 + ai2x2 + · · ·+ ainxn = bi
...aj1x1 + aj2x2 + · · ·+ ajnxn = bj
...
⇒
...aj1x1 + aj2x2 + · · ·+ ajnxn = bj
...ai1x1 + ai2x2 + · · ·+ ainxn = bi
...
To undo the operation,
apply it once more .
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 424: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/424.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Gaussian Elimination. Gauss-Jordan Reduction
Operation 3: interchange the ith and jth equations.
...ai1x1 + ai2x2 + · · ·+ ainxn = bi
...aj1x1 + aj2x2 + · · ·+ ajnxn = bj
...
⇒
...aj1x1 + aj2x2 + · · ·+ ajnxn = bj
...ai1x1 + ai2x2 + · · ·+ ainxn = bi
...
To undo the operation, apply it
once more .
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 425: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/425.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Gaussian Elimination. Gauss-Jordan Reduction
Operation 3: interchange the ith and jth equations.
...ai1x1 + ai2x2 + · · ·+ ainxn = bi
...aj1x1 + aj2x2 + · · ·+ ajnxn = bj
...
⇒
...aj1x1 + aj2x2 + · · ·+ ajnxn = bj
...ai1x1 + ai2x2 + · · ·+ ainxn = bi
...
To undo the operation, apply it once more .Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 426: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/426.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Gaussian Elimination. Gauss-Jordan Reduction
Example 1.5 x − y = 2
2x − y − z = 3x + y + z = 6
Add −2 times the 1st equation to the 2nd equation:x − y = 2
y − z = −1x + y + z = 6
R2 := R2 − 2 ∗ R1
Add −1 times the 1st equation to the 3rd equation:x − y = 2
y − z = −12y + z = 4
R3 := (−1) ∗ R1 + R3
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 427: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/427.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Gaussian Elimination. Gauss-Jordan Reduction
Example 1.5
x − y = 2
2x − y − z = 3x + y + z = 6
Add −2 times the 1st equation to the 2nd equation:x − y = 2
y − z = −1x + y + z = 6
R2 := R2 − 2 ∗ R1
Add −1 times the 1st equation to the 3rd equation:x − y = 2
y − z = −12y + z = 4
R3 := (−1) ∗ R1 + R3
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 428: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/428.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Gaussian Elimination. Gauss-Jordan Reduction
Example 1.5 x − y = 2
2x − y − z = 3x + y + z = 6
Add −2 times the 1st equation to the 2nd equation:x − y = 2
y − z = −1x + y + z = 6
R2 := R2 − 2 ∗ R1
Add −1 times the 1st equation to the 3rd equation:x − y = 2
y − z = −12y + z = 4
R3 := (−1) ∗ R1 + R3
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 429: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/429.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Gaussian Elimination. Gauss-Jordan Reduction
Example 1.5 x − y = 2
2x − y − z = 3x + y + z = 6
Add
−2 times the 1st equation to the 2nd equation:x − y = 2
y − z = −1x + y + z = 6
R2 := R2 − 2 ∗ R1
Add −1 times the 1st equation to the 3rd equation:x − y = 2
y − z = −12y + z = 4
R3 := (−1) ∗ R1 + R3
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 430: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/430.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Gaussian Elimination. Gauss-Jordan Reduction
Example 1.5 x − y = 2
2x − y − z = 3x + y + z = 6
Add −2 times
the 1st equation to the 2nd equation:x − y = 2
y − z = −1x + y + z = 6
R2 := R2 − 2 ∗ R1
Add −1 times the 1st equation to the 3rd equation:x − y = 2
y − z = −12y + z = 4
R3 := (−1) ∗ R1 + R3
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 431: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/431.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Gaussian Elimination. Gauss-Jordan Reduction
Example 1.5 x − y = 2
2x − y − z = 3x + y + z = 6
Add −2 times the 1st equation to
the 2nd equation:x − y = 2
y − z = −1x + y + z = 6
R2 := R2 − 2 ∗ R1
Add −1 times the 1st equation to the 3rd equation:x − y = 2
y − z = −12y + z = 4
R3 := (−1) ∗ R1 + R3
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 432: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/432.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Gaussian Elimination. Gauss-Jordan Reduction
Example 1.5 x − y = 2
2x − y − z = 3x + y + z = 6
Add −2 times the 1st equation to the 2nd equation:
x − y = 2
y − z = −1x + y + z = 6
R2 := R2 − 2 ∗ R1
Add −1 times the 1st equation to the 3rd equation:x − y = 2
y − z = −12y + z = 4
R3 := (−1) ∗ R1 + R3
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 433: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/433.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Gaussian Elimination. Gauss-Jordan Reduction
Example 1.5 x − y = 2
2x − y − z = 3x + y + z = 6
Add −2 times the 1st equation to the 2nd equation:x − y = 2
y − z = −1x + y + z = 6
R2 := R2 − 2 ∗ R1
Add −1 times the 1st equation to the 3rd equation:x − y = 2
y − z = −12y + z = 4
R3 := (−1) ∗ R1 + R3
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 434: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/434.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Gaussian Elimination. Gauss-Jordan Reduction
Example 1.5 x − y = 2
2x − y − z = 3x + y + z = 6
Add −2 times the 1st equation to the 2nd equation:x − y = 2
y − z = −1x + y + z = 6
R2 := R2 − 2 ∗ R1
Add
−1 times the 1st equation to the 3rd equation:x − y = 2
y − z = −12y + z = 4
R3 := (−1) ∗ R1 + R3
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 435: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/435.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Gaussian Elimination. Gauss-Jordan Reduction
Example 1.5 x − y = 2
2x − y − z = 3x + y + z = 6
Add −2 times the 1st equation to the 2nd equation:x − y = 2
y − z = −1x + y + z = 6
R2 := R2 − 2 ∗ R1
Add −1 times
the 1st equation to the 3rd equation:x − y = 2
y − z = −12y + z = 4
R3 := (−1) ∗ R1 + R3
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 436: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/436.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Gaussian Elimination. Gauss-Jordan Reduction
Example 1.5 x − y = 2
2x − y − z = 3x + y + z = 6
Add −2 times the 1st equation to the 2nd equation:x − y = 2
y − z = −1x + y + z = 6
R2 := R2 − 2 ∗ R1
Add −1 times the 1st equation to
the 3rd equation:x − y = 2
y − z = −12y + z = 4
R3 := (−1) ∗ R1 + R3
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 437: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/437.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Gaussian Elimination. Gauss-Jordan Reduction
Example 1.5 x − y = 2
2x − y − z = 3x + y + z = 6
Add −2 times the 1st equation to the 2nd equation:x − y = 2
y − z = −1x + y + z = 6
R2 := R2 − 2 ∗ R1
Add −1 times the 1st equation to the 3rd equation:
x − y = 2
y − z = −12y + z = 4
R3 := (−1) ∗ R1 + R3
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 438: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/438.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Gaussian Elimination. Gauss-Jordan Reduction
Example 1.5 x − y = 2
2x − y − z = 3x + y + z = 6
Add −2 times the 1st equation to the 2nd equation:x − y = 2
y − z = −1x + y + z = 6
R2 := R2 − 2 ∗ R1
Add −1 times the 1st equation to the 3rd equation:x − y = 2
y − z = −12y + z = 4
R3 := (−1) ∗ R1 + R3
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 439: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/439.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Gaussian Elimination. Gauss-Jordan Reduction
Add −2 times the 2nd equation to the 3rd equation:x − y = 2
y − z = −13z = 6
R3 := (−2) ∗ R2 + R3
Note: At this point, the elimination process is completed, and wecan solve the system by back substitution. However, we cancontinue with elementary operations
Multiply the 3rd equation by 1/3:x − y = 2
y − z = −1z = 2
R3 := (1/3) ∗ R3
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 440: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/440.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Gaussian Elimination. Gauss-Jordan Reduction
Add
−2 times the 2nd equation to the 3rd equation:x − y = 2
y − z = −13z = 6
R3 := (−2) ∗ R2 + R3
Note: At this point, the elimination process is completed, and wecan solve the system by back substitution. However, we cancontinue with elementary operations
Multiply the 3rd equation by 1/3:x − y = 2
y − z = −1z = 2
R3 := (1/3) ∗ R3
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 441: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/441.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Gaussian Elimination. Gauss-Jordan Reduction
Add −2 times
the 2nd equation to the 3rd equation:x − y = 2
y − z = −13z = 6
R3 := (−2) ∗ R2 + R3
Note: At this point, the elimination process is completed, and wecan solve the system by back substitution. However, we cancontinue with elementary operations
Multiply the 3rd equation by 1/3:x − y = 2
y − z = −1z = 2
R3 := (1/3) ∗ R3
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 442: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/442.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Gaussian Elimination. Gauss-Jordan Reduction
Add −2 times the 2nd equation to
the 3rd equation:x − y = 2
y − z = −13z = 6
R3 := (−2) ∗ R2 + R3
Note: At this point, the elimination process is completed, and wecan solve the system by back substitution. However, we cancontinue with elementary operations
Multiply the 3rd equation by 1/3:x − y = 2
y − z = −1z = 2
R3 := (1/3) ∗ R3
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 443: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/443.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Gaussian Elimination. Gauss-Jordan Reduction
Add −2 times the 2nd equation to the 3rd equation:
x − y = 2
y − z = −13z = 6
R3 := (−2) ∗ R2 + R3
Note: At this point, the elimination process is completed, and wecan solve the system by back substitution. However, we cancontinue with elementary operations
Multiply the 3rd equation by 1/3:x − y = 2
y − z = −1z = 2
R3 := (1/3) ∗ R3
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 444: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/444.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Gaussian Elimination. Gauss-Jordan Reduction
Add −2 times the 2nd equation to the 3rd equation:x − y = 2
y − z = −13z = 6
R3 := (−2) ∗ R2 + R3
Note: At this point, the elimination process is completed, and wecan solve the system by back substitution. However, we cancontinue with elementary operations
Multiply the 3rd equation by 1/3:x − y = 2
y − z = −1z = 2
R3 := (1/3) ∗ R3
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 445: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/445.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Gaussian Elimination. Gauss-Jordan Reduction
Add −2 times the 2nd equation to the 3rd equation:x − y = 2
y − z = −13z = 6
R3 := (−2) ∗ R2 + R3
Note:
At this point, the elimination process is completed, and wecan solve the system by back substitution. However, we cancontinue with elementary operations
Multiply the 3rd equation by 1/3:x − y = 2
y − z = −1z = 2
R3 := (1/3) ∗ R3
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 446: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/446.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Gaussian Elimination. Gauss-Jordan Reduction
Add −2 times the 2nd equation to the 3rd equation:x − y = 2
y − z = −13z = 6
R3 := (−2) ∗ R2 + R3
Note: At this point,
the elimination process is completed, and wecan solve the system by back substitution. However, we cancontinue with elementary operations
Multiply the 3rd equation by 1/3:x − y = 2
y − z = −1z = 2
R3 := (1/3) ∗ R3
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 447: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/447.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Gaussian Elimination. Gauss-Jordan Reduction
Add −2 times the 2nd equation to the 3rd equation:x − y = 2
y − z = −13z = 6
R3 := (−2) ∗ R2 + R3
Note: At this point, the elimination process
is completed, and wecan solve the system by back substitution. However, we cancontinue with elementary operations
Multiply the 3rd equation by 1/3:x − y = 2
y − z = −1z = 2
R3 := (1/3) ∗ R3
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 448: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/448.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Gaussian Elimination. Gauss-Jordan Reduction
Add −2 times the 2nd equation to the 3rd equation:x − y = 2
y − z = −13z = 6
R3 := (−2) ∗ R2 + R3
Note: At this point, the elimination process is completed, and
wecan solve the system by back substitution. However, we cancontinue with elementary operations
Multiply the 3rd equation by 1/3:x − y = 2
y − z = −1z = 2
R3 := (1/3) ∗ R3
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 449: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/449.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Gaussian Elimination. Gauss-Jordan Reduction
Add −2 times the 2nd equation to the 3rd equation:x − y = 2
y − z = −13z = 6
R3 := (−2) ∗ R2 + R3
Note: At this point, the elimination process is completed, and wecan solve
the system by back substitution. However, we cancontinue with elementary operations
Multiply the 3rd equation by 1/3:x − y = 2
y − z = −1z = 2
R3 := (1/3) ∗ R3
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 450: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/450.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Gaussian Elimination. Gauss-Jordan Reduction
Add −2 times the 2nd equation to the 3rd equation:x − y = 2
y − z = −13z = 6
R3 := (−2) ∗ R2 + R3
Note: At this point, the elimination process is completed, and wecan solve the system by
back substitution. However, we cancontinue with elementary operations
Multiply the 3rd equation by 1/3:x − y = 2
y − z = −1z = 2
R3 := (1/3) ∗ R3
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 451: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/451.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Gaussian Elimination. Gauss-Jordan Reduction
Add −2 times the 2nd equation to the 3rd equation:x − y = 2
y − z = −13z = 6
R3 := (−2) ∗ R2 + R3
Note: At this point, the elimination process is completed, and wecan solve the system by back substitution.
However, we cancontinue with elementary operations
Multiply the 3rd equation by 1/3:x − y = 2
y − z = −1z = 2
R3 := (1/3) ∗ R3
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 452: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/452.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Gaussian Elimination. Gauss-Jordan Reduction
Add −2 times the 2nd equation to the 3rd equation:x − y = 2
y − z = −13z = 6
R3 := (−2) ∗ R2 + R3
Note: At this point, the elimination process is completed, and wecan solve the system by back substitution. However,
we cancontinue with elementary operations
Multiply the 3rd equation by 1/3:x − y = 2
y − z = −1z = 2
R3 := (1/3) ∗ R3
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 453: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/453.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Gaussian Elimination. Gauss-Jordan Reduction
Add −2 times the 2nd equation to the 3rd equation:x − y = 2
y − z = −13z = 6
R3 := (−2) ∗ R2 + R3
Note: At this point, the elimination process is completed, and wecan solve the system by back substitution. However, we cancontinue
with elementary operations
Multiply the 3rd equation by 1/3:x − y = 2
y − z = −1z = 2
R3 := (1/3) ∗ R3
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 454: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/454.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Gaussian Elimination. Gauss-Jordan Reduction
Add −2 times the 2nd equation to the 3rd equation:x − y = 2
y − z = −13z = 6
R3 := (−2) ∗ R2 + R3
Note: At this point, the elimination process is completed, and wecan solve the system by back substitution. However, we cancontinue with elementary operations
Multiply the 3rd equation by 1/3:x − y = 2
y − z = −1z = 2
R3 := (1/3) ∗ R3
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 455: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/455.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Gaussian Elimination. Gauss-Jordan Reduction
Add −2 times the 2nd equation to the 3rd equation:x − y = 2
y − z = −13z = 6
R3 := (−2) ∗ R2 + R3
Note: At this point, the elimination process is completed, and wecan solve the system by back substitution. However, we cancontinue with elementary operations
Multiply
the 3rd equation by 1/3:x − y = 2
y − z = −1z = 2
R3 := (1/3) ∗ R3
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 456: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/456.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Gaussian Elimination. Gauss-Jordan Reduction
Add −2 times the 2nd equation to the 3rd equation:x − y = 2
y − z = −13z = 6
R3 := (−2) ∗ R2 + R3
Note: At this point, the elimination process is completed, and wecan solve the system by back substitution. However, we cancontinue with elementary operations
Multiply the 3rd equation by 1/3:
x − y = 2
y − z = −1z = 2
R3 := (1/3) ∗ R3
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 457: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/457.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Gaussian Elimination. Gauss-Jordan Reduction
Add −2 times the 2nd equation to the 3rd equation:x − y = 2
y − z = −13z = 6
R3 := (−2) ∗ R2 + R3
Note: At this point, the elimination process is completed, and wecan solve the system by back substitution. However, we cancontinue with elementary operations
Multiply the 3rd equation by 1/3:x − y = 2
y − z = −1z = 2
R3 := (1/3) ∗ R3
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 458: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/458.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Gaussian Elimination. Gauss-Jordan Reduction
Add the 3rd equation to the 2nd equation:x − y = 2
y = 1z = 2
R2 := R2 + R3
Add the 2nd equation to the 1st equation:x = 3
y = 1z = 2
R1 := R2 + R1
Thus, the system x − y = 2
2x − y − z = 3x + y + z = 6
has as the solution set, the point (3, 1, 2)
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 459: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/459.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Gaussian Elimination. Gauss-Jordan Reduction
Add
the 3rd equation to the 2nd equation:x − y = 2
y = 1z = 2
R2 := R2 + R3
Add the 2nd equation to the 1st equation:x = 3
y = 1z = 2
R1 := R2 + R1
Thus, the system x − y = 2
2x − y − z = 3x + y + z = 6
has as the solution set, the point (3, 1, 2)
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 460: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/460.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Gaussian Elimination. Gauss-Jordan Reduction
Add the 3rd equation to
the 2nd equation:x − y = 2
y = 1z = 2
R2 := R2 + R3
Add the 2nd equation to the 1st equation:x = 3
y = 1z = 2
R1 := R2 + R1
Thus, the system x − y = 2
2x − y − z = 3x + y + z = 6
has as the solution set, the point (3, 1, 2)
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 461: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/461.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Gaussian Elimination. Gauss-Jordan Reduction
Add the 3rd equation to the 2nd equation:
x − y = 2
y = 1z = 2
R2 := R2 + R3
Add the 2nd equation to the 1st equation:x = 3
y = 1z = 2
R1 := R2 + R1
Thus, the system x − y = 2
2x − y − z = 3x + y + z = 6
has as the solution set, the point (3, 1, 2)
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 462: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/462.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Gaussian Elimination. Gauss-Jordan Reduction
Add the 3rd equation to the 2nd equation:x − y = 2
y = 1z = 2
R2 := R2 + R3
Add the 2nd equation to the 1st equation:x = 3
y = 1z = 2
R1 := R2 + R1
Thus, the system x − y = 2
2x − y − z = 3x + y + z = 6
has as the solution set, the point (3, 1, 2)
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 463: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/463.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Gaussian Elimination. Gauss-Jordan Reduction
Add the 3rd equation to the 2nd equation:x − y = 2
y = 1z = 2
R2 := R2 + R3
Add
the 2nd equation to the 1st equation:x = 3
y = 1z = 2
R1 := R2 + R1
Thus, the system x − y = 2
2x − y − z = 3x + y + z = 6
has as the solution set, the point (3, 1, 2)
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 464: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/464.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Gaussian Elimination. Gauss-Jordan Reduction
Add the 3rd equation to the 2nd equation:x − y = 2
y = 1z = 2
R2 := R2 + R3
Add the 2nd equation to
the 1st equation:x = 3
y = 1z = 2
R1 := R2 + R1
Thus, the system x − y = 2
2x − y − z = 3x + y + z = 6
has as the solution set, the point (3, 1, 2)
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 465: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/465.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Gaussian Elimination. Gauss-Jordan Reduction
Add the 3rd equation to the 2nd equation:x − y = 2
y = 1z = 2
R2 := R2 + R3
Add the 2nd equation to the 1st equation:
x = 3
y = 1z = 2
R1 := R2 + R1
Thus, the system x − y = 2
2x − y − z = 3x + y + z = 6
has as the solution set, the point (3, 1, 2)
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 466: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/466.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Gaussian Elimination. Gauss-Jordan Reduction
Add the 3rd equation to the 2nd equation:x − y = 2
y = 1z = 2
R2 := R2 + R3
Add the 2nd equation to the 1st equation:x = 3
y = 1z = 2
R1 := R2 + R1
Thus, the system x − y = 2
2x − y − z = 3x + y + z = 6
has as the solution set, the point (3, 1, 2)
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 467: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/467.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Gaussian Elimination. Gauss-Jordan Reduction
Add the 3rd equation to the 2nd equation:x − y = 2
y = 1z = 2
R2 := R2 + R3
Add the 2nd equation to the 1st equation:x = 3
y = 1z = 2
R1 := R2 + R1
Thus,
the system x − y = 2
2x − y − z = 3x + y + z = 6
has as the solution set, the point (3, 1, 2)
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 468: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/468.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Gaussian Elimination. Gauss-Jordan Reduction
Add the 3rd equation to the 2nd equation:x − y = 2
y = 1z = 2
R2 := R2 + R3
Add the 2nd equation to the 1st equation:x = 3
y = 1z = 2
R1 := R2 + R1
Thus, the system
x − y = 2
2x − y − z = 3x + y + z = 6
has as the solution set, the point (3, 1, 2)
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 469: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/469.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Gaussian Elimination. Gauss-Jordan Reduction
Add the 3rd equation to the 2nd equation:x − y = 2
y = 1z = 2
R2 := R2 + R3
Add the 2nd equation to the 1st equation:x = 3
y = 1z = 2
R1 := R2 + R1
Thus, the system x − y = 2
2x − y − z = 3x + y + z = 6
has as the solution set, the point (3, 1, 2)
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 470: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/470.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Gaussian Elimination. Gauss-Jordan Reduction
Add the 3rd equation to the 2nd equation:x − y = 2
y = 1z = 2
R2 := R2 + R3
Add the 2nd equation to the 1st equation:x = 3
y = 1z = 2
R1 := R2 + R1
Thus, the system x − y = 2
2x − y − z = 3x + y + z = 6
has
as the solution set, the point (3, 1, 2)
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 471: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/471.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Gaussian Elimination. Gauss-Jordan Reduction
Add the 3rd equation to the 2nd equation:x − y = 2
y = 1z = 2
R2 := R2 + R3
Add the 2nd equation to the 1st equation:x = 3
y = 1z = 2
R1 := R2 + R1
Thus, the system x − y = 2
2x − y − z = 3x + y + z = 6
has as the solution set,
the point (3, 1, 2)
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 472: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/472.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Gaussian Elimination. Gauss-Jordan Reduction
Add the 3rd equation to the 2nd equation:x − y = 2
y = 1z = 2
R2 := R2 + R3
Add the 2nd equation to the 1st equation:x = 3
y = 1z = 2
R1 := R2 + R1
Thus, the system x − y = 2
2x − y − z = 3x + y + z = 6
has as the solution set, the point (3, 1, 2)Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 473: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/473.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Row echelon form. Gauss-Jordan Reduction
Matrices
Definition. A matrix is a rectangular array of numbers.
The dimension of a matrix is given by
dimensions = (number of rows) X ( number of columns)
Thus we have
n × n : Square matrixn × 1 : Column vector1× n : Row vector
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 474: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/474.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Row echelon form. Gauss-Jordan Reduction
Matrices
Definition. A matrix is a rectangular array of numbers.
The dimension of a matrix is given by
dimensions = (number of rows) X ( number of columns)
Thus we have
n × n : Square matrixn × 1 : Column vector1× n : Row vector
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 475: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/475.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Row echelon form. Gauss-Jordan Reduction
Matrices
Definition. A matrix is a rectangular array of numbers.
The dimension of a matrix is given by
dimensions = (number of rows) X ( number of columns)
Thus we have
n × n : Square matrixn × 1 : Column vector1× n : Row vector
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 476: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/476.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Row echelon form. Gauss-Jordan Reduction
Matrices
Definition. A matrix is a rectangular array of numbers.
The dimension of a matrix is given by
dimensions = (number of rows) X ( number of columns)
Thus we have
n × n : Square matrixn × 1 : Column vector1× n : Row vector
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 477: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/477.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Row echelon form. Gauss-Jordan Reduction
Matrices
Definition. A matrix is a rectangular array of numbers.
The dimension of a matrix is given by
dimensions = (number of rows) X ( number of columns)
Thus we have
n × n : Square matrixn × 1 : Column vector1× n : Row vector
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 478: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/478.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Row echelon form. Gauss-Jordan Reduction
Matrices
Definition. A matrix is a rectangular array of numbers.
The dimension of a matrix is given by
dimensions = (number of rows) X ( number of columns)
Thus we have
n × n : Square matrixn × 1 : Column vector1× n : Row vector
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 479: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/479.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Row echelon form. Gauss-Jordan Reduction
Matrices
Definition. A matrix is a rectangular array of numbers.
The dimension of a matrix is given by
dimensions = (number of rows) X ( number of columns)
Thus we have
n × n : Square matrix
n × 1 : Column vector1× n : Row vector
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 480: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/480.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Row echelon form. Gauss-Jordan Reduction
Matrices
Definition. A matrix is a rectangular array of numbers.
The dimension of a matrix is given by
dimensions = (number of rows) X ( number of columns)
Thus we have
n × n : Square matrixn × 1 : Column vector
1× n : Row vector
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 481: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/481.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Row echelon form. Gauss-Jordan Reduction
Matrices
Definition. A matrix is a rectangular array of numbers.
The dimension of a matrix is given by
dimensions = (number of rows) X ( number of columns)
Thus we have
n × n : Square matrixn × 1 : Column vector1× n : Row vector
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 482: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/482.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Row echelon form. Gauss-Jordan Reduction
Matrices
Definition. A matrix is a rectangular array of numbers.
The dimension of a matrix is given by
dimensions = (number of rows) X ( number of columns)
Thus we have
n × n : Square matrixn × 1 : Column vector1× n : Row vector
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 483: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/483.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Row echelon form. Gauss-Jordan Reduction
Examples of matrices are:
2 7−1 03 3
(3× 2)
(2 7 0−1 1 5
)(2× 3)
358
(3× 1)
(2 4 9
)(1× 3)
(−2 01 5
)(2× 2)
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 484: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/484.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Row echelon form. Gauss-Jordan Reduction
Examples of matrices are:
2 7−1 03 3
(3× 2)
(2 7 0−1 1 5
)(2× 3)
358
(3× 1)
(2 4 9
)(1× 3)
(−2 01 5
)(2× 2)
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 485: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/485.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Row echelon form. Gauss-Jordan Reduction
Examples of matrices are:
2 7−1 03 3
(3× 2)
(2 7 0−1 1 5
)(2× 3)
358
(3× 1)
(2 4 9
)(1× 3)
(−2 01 5
)(2× 2)
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 486: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/486.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Row echelon form. Gauss-Jordan Reduction
Examples of matrices are:
2 7−1 03 3
(3× 2)
(2 7 0−1 1 5
)(2× 3)
358
(3× 1)
(2 4 9
)(1× 3)
(−2 01 5
)(2× 2)
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 487: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/487.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Row echelon form. Gauss-Jordan Reduction
Examples of matrices are:
2 7−1 03 3
(3× 2)
(2 7 0−1 1 5
)
(2× 3)
358
(3× 1)
(2 4 9
)(1× 3)
(−2 01 5
)(2× 2)
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 488: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/488.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Row echelon form. Gauss-Jordan Reduction
Examples of matrices are:
2 7−1 03 3
(3× 2)
(2 7 0−1 1 5
)(2× 3)
358
(3× 1)
(2 4 9
)(1× 3)
(−2 01 5
)(2× 2)
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 489: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/489.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Row echelon form. Gauss-Jordan Reduction
Examples of matrices are:
2 7−1 03 3
(3× 2)
(2 7 0−1 1 5
)(2× 3)
358
(3× 1)
(2 4 9
)(1× 3)
(−2 01 5
)(2× 2)
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 490: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/490.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Row echelon form. Gauss-Jordan Reduction
Examples of matrices are:
2 7−1 03 3
(3× 2)
(2 7 0−1 1 5
)(2× 3)
358
(3× 1)
(2 4 9
)(1× 3)
(−2 01 5
)(2× 2)
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 491: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/491.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Row echelon form. Gauss-Jordan Reduction
Examples of matrices are:
2 7−1 03 3
(3× 2)
(2 7 0−1 1 5
)(2× 3)
358
(3× 1)
(2 4 9
)
(1× 3)
(−2 01 5
)(2× 2)
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 492: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/492.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Row echelon form. Gauss-Jordan Reduction
Examples of matrices are:
2 7−1 03 3
(3× 2)
(2 7 0−1 1 5
)(2× 3)
358
(3× 1)
(2 4 9
)(1× 3)
(−2 01 5
)(2× 2)
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 493: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/493.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Row echelon form. Gauss-Jordan Reduction
Examples of matrices are:
2 7−1 03 3
(3× 2)
(2 7 0−1 1 5
)(2× 3)
358
(3× 1)
(2 4 9
)(1× 3)
(−2 01 5
)
(2× 2)
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 494: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/494.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Row echelon form. Gauss-Jordan Reduction
Examples of matrices are:
2 7−1 03 3
(3× 2)
(2 7 0−1 1 5
)(2× 3)
358
(3× 1)
(2 4 9
)(1× 3)
(−2 01 5
)(2× 2)
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 495: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/495.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Row echelon form. Gauss-Jordan Reduction
From a system of linear equations:a11x1 + a12x2 + · · ·+ a1nxn = b1
a21x1 + a22x2 + · · ·+ a2nxn = b2...
am1x1 + am2x2 + · · ·+ amnxn = bm
We can find the coefficient matrix and column vector of theright-hand sides:
a11 a12 · · · a1na21 a22 · · · a2n
...am1 am2 · · · amn
b1
b2...
bm
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 496: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/496.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Row echelon form. Gauss-Jordan Reduction
From a system of linear equations:
a11x1 + a12x2 + · · ·+ a1nxn = b1
a21x1 + a22x2 + · · ·+ a2nxn = b2...
am1x1 + am2x2 + · · ·+ amnxn = bm
We can find the coefficient matrix and column vector of theright-hand sides:
a11 a12 · · · a1na21 a22 · · · a2n
...am1 am2 · · · amn
b1
b2...
bm
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 497: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/497.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Row echelon form. Gauss-Jordan Reduction
From a system of linear equations:a11x1 + a12x2 + · · ·+ a1nxn = b1
a21x1 + a22x2 + · · ·+ a2nxn = b2...
am1x1 + am2x2 + · · ·+ amnxn = bm
We can find the coefficient matrix and column vector of theright-hand sides:
a11 a12 · · · a1na21 a22 · · · a2n
...am1 am2 · · · amn
b1
b2...
bm
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 498: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/498.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Row echelon form. Gauss-Jordan Reduction
From a system of linear equations:a11x1 + a12x2 + · · ·+ a1nxn = b1
a21x1 + a22x2 + · · ·+ a2nxn = b2...
am1x1 + am2x2 + · · ·+ amnxn = bm
We can find
the coefficient matrix and column vector of theright-hand sides:
a11 a12 · · · a1na21 a22 · · · a2n
...am1 am2 · · · amn
b1
b2...
bm
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 499: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/499.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Row echelon form. Gauss-Jordan Reduction
From a system of linear equations:a11x1 + a12x2 + · · ·+ a1nxn = b1
a21x1 + a22x2 + · · ·+ a2nxn = b2...
am1x1 + am2x2 + · · ·+ amnxn = bm
We can find the coefficient matrix and
column vector of theright-hand sides:
a11 a12 · · · a1na21 a22 · · · a2n
...am1 am2 · · · amn
b1
b2...
bm
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 500: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/500.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Row echelon form. Gauss-Jordan Reduction
From a system of linear equations:a11x1 + a12x2 + · · ·+ a1nxn = b1
a21x1 + a22x2 + · · ·+ a2nxn = b2...
am1x1 + am2x2 + · · ·+ amnxn = bm
We can find the coefficient matrix and column vector
of theright-hand sides:
a11 a12 · · · a1na21 a22 · · · a2n
...am1 am2 · · · amn
b1
b2...
bm
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 501: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/501.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Row echelon form. Gauss-Jordan Reduction
From a system of linear equations:a11x1 + a12x2 + · · ·+ a1nxn = b1
a21x1 + a22x2 + · · ·+ a2nxn = b2...
am1x1 + am2x2 + · · ·+ amnxn = bm
We can find the coefficient matrix and column vector of theright-hand sides:
a11 a12 · · · a1na21 a22 · · · a2n
...am1 am2 · · · amn
b1
b2...
bm
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 502: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/502.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Row echelon form. Gauss-Jordan Reduction
From a system of linear equations:a11x1 + a12x2 + · · ·+ a1nxn = b1
a21x1 + a22x2 + · · ·+ a2nxn = b2...
am1x1 + am2x2 + · · ·+ amnxn = bm
We can find the coefficient matrix and column vector of theright-hand sides:
a11 a12 · · · a1na21 a22 · · · a2n
...am1 am2 · · · amn
b1
b2...
bm
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 503: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/503.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Row echelon form. Gauss-Jordan Reduction
From a system of linear equations:a11x1 + a12x2 + · · ·+ a1nxn = b1
a21x1 + a22x2 + · · ·+ a2nxn = b2...
am1x1 + am2x2 + · · ·+ amnxn = bm
We can find the coefficient matrix and column vector of theright-hand sides:
a11 a12 · · · a1na21 a22 · · · a2n
...am1 am2 · · · amn
b1
b2...
bm
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 504: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/504.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Row echelon form. Gauss-Jordan Reduction
and also associated to the linear system we have the Augmentedmatrix:
a11 a12 · · · a1n b1
a21 a22 · · · a2n b2...
am1 am2 · · · amn bm
Now,remember that using Gaussian elimination, the solution of asystem of linear equations splits into two parts:(A) Elimination and(B) Back substitution.
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 505: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/505.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Row echelon form. Gauss-Jordan Reduction
and also
associated to the linear system we have the Augmentedmatrix:
a11 a12 · · · a1n b1
a21 a22 · · · a2n b2...
am1 am2 · · · amn bm
Now,remember that using Gaussian elimination, the solution of asystem of linear equations splits into two parts:(A) Elimination and(B) Back substitution.
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 506: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/506.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Row echelon form. Gauss-Jordan Reduction
and also associated to the linear system
we have the Augmentedmatrix:
a11 a12 · · · a1n b1
a21 a22 · · · a2n b2...
am1 am2 · · · amn bm
Now,remember that using Gaussian elimination, the solution of asystem of linear equations splits into two parts:(A) Elimination and(B) Back substitution.
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 507: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/507.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Row echelon form. Gauss-Jordan Reduction
and also associated to the linear system we have the Augmentedmatrix:
a11 a12 · · · a1n b1
a21 a22 · · · a2n b2...
am1 am2 · · · amn bm
Now,remember that using Gaussian elimination, the solution of asystem of linear equations splits into two parts:(A) Elimination and(B) Back substitution.
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 508: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/508.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Row echelon form. Gauss-Jordan Reduction
and also associated to the linear system we have the Augmentedmatrix:
a11 a12 · · · a1n b1
a21 a22 · · · a2n b2...
am1 am2 · · · amn bm
Now,remember that using Gaussian elimination, the solution of asystem of linear equations splits into two parts:(A) Elimination and(B) Back substitution.
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 509: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/509.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Row echelon form. Gauss-Jordan Reduction
and also associated to the linear system we have the Augmentedmatrix:
a11 a12 · · · a1n b1
a21 a22 · · · a2n b2...
am1 am2 · · · amn bm
Now,
remember that using Gaussian elimination, the solution of asystem of linear equations splits into two parts:(A) Elimination and(B) Back substitution.
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 510: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/510.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Row echelon form. Gauss-Jordan Reduction
and also associated to the linear system we have the Augmentedmatrix:
a11 a12 · · · a1n b1
a21 a22 · · · a2n b2...
am1 am2 · · · amn bm
Now,remember that using Gaussian elimination,
the solution of asystem of linear equations splits into two parts:(A) Elimination and(B) Back substitution.
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 511: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/511.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Row echelon form. Gauss-Jordan Reduction
and also associated to the linear system we have the Augmentedmatrix:
a11 a12 · · · a1n b1
a21 a22 · · · a2n b2...
am1 am2 · · · amn bm
Now,remember that using Gaussian elimination, the solution of asystem of linear equations
splits into two parts:(A) Elimination and(B) Back substitution.
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 512: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/512.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Row echelon form. Gauss-Jordan Reduction
and also associated to the linear system we have the Augmentedmatrix:
a11 a12 · · · a1n b1
a21 a22 · · · a2n b2...
am1 am2 · · · amn bm
Now,remember that using Gaussian elimination, the solution of asystem of linear equations splits into two parts:
(A) Elimination and(B) Back substitution.
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 513: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/513.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Row echelon form. Gauss-Jordan Reduction
and also associated to the linear system we have the Augmentedmatrix:
a11 a12 · · · a1n b1
a21 a22 · · · a2n b2...
am1 am2 · · · amn bm
Now,remember that using Gaussian elimination, the solution of asystem of linear equations splits into two parts:(A) Elimination and
(B) Back substitution.
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 514: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/514.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Row echelon form. Gauss-Jordan Reduction
and also associated to the linear system we have the Augmentedmatrix:
a11 a12 · · · a1n b1
a21 a22 · · · a2n b2...
am1 am2 · · · amn bm
Now,remember that using Gaussian elimination, the solution of asystem of linear equations splits into two parts:(A) Elimination and(B) Back substitution.
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 515: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/515.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Row echelon form. Gauss-Jordan Reduction
and also associated to the linear system we have the Augmentedmatrix:
a11 a12 · · · a1n b1
a21 a22 · · · a2n b2...
am1 am2 · · · amn bm
Now,remember that using Gaussian elimination, the solution of asystem of linear equations splits into two parts:(A) Elimination and(B) Back substitution.
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 516: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/516.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Row echelon form. Gauss-Jordan Reduction
Both parts can be done by applying a finite number of elementaryoperations:
(1) to multiply a row by a nonzero scalar;
(2) to add the ith row multiplied by some r ∈ R to the jth row;
(3) to interchange two rows.
Notation
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 517: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/517.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Row echelon form. Gauss-Jordan Reduction
Both parts
can be done by applying a finite number of elementaryoperations:
(1) to multiply a row by a nonzero scalar;
(2) to add the ith row multiplied by some r ∈ R to the jth row;
(3) to interchange two rows.
Notation
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 518: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/518.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Row echelon form. Gauss-Jordan Reduction
Both parts can be done
by applying a finite number of elementaryoperations:
(1) to multiply a row by a nonzero scalar;
(2) to add the ith row multiplied by some r ∈ R to the jth row;
(3) to interchange two rows.
Notation
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 519: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/519.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Row echelon form. Gauss-Jordan Reduction
Both parts can be done by applying a finite number
of elementaryoperations:
(1) to multiply a row by a nonzero scalar;
(2) to add the ith row multiplied by some r ∈ R to the jth row;
(3) to interchange two rows.
Notation
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 520: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/520.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Row echelon form. Gauss-Jordan Reduction
Both parts can be done by applying a finite number of elementaryoperations:
(1) to multiply a row by a nonzero scalar;
(2) to add the ith row multiplied by some r ∈ R to the jth row;
(3) to interchange two rows.
Notation
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 521: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/521.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Row echelon form. Gauss-Jordan Reduction
Both parts can be done by applying a finite number of elementaryoperations:
(1) to multiply a row by a nonzero scalar;
(2) to add the ith row multiplied by some r ∈ R to the jth row;
(3) to interchange two rows.
Notation
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 522: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/522.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Row echelon form. Gauss-Jordan Reduction
Both parts can be done by applying a finite number of elementaryoperations:
(1) to multiply a row by a nonzero scalar;
(2) to add the ith row multiplied by some r ∈ R to the jth row;
(3) to interchange two rows.
Notation
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 523: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/523.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Row echelon form. Gauss-Jordan Reduction
Both parts can be done by applying a finite number of elementaryoperations:
(1) to multiply a row by a nonzero scalar;
(2) to add the ith row multiplied by some r ∈ R to the jth row;
(3) to interchange two rows.
Notation
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 524: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/524.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Row echelon form. Gauss-Jordan Reduction
Both parts can be done by applying a finite number of elementaryoperations:
(1) to multiply a row by a nonzero scalar;
(2) to add the ith row multiplied by some r ∈ R to the jth row;
(3) to interchange two rows.
Notation
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 525: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/525.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Row echelon form. Gauss-Jordan Reduction
Augmented matrix:
a11 a12 · · · a1n b1
a21 a22 · · · a2n b2...
am1 am2 · · · amn bm
=
v1v2...
vm
where vi = (ai1 ai1 ai1 · · · ai1) is a row vector.
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 526: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/526.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Row echelon form. Gauss-Jordan Reduction
Augmented matrix:
a11 a12 · · · a1n b1
a21 a22 · · · a2n b2...
am1 am2 · · · amn bm
=
v1v2...
vm
where vi = (ai1 ai1 ai1 · · · ai1) is a row vector.
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 527: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/527.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Row echelon form. Gauss-Jordan Reduction
Augmented matrix:
a11 a12 · · · a1n b1
a21 a22 · · · a2n b2...
am1 am2 · · · amn bm
=
v1v2...
vm
where vi = (ai1 ai1 ai1 · · · ai1) is a row vector.
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 528: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/528.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Row echelon form. Gauss-Jordan Reduction
Augmented matrix:
a11 a12 · · · a1n b1
a21 a22 · · · a2n b2...
am1 am2 · · · amn bm
=
v1v2...
vm
where
vi = (ai1 ai1 ai1 · · · ai1) is a row vector.
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 529: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/529.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Row echelon form. Gauss-Jordan Reduction
Augmented matrix:
a11 a12 · · · a1n b1
a21 a22 · · · a2n b2...
am1 am2 · · · amn bm
=
v1v2...
vm
where vi = (ai1 ai1 ai1 · · · ai1)
is a row vector.
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 530: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/530.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Row echelon form. Gauss-Jordan Reduction
Augmented matrix:
a11 a12 · · · a1n b1
a21 a22 · · · a2n b2...
am1 am2 · · · amn bm
=
v1v2...
vm
where vi = (ai1 ai1 ai1 · · · ai1) is a row vector.
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 531: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/531.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Row echelon form. Gauss-Jordan Reduction
Elementary row operations on Matrices
Operation 1: To multiply the ith row by r 6= 0
v1...
vi...
vm
⇒
v1...
rvi...
vm
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 532: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/532.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Row echelon form. Gauss-Jordan Reduction
Elementary row operations on Matrices
Operation 1: To multiply the ith row by r 6= 0
v1...
vi...
vm
⇒
v1...
rvi...
vm
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 533: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/533.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Row echelon form. Gauss-Jordan Reduction
Elementary row operations on Matrices
Operation 1:
To multiply the ith row by r 6= 0
v1...
vi...
vm
⇒
v1...
rvi...
vm
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 534: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/534.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Row echelon form. Gauss-Jordan Reduction
Elementary row operations on Matrices
Operation 1: To multiply the ith row by r 6= 0
v1...
vi...
vm
⇒
v1...
rvi...
vm
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 535: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/535.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Row echelon form. Gauss-Jordan Reduction
Elementary row operations on Matrices
Operation 1: To multiply the ith row by r 6= 0
v1...
vi...
vm
⇒
v1...
rvi...
vm
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 536: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/536.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Row echelon form. Gauss-Jordan Reduction
Elementary row operations on Matrices
Operation 1: To multiply the ith row by r 6= 0
v1...
vi...
vm
⇒
v1...
rvi...
vm
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 537: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/537.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Row echelon form. Gauss-Jordan Reduction
Operation 2: To add the ith row multiplied by r to the jth row
v1...
vi...
vj...
vm
⇒
v1...
vi...
vj + rvi...
vm
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 538: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/538.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Row echelon form. Gauss-Jordan Reduction
Operation 2:
To add the ith row multiplied by r to the jth row
v1...
vi...
vj...
vm
⇒
v1...
vi...
vj + rvi...
vm
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 539: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/539.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Row echelon form. Gauss-Jordan Reduction
Operation 2: To add the ith row multiplied by r to the jth row
v1...
vi...
vj...
vm
⇒
v1...
vi...
vj + rvi...
vm
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 540: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/540.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Row echelon form. Gauss-Jordan Reduction
Operation 2: To add the ith row multiplied by r to the jth row
v1...
vi...
vj...
vm
⇒
v1...
vi...
vj + rvi...
vm
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 541: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/541.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Row echelon form. Gauss-Jordan Reduction
Operation 2: To add the ith row multiplied by r to the jth row
v1...
vi...
vj...
vm
⇒
v1...
vi...
vj + rvi...
vm
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 542: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/542.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Row echelon form. Gauss-Jordan Reduction
Operation 3: To interchange the ith row with the jth row
v1...
vi...
vj...
vm
⇒
v1...
vj...
vi...
vm
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 543: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/543.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Row echelon form. Gauss-Jordan Reduction
Operation 3:
To interchange the ith row with the jth row
v1...
vi...
vj...
vm
⇒
v1...
vj...
vi...
vm
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 544: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/544.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Row echelon form. Gauss-Jordan Reduction
Operation 3: To interchange the ith row with the jth row
v1...
vi...
vj...
vm
⇒
v1...
vj...
vi...
vm
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 545: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/545.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Row echelon form. Gauss-Jordan Reduction
Operation 3: To interchange the ith row with the jth row
v1...
vi...
vj...
vm
⇒
v1...
vj...
vi...
vm
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 546: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/546.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Row echelon form. Gauss-Jordan Reduction
Operation 3: To interchange the ith row with the jth row
v1...
vi...
vj...
vm
⇒
v1...
vj...
vi...
vm
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 547: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/547.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Row echelon form. Gauss-Jordan Reduction
Row echelon form
Definition. Leading entry of a matrix is the first nonzero entry ina row.
The goal of the Gaussian elimination is to convert the augmentedmatrix into row echelon form
1) Leading entries shift to the right as we go from the first row tothe last one;
2) Each leading entry is equal to 1.
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 548: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/548.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Row echelon form. Gauss-Jordan Reduction
Row echelon form
Definition. Leading entry of a matrix is the first nonzero entry ina row.
The goal of the Gaussian elimination is to convert the augmentedmatrix into row echelon form
1) Leading entries shift to the right as we go from the first row tothe last one;
2) Each leading entry is equal to 1.
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 549: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/549.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Row echelon form. Gauss-Jordan Reduction
Row echelon form
Definition. Leading entry
of a matrix is the first nonzero entry ina row.
The goal of the Gaussian elimination is to convert the augmentedmatrix into row echelon form
1) Leading entries shift to the right as we go from the first row tothe last one;
2) Each leading entry is equal to 1.
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 550: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/550.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Row echelon form. Gauss-Jordan Reduction
Row echelon form
Definition. Leading entry of a matrix is
the first nonzero entry ina row.
The goal of the Gaussian elimination is to convert the augmentedmatrix into row echelon form
1) Leading entries shift to the right as we go from the first row tothe last one;
2) Each leading entry is equal to 1.
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 551: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/551.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Row echelon form. Gauss-Jordan Reduction
Row echelon form
Definition. Leading entry of a matrix is the first nonzero entry ina row.
The goal of the Gaussian elimination is to convert the augmentedmatrix into row echelon form
1) Leading entries shift to the right as we go from the first row tothe last one;
2) Each leading entry is equal to 1.
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 552: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/552.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Row echelon form. Gauss-Jordan Reduction
Row echelon form
Definition. Leading entry of a matrix is the first nonzero entry ina row.
The goal of
the Gaussian elimination is to convert the augmentedmatrix into row echelon form
1) Leading entries shift to the right as we go from the first row tothe last one;
2) Each leading entry is equal to 1.
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 553: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/553.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Row echelon form. Gauss-Jordan Reduction
Row echelon form
Definition. Leading entry of a matrix is the first nonzero entry ina row.
The goal of the Gaussian elimination
is to convert the augmentedmatrix into row echelon form
1) Leading entries shift to the right as we go from the first row tothe last one;
2) Each leading entry is equal to 1.
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 554: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/554.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Row echelon form. Gauss-Jordan Reduction
Row echelon form
Definition. Leading entry of a matrix is the first nonzero entry ina row.
The goal of the Gaussian elimination is to convert
the augmentedmatrix into row echelon form
1) Leading entries shift to the right as we go from the first row tothe last one;
2) Each leading entry is equal to 1.
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 555: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/555.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Row echelon form. Gauss-Jordan Reduction
Row echelon form
Definition. Leading entry of a matrix is the first nonzero entry ina row.
The goal of the Gaussian elimination is to convert the augmentedmatrix
into row echelon form
1) Leading entries shift to the right as we go from the first row tothe last one;
2) Each leading entry is equal to 1.
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 556: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/556.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Row echelon form. Gauss-Jordan Reduction
Row echelon form
Definition. Leading entry of a matrix is the first nonzero entry ina row.
The goal of the Gaussian elimination is to convert the augmentedmatrix into row echelon form
1) Leading entries shift to the right as we go from the first row tothe last one;
2) Each leading entry is equal to 1.
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 557: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/557.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Row echelon form. Gauss-Jordan Reduction
Row echelon form
Definition. Leading entry of a matrix is the first nonzero entry ina row.
The goal of the Gaussian elimination is to convert the augmentedmatrix into row echelon form
1) Leading entries
shift to the right as we go from the first row tothe last one;
2) Each leading entry is equal to 1.
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 558: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/558.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Row echelon form. Gauss-Jordan Reduction
Row echelon form
Definition. Leading entry of a matrix is the first nonzero entry ina row.
The goal of the Gaussian elimination is to convert the augmentedmatrix into row echelon form
1) Leading entries shift to the right
as we go from the first row tothe last one;
2) Each leading entry is equal to 1.
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 559: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/559.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Row echelon form. Gauss-Jordan Reduction
Row echelon form
Definition. Leading entry of a matrix is the first nonzero entry ina row.
The goal of the Gaussian elimination is to convert the augmentedmatrix into row echelon form
1) Leading entries shift to the right as we go from
the first row tothe last one;
2) Each leading entry is equal to 1.
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 560: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/560.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Row echelon form. Gauss-Jordan Reduction
Row echelon form
Definition. Leading entry of a matrix is the first nonzero entry ina row.
The goal of the Gaussian elimination is to convert the augmentedmatrix into row echelon form
1) Leading entries shift to the right as we go from the first row to
the last one;
2) Each leading entry is equal to 1.
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 561: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/561.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Row echelon form. Gauss-Jordan Reduction
Row echelon form
Definition. Leading entry of a matrix is the first nonzero entry ina row.
The goal of the Gaussian elimination is to convert the augmentedmatrix into row echelon form
1) Leading entries shift to the right as we go from the first row tothe last one;
2) Each leading entry is equal to 1.
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 562: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/562.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Row echelon form. Gauss-Jordan Reduction
Row echelon form
Definition. Leading entry of a matrix is the first nonzero entry ina row.
The goal of the Gaussian elimination is to convert the augmentedmatrix into row echelon form
1) Leading entries shift to the right as we go from the first row tothe last one;
2) Each
leading entry is equal to 1.
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 563: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/563.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Row echelon form. Gauss-Jordan Reduction
Row echelon form
Definition. Leading entry of a matrix is the first nonzero entry ina row.
The goal of the Gaussian elimination is to convert the augmentedmatrix into row echelon form
1) Leading entries shift to the right as we go from the first row tothe last one;
2) Each leading entry
is equal to 1.
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 564: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/564.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Row echelon form. Gauss-Jordan Reduction
Row echelon form
Definition. Leading entry of a matrix is the first nonzero entry ina row.
The goal of the Gaussian elimination is to convert the augmentedmatrix into row echelon form
1) Leading entries shift to the right as we go from the first row tothe last one;
2) Each leading entry is equal to
1.
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 565: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/565.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Row echelon form. Gauss-Jordan Reduction
Row echelon form
Definition. Leading entry of a matrix is the first nonzero entry ina row.
The goal of the Gaussian elimination is to convert the augmentedmatrix into row echelon form
1) Leading entries shift to the right as we go from the first row tothe last one;
2) Each leading entry is equal to 1.
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 566: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/566.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Row echelon form. Gauss-Jordan Reduction
Thus, we have this example in the echelon form
1 −1 4 1 7 9 6 1 3 9 −50 1 3 −2 7 1 5 3 3 4 −10 0 0 0 1 5 6 −3 1 2 10 0 0 0 0 0 1 8 1 7 20 0 0 0 0 0 0 1 −2 7 −30 0 0 0 0 0 0 0 1 −2 90 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 567: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/567.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Row echelon form. Gauss-Jordan Reduction
Thus,
we have this example in the echelon form
1 −1 4 1 7 9 6 1 3 9 −50 1 3 −2 7 1 5 3 3 4 −10 0 0 0 1 5 6 −3 1 2 10 0 0 0 0 0 1 8 1 7 20 0 0 0 0 0 0 1 −2 7 −30 0 0 0 0 0 0 0 1 −2 90 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 568: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/568.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Row echelon form. Gauss-Jordan Reduction
Thus, we have this example
in the echelon form
1 −1 4 1 7 9 6 1 3 9 −50 1 3 −2 7 1 5 3 3 4 −10 0 0 0 1 5 6 −3 1 2 10 0 0 0 0 0 1 8 1 7 20 0 0 0 0 0 0 1 −2 7 −30 0 0 0 0 0 0 0 1 −2 90 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 569: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/569.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Row echelon form. Gauss-Jordan Reduction
Thus, we have this example in the echelon form
1 −1 4 1 7 9 6 1 3 9 −50 1 3 −2 7 1 5 3 3 4 −10 0 0 0 1 5 6 −3 1 2 10 0 0 0 0 0 1 8 1 7 20 0 0 0 0 0 0 1 −2 7 −30 0 0 0 0 0 0 0 1 −2 90 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 570: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/570.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Row echelon form. Gauss-Jordan Reduction
Thus, we have this example in the echelon form
1 −1 4 1 7 9 6 1 3 9 −50 1 3 −2 7 1 5 3 3 4 −10 0 0 0 1 5 6 −3 1 2 10 0 0 0 0 0 1 8 1 7 20 0 0 0 0 0 0 1 −2 7 −30 0 0 0 0 0 0 0 1 −2 90 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 571: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/571.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Row echelon form. Gauss-Jordan Reduction
Row echelon form
General augmented matrix in row echelon form
� ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗� � � ∗ ∗ ∗ ∗ ∗ ∗ ∗
� � ∗ ∗ ∗ ∗ ∗� ∗ ∗ ∗ ∗
� � ∗ ∗� ∗
�
1) Leading entries are boxed (all equal to 1);
2) All the entries below, the (imaginary) staircase line are zero;
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 572: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/572.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Row echelon form. Gauss-Jordan Reduction
Row echelon form
General augmented matrix in row echelon form
� ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗� � � ∗ ∗ ∗ ∗ ∗ ∗ ∗
� � ∗ ∗ ∗ ∗ ∗� ∗ ∗ ∗ ∗
� � ∗ ∗� ∗
�
1) Leading entries are boxed (all equal to 1);
2) All the entries below, the (imaginary) staircase line are zero;
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 573: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/573.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Row echelon form. Gauss-Jordan Reduction
Row echelon form
General augmented matrix in row echelon form
� ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗� � � ∗ ∗ ∗ ∗ ∗ ∗ ∗
� � ∗ ∗ ∗ ∗ ∗� ∗ ∗ ∗ ∗
� � ∗ ∗� ∗
�
1) Leading entries are boxed (all equal to 1);
2) All the entries below, the (imaginary) staircase line are zero;
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 574: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/574.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Row echelon form. Gauss-Jordan Reduction
Row echelon form
General augmented matrix in row echelon form
� ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗� � � ∗ ∗ ∗ ∗ ∗ ∗ ∗
� � ∗ ∗ ∗ ∗ ∗� ∗ ∗ ∗ ∗
� � ∗ ∗� ∗
�
1) Leading entries are boxed (all equal to 1);
2) All the entries below, the (imaginary) staircase line are zero;
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 575: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/575.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Row echelon form. Gauss-Jordan Reduction
Row echelon form
General augmented matrix in row echelon form
� ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗� � � ∗ ∗ ∗ ∗ ∗ ∗ ∗
� � ∗ ∗ ∗ ∗ ∗� ∗ ∗ ∗ ∗
� � ∗ ∗� ∗
�
1) Leading entries
are boxed (all equal to 1);
2) All the entries below, the (imaginary) staircase line are zero;
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 576: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/576.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Row echelon form. Gauss-Jordan Reduction
Row echelon form
General augmented matrix in row echelon form
� ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗� � � ∗ ∗ ∗ ∗ ∗ ∗ ∗
� � ∗ ∗ ∗ ∗ ∗� ∗ ∗ ∗ ∗
� � ∗ ∗� ∗
�
1) Leading entries are boxed
(all equal to 1);
2) All the entries below, the (imaginary) staircase line are zero;
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 577: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/577.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Row echelon form. Gauss-Jordan Reduction
Row echelon form
General augmented matrix in row echelon form
� ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗� � � ∗ ∗ ∗ ∗ ∗ ∗ ∗
� � ∗ ∗ ∗ ∗ ∗� ∗ ∗ ∗ ∗
� � ∗ ∗� ∗
�
1) Leading entries are boxed (all equal to 1);
2) All the entries below, the (imaginary) staircase line are zero;
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 578: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/578.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Row echelon form. Gauss-Jordan Reduction
Row echelon form
General augmented matrix in row echelon form
� ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗� � � ∗ ∗ ∗ ∗ ∗ ∗ ∗
� � ∗ ∗ ∗ ∗ ∗� ∗ ∗ ∗ ∗
� � ∗ ∗� ∗
�
1) Leading entries are boxed (all equal to 1);
2) All
the entries below, the (imaginary) staircase line are zero;
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 579: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/579.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Row echelon form. Gauss-Jordan Reduction
Row echelon form
General augmented matrix in row echelon form
� ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗� � � ∗ ∗ ∗ ∗ ∗ ∗ ∗
� � ∗ ∗ ∗ ∗ ∗� ∗ ∗ ∗ ∗
� � ∗ ∗� ∗
�
1) Leading entries are boxed (all equal to 1);
2) All the entries below,
the (imaginary) staircase line are zero;
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 580: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/580.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Row echelon form. Gauss-Jordan Reduction
Row echelon form
General augmented matrix in row echelon form
� ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗� � � ∗ ∗ ∗ ∗ ∗ ∗ ∗
� � ∗ ∗ ∗ ∗ ∗� ∗ ∗ ∗ ∗
� � ∗ ∗� ∗
�
1) Leading entries are boxed (all equal to 1);
2) All the entries below, the (imaginary)
staircase line are zero;
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 581: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/581.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Row echelon form. Gauss-Jordan Reduction
Row echelon form
General augmented matrix in row echelon form
� ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗� � � ∗ ∗ ∗ ∗ ∗ ∗ ∗
� � ∗ ∗ ∗ ∗ ∗� ∗ ∗ ∗ ∗
� � ∗ ∗� ∗
�
1) Leading entries are boxed (all equal to 1);
2) All the entries below, the (imaginary) staircase line
are zero;
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 582: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/582.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Row echelon form. Gauss-Jordan Reduction
Row echelon form
General augmented matrix in row echelon form
� ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗� � � ∗ ∗ ∗ ∗ ∗ ∗ ∗
� � ∗ ∗ ∗ ∗ ∗� ∗ ∗ ∗ ∗
� � ∗ ∗� ∗
�
1) Leading entries are boxed (all equal to 1);
2) All the entries below, the (imaginary) staircase line are zero;
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 583: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/583.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Row echelon form. Gauss-Jordan Reduction
3) Each step of the (imaginary) staircase has height 1;
4) Each circle marks a column without a leading entry thatcorresponds to a free variable.
Strict triangular form
Is a particular case of row echelon form that can occur for systemsof n variables :
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 584: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/584.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Row echelon form. Gauss-Jordan Reduction
3) Each step
of the (imaginary) staircase has height 1;
4) Each circle marks a column without a leading entry thatcorresponds to a free variable.
Strict triangular form
Is a particular case of row echelon form that can occur for systemsof n variables :
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 585: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/585.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Row echelon form. Gauss-Jordan Reduction
3) Each step of the
(imaginary) staircase has height 1;
4) Each circle marks a column without a leading entry thatcorresponds to a free variable.
Strict triangular form
Is a particular case of row echelon form that can occur for systemsof n variables :
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 586: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/586.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Row echelon form. Gauss-Jordan Reduction
3) Each step of the (imaginary) staircase
has height 1;
4) Each circle marks a column without a leading entry thatcorresponds to a free variable.
Strict triangular form
Is a particular case of row echelon form that can occur for systemsof n variables :
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 587: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/587.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Row echelon form. Gauss-Jordan Reduction
3) Each step of the (imaginary) staircase has height
1;
4) Each circle marks a column without a leading entry thatcorresponds to a free variable.
Strict triangular form
Is a particular case of row echelon form that can occur for systemsof n variables :
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 588: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/588.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Row echelon form. Gauss-Jordan Reduction
3) Each step of the (imaginary) staircase has height 1;
4) Each circle marks a column without a leading entry thatcorresponds to a free variable.
Strict triangular form
Is a particular case of row echelon form that can occur for systemsof n variables :
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 589: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/589.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Row echelon form. Gauss-Jordan Reduction
3) Each step of the (imaginary) staircase has height 1;
4) Each circle marks a column without a leading entry thatcorresponds to a free variable.
Strict triangular form
Is a particular case of row echelon form that can occur for systemsof n variables :
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 590: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/590.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Row echelon form. Gauss-Jordan Reduction
3) Each step of the (imaginary) staircase has height 1;
4) Each circle marks a column without a leading entry thatcorresponds to a free variable.
Strict triangular form
Is a particular case of row echelon form that can occur for systemsof n variables :
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 591: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/591.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Row echelon form. Gauss-Jordan Reduction
3) Each step of the (imaginary) staircase has height 1;
4) Each circle marks a column without a leading entry thatcorresponds to a free variable.
Strict triangular form
Is a particular case
of row echelon form that can occur for systemsof n variables :
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 592: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/592.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Row echelon form. Gauss-Jordan Reduction
3) Each step of the (imaginary) staircase has height 1;
4) Each circle marks a column without a leading entry thatcorresponds to a free variable.
Strict triangular form
Is a particular case of row echelon form
that can occur for systemsof n variables :
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 593: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/593.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Row echelon form. Gauss-Jordan Reduction
3) Each step of the (imaginary) staircase has height 1;
4) Each circle marks a column without a leading entry thatcorresponds to a free variable.
Strict triangular form
Is a particular case of row echelon form that can occur for systemsof n variables :
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 594: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/594.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Row echelon form. Gauss-Jordan Reduction
� ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗� ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗
� ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗� ∗ ∗ ∗ ∗ ∗ ∗ ∗
� ∗ ∗ ∗ ∗ ∗ ∗� ∗ ∗ ∗ ∗ ∗
� ∗ ∗ ∗ ∗� ∗ ∗ ∗
� ∗ ∗� ∗
1) No zero rows.
2) No free variables.
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 595: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/595.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Row echelon form. Gauss-Jordan Reduction
� ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗� ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗
� ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗� ∗ ∗ ∗ ∗ ∗ ∗ ∗
� ∗ ∗ ∗ ∗ ∗ ∗� ∗ ∗ ∗ ∗ ∗
� ∗ ∗ ∗ ∗� ∗ ∗ ∗
� ∗ ∗� ∗
1) No zero rows.
2) No free variables.
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 596: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/596.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Row echelon form. Gauss-Jordan Reduction
� ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗� ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗
� ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗� ∗ ∗ ∗ ∗ ∗ ∗ ∗
� ∗ ∗ ∗ ∗ ∗ ∗� ∗ ∗ ∗ ∗ ∗
� ∗ ∗ ∗ ∗� ∗ ∗ ∗
� ∗ ∗� ∗
1) No zero rows.
2) No free variables.
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 597: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/597.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Row echelon form. Gauss-Jordan Reduction
� ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗� ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗
� ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗� ∗ ∗ ∗ ∗ ∗ ∗ ∗
� ∗ ∗ ∗ ∗ ∗ ∗� ∗ ∗ ∗ ∗ ∗
� ∗ ∗ ∗ ∗� ∗ ∗ ∗
� ∗ ∗� ∗
1) No zero rows.
2) No free variables.Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 598: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/598.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Row echelon form. Gauss-Jordan Reduction
Consistency check
The original system of linear equations is consistent if there is noleading entry in the rightmost column of the augmented matrix inrow echelon form.
Augmented matrix of an inconsistent system
� ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗� � � ∗ ∗ ∗ ∗ ∗ ∗ ∗
� � ∗ ∗ ∗ ∗ ∗� ∗ ∗ ∗ ∗
� � ∗ ∗� ∗
�
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 599: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/599.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Row echelon form. Gauss-Jordan Reduction
Consistency check
The original system of linear equations is consistent if there is noleading entry in the rightmost column of the augmented matrix inrow echelon form.
Augmented matrix of an inconsistent system
� ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗� � � ∗ ∗ ∗ ∗ ∗ ∗ ∗
� � ∗ ∗ ∗ ∗ ∗� ∗ ∗ ∗ ∗
� � ∗ ∗� ∗
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Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 600: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/600.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Row echelon form. Gauss-Jordan Reduction
Consistency check
The original system
of linear equations is consistent if there is noleading entry in the rightmost column of the augmented matrix inrow echelon form.
Augmented matrix of an inconsistent system
� ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗� � � ∗ ∗ ∗ ∗ ∗ ∗ ∗
� � ∗ ∗ ∗ ∗ ∗� ∗ ∗ ∗ ∗
� � ∗ ∗� ∗
�
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 601: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/601.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Row echelon form. Gauss-Jordan Reduction
Consistency check
The original system of linear equations
is consistent if there is noleading entry in the rightmost column of the augmented matrix inrow echelon form.
Augmented matrix of an inconsistent system
� ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗� � � ∗ ∗ ∗ ∗ ∗ ∗ ∗
� � ∗ ∗ ∗ ∗ ∗� ∗ ∗ ∗ ∗
� � ∗ ∗� ∗
�
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 602: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/602.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Row echelon form. Gauss-Jordan Reduction
Consistency check
The original system of linear equations is consistent
if there is noleading entry in the rightmost column of the augmented matrix inrow echelon form.
Augmented matrix of an inconsistent system
� ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗� � � ∗ ∗ ∗ ∗ ∗ ∗ ∗
� � ∗ ∗ ∗ ∗ ∗� ∗ ∗ ∗ ∗
� � ∗ ∗� ∗
�
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 603: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/603.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Row echelon form. Gauss-Jordan Reduction
Consistency check
The original system of linear equations is consistent if there is noleading entry
in the rightmost column of the augmented matrix inrow echelon form.
Augmented matrix of an inconsistent system
� ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗� � � ∗ ∗ ∗ ∗ ∗ ∗ ∗
� � ∗ ∗ ∗ ∗ ∗� ∗ ∗ ∗ ∗
� � ∗ ∗� ∗
�
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 604: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/604.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Row echelon form. Gauss-Jordan Reduction
Consistency check
The original system of linear equations is consistent if there is noleading entry in the rightmost column
of the augmented matrix inrow echelon form.
Augmented matrix of an inconsistent system
� ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗� � � ∗ ∗ ∗ ∗ ∗ ∗ ∗
� � ∗ ∗ ∗ ∗ ∗� ∗ ∗ ∗ ∗
� � ∗ ∗� ∗
�
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 605: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/605.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Row echelon form. Gauss-Jordan Reduction
Consistency check
The original system of linear equations is consistent if there is noleading entry in the rightmost column of the augmented matrix
inrow echelon form.
Augmented matrix of an inconsistent system
� ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗� � � ∗ ∗ ∗ ∗ ∗ ∗ ∗
� � ∗ ∗ ∗ ∗ ∗� ∗ ∗ ∗ ∗
� � ∗ ∗� ∗
�
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 606: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/606.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Row echelon form. Gauss-Jordan Reduction
Consistency check
The original system of linear equations is consistent if there is noleading entry in the rightmost column of the augmented matrix inrow echelon form.
Augmented matrix of an inconsistent system
� ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗� � � ∗ ∗ ∗ ∗ ∗ ∗ ∗
� � ∗ ∗ ∗ ∗ ∗� ∗ ∗ ∗ ∗
� � ∗ ∗� ∗
�
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 607: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/607.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Row echelon form. Gauss-Jordan Reduction
Consistency check
The original system of linear equations is consistent if there is noleading entry in the rightmost column of the augmented matrix inrow echelon form.
Augmented matrix of an inconsistent system
� ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗� � � ∗ ∗ ∗ ∗ ∗ ∗ ∗
� � ∗ ∗ ∗ ∗ ∗� ∗ ∗ ∗ ∗
� � ∗ ∗� ∗
�
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 608: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/608.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Row echelon form. Gauss-Jordan Reduction
Consistency check
The original system of linear equations is consistent if there is noleading entry in the rightmost column of the augmented matrix inrow echelon form.
Augmented matrix of an inconsistent system
� ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗� � � ∗ ∗ ∗ ∗ ∗ ∗ ∗
� � ∗ ∗ ∗ ∗ ∗� ∗ ∗ ∗ ∗
� � ∗ ∗� ∗
�
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 609: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/609.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Row echelon form. Gauss-Jordan Reduction
The goal of the Gauss-Jordan reduction is to convert theaugmented matrix into reduced row echelon form :
1 ∗ ∗ ∗ ∗ ∗1 � � ∗ ∗ ∗
1 � ∗ ∗1 ∗ ∗
1 � ∗1 ∗
1) All entries below the staircase line are zero; 2) Each boxed entryis 1, the other entries in its column are zero; 3) Each circlecorresponds to a free variable.
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 610: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/610.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Row echelon form. Gauss-Jordan Reduction
The goal
of the Gauss-Jordan reduction is to convert theaugmented matrix into reduced row echelon form :
1 ∗ ∗ ∗ ∗ ∗1 � � ∗ ∗ ∗
1 � ∗ ∗1 ∗ ∗
1 � ∗1 ∗
1) All entries below the staircase line are zero; 2) Each boxed entryis 1, the other entries in its column are zero; 3) Each circlecorresponds to a free variable.
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 611: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/611.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Row echelon form. Gauss-Jordan Reduction
The goal of the Gauss-Jordan reduction
is to convert theaugmented matrix into reduced row echelon form :
1 ∗ ∗ ∗ ∗ ∗1 � � ∗ ∗ ∗
1 � ∗ ∗1 ∗ ∗
1 � ∗1 ∗
1) All entries below the staircase line are zero; 2) Each boxed entryis 1, the other entries in its column are zero; 3) Each circlecorresponds to a free variable.
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 612: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/612.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Row echelon form. Gauss-Jordan Reduction
The goal of the Gauss-Jordan reduction is to convert
theaugmented matrix into reduced row echelon form :
1 ∗ ∗ ∗ ∗ ∗1 � � ∗ ∗ ∗
1 � ∗ ∗1 ∗ ∗
1 � ∗1 ∗
1) All entries below the staircase line are zero; 2) Each boxed entryis 1, the other entries in its column are zero; 3) Each circlecorresponds to a free variable.
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 613: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/613.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Row echelon form. Gauss-Jordan Reduction
The goal of the Gauss-Jordan reduction is to convert theaugmented matrix
into reduced row echelon form :
1 ∗ ∗ ∗ ∗ ∗1 � � ∗ ∗ ∗
1 � ∗ ∗1 ∗ ∗
1 � ∗1 ∗
1) All entries below the staircase line are zero; 2) Each boxed entryis 1, the other entries in its column are zero; 3) Each circlecorresponds to a free variable.
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 614: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/614.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Row echelon form. Gauss-Jordan Reduction
The goal of the Gauss-Jordan reduction is to convert theaugmented matrix into reduced row echelon form :
1 ∗ ∗ ∗ ∗ ∗1 � � ∗ ∗ ∗
1 � ∗ ∗1 ∗ ∗
1 � ∗1 ∗
1) All entries below the staircase line are zero; 2) Each boxed entryis 1, the other entries in its column are zero; 3) Each circlecorresponds to a free variable.
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 615: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/615.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Row echelon form. Gauss-Jordan Reduction
The goal of the Gauss-Jordan reduction is to convert theaugmented matrix into reduced row echelon form :
1 ∗ ∗ ∗ ∗ ∗1 � � ∗ ∗ ∗
1 � ∗ ∗1 ∗ ∗
1 � ∗1 ∗
1) All entries below the staircase line are zero; 2) Each boxed entryis 1, the other entries in its column are zero; 3) Each circlecorresponds to a free variable.
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 616: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/616.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Row echelon form. Gauss-Jordan Reduction
The goal of the Gauss-Jordan reduction is to convert theaugmented matrix into reduced row echelon form :
1 ∗ ∗ ∗ ∗ ∗1 � � ∗ ∗ ∗
1 � ∗ ∗1 ∗ ∗
1 � ∗1 ∗
1) All entries
below the staircase line are zero; 2) Each boxed entryis 1, the other entries in its column are zero; 3) Each circlecorresponds to a free variable.
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 617: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/617.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Row echelon form. Gauss-Jordan Reduction
The goal of the Gauss-Jordan reduction is to convert theaugmented matrix into reduced row echelon form :
1 ∗ ∗ ∗ ∗ ∗1 � � ∗ ∗ ∗
1 � ∗ ∗1 ∗ ∗
1 � ∗1 ∗
1) All entries below the staircase line
are zero; 2) Each boxed entryis 1, the other entries in its column are zero; 3) Each circlecorresponds to a free variable.
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 618: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/618.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Row echelon form. Gauss-Jordan Reduction
The goal of the Gauss-Jordan reduction is to convert theaugmented matrix into reduced row echelon form :
1 ∗ ∗ ∗ ∗ ∗1 � � ∗ ∗ ∗
1 � ∗ ∗1 ∗ ∗
1 � ∗1 ∗
1) All entries below the staircase line are zero;
2) Each boxed entryis 1, the other entries in its column are zero; 3) Each circlecorresponds to a free variable.
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 619: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/619.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Row echelon form. Gauss-Jordan Reduction
The goal of the Gauss-Jordan reduction is to convert theaugmented matrix into reduced row echelon form :
1 ∗ ∗ ∗ ∗ ∗1 � � ∗ ∗ ∗
1 � ∗ ∗1 ∗ ∗
1 � ∗1 ∗
1) All entries below the staircase line are zero; 2) Each boxed entry
is 1, the other entries in its column are zero; 3) Each circlecorresponds to a free variable.
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 620: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/620.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Row echelon form. Gauss-Jordan Reduction
The goal of the Gauss-Jordan reduction is to convert theaugmented matrix into reduced row echelon form :
1 ∗ ∗ ∗ ∗ ∗1 � � ∗ ∗ ∗
1 � ∗ ∗1 ∗ ∗
1 � ∗1 ∗
1) All entries below the staircase line are zero; 2) Each boxed entryis 1,
the other entries in its column are zero; 3) Each circlecorresponds to a free variable.
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 621: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/621.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Row echelon form. Gauss-Jordan Reduction
The goal of the Gauss-Jordan reduction is to convert theaugmented matrix into reduced row echelon form :
1 ∗ ∗ ∗ ∗ ∗1 � � ∗ ∗ ∗
1 � ∗ ∗1 ∗ ∗
1 � ∗1 ∗
1) All entries below the staircase line are zero; 2) Each boxed entryis 1, the other entries
in its column are zero; 3) Each circlecorresponds to a free variable.
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 622: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/622.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Row echelon form. Gauss-Jordan Reduction
The goal of the Gauss-Jordan reduction is to convert theaugmented matrix into reduced row echelon form :
1 ∗ ∗ ∗ ∗ ∗1 � � ∗ ∗ ∗
1 � ∗ ∗1 ∗ ∗
1 � ∗1 ∗
1) All entries below the staircase line are zero; 2) Each boxed entryis 1, the other entries in its column
are zero; 3) Each circlecorresponds to a free variable.
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 623: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/623.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Row echelon form. Gauss-Jordan Reduction
The goal of the Gauss-Jordan reduction is to convert theaugmented matrix into reduced row echelon form :
1 ∗ ∗ ∗ ∗ ∗1 � � ∗ ∗ ∗
1 � ∗ ∗1 ∗ ∗
1 � ∗1 ∗
1) All entries below the staircase line are zero; 2) Each boxed entryis 1, the other entries in its column are zero;
3) Each circlecorresponds to a free variable.
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 624: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/624.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Row echelon form. Gauss-Jordan Reduction
The goal of the Gauss-Jordan reduction is to convert theaugmented matrix into reduced row echelon form :
1 ∗ ∗ ∗ ∗ ∗1 � � ∗ ∗ ∗
1 � ∗ ∗1 ∗ ∗
1 � ∗1 ∗
1) All entries below the staircase line are zero; 2) Each boxed entryis 1, the other entries in its column are zero; 3) Each circle
corresponds to a free variable.
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 625: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/625.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Row echelon form. Gauss-Jordan Reduction
The goal of the Gauss-Jordan reduction is to convert theaugmented matrix into reduced row echelon form :
1 ∗ ∗ ∗ ∗ ∗1 � � ∗ ∗ ∗
1 � ∗ ∗1 ∗ ∗
1 � ∗1 ∗
1) All entries below the staircase line are zero; 2) Each boxed entryis 1, the other entries in its column are zero; 3) Each circlecorresponds to
a free variable.
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 626: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/626.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Row echelon form. Gauss-Jordan Reduction
The goal of the Gauss-Jordan reduction is to convert theaugmented matrix into reduced row echelon form :
1 ∗ ∗ ∗ ∗ ∗1 � � ∗ ∗ ∗
1 � ∗ ∗1 ∗ ∗
1 � ∗1 ∗
1) All entries below the staircase line are zero; 2) Each boxed entryis 1, the other entries in its column are zero; 3) Each circlecorresponds to a free variable.
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 627: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/627.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Row echelon form. Gauss-Jordan Reduction
Example 1.6From a previous example, we have
x − y = 1
2x − y− z = 3x + y+ z = 6
⇒
1 −1 0 22 −1 −1 31 1 1 6
Row echelon form (also strict triangular):
x − y = 2
y − z = −1x + y+ z = 2
⇒
1 −1 0 2
0 1 −1 −1
0 0 1 2
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 628: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/628.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Row echelon form. Gauss-Jordan Reduction
Example 1.6
From a previous example, we have
x − y = 1
2x − y− z = 3x + y+ z = 6
⇒
1 −1 0 22 −1 −1 31 1 1 6
Row echelon form (also strict triangular):
x − y = 2
y − z = −1x + y+ z = 2
⇒
1 −1 0 2
0 1 −1 −1
0 0 1 2
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 629: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/629.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Row echelon form. Gauss-Jordan Reduction
Example 1.6From a previous example,
we have
x − y = 1
2x − y− z = 3x + y+ z = 6
⇒
1 −1 0 22 −1 −1 31 1 1 6
Row echelon form (also strict triangular):
x − y = 2
y − z = −1x + y+ z = 2
⇒
1 −1 0 2
0 1 −1 −1
0 0 1 2
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 630: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/630.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Row echelon form. Gauss-Jordan Reduction
Example 1.6From a previous example, we have
x − y = 1
2x − y− z = 3x + y+ z = 6
⇒
1 −1 0 22 −1 −1 31 1 1 6
Row echelon form (also strict triangular):
x − y = 2
y − z = −1x + y+ z = 2
⇒
1 −1 0 2
0 1 −1 −1
0 0 1 2
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 631: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/631.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Row echelon form. Gauss-Jordan Reduction
Example 1.6From a previous example, we have
x − y = 1
2x − y− z = 3x + y+ z = 6
⇒
1 −1 0 22 −1 −1 31 1 1 6
Row echelon form (also strict triangular):
x − y = 2
y − z = −1x + y+ z = 2
⇒
1 −1 0 2
0 1 −1 −1
0 0 1 2
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 632: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/632.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Row echelon form. Gauss-Jordan Reduction
Example 1.6From a previous example, we have
x − y = 1
2x − y− z = 3x + y+ z = 6
⇒
1 −1 0 22 −1 −1 31 1 1 6
Row echelon form (also strict triangular):
x − y = 2
y − z = −1x + y+ z = 2
⇒
1 −1 0 2
0 1 −1 −1
0 0 1 2
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 633: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/633.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Row echelon form. Gauss-Jordan Reduction
Example 1.6From a previous example, we have
x − y = 1
2x − y− z = 3x + y+ z = 6
⇒
1 −1 0 22 −1 −1 31 1 1 6
Row echelon form (also strict triangular):
x − y = 2
y − z = −1x + y+ z = 2
⇒
1 −1 0 2
0 1 −1 −1
0 0 1 2
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 634: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/634.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Row echelon form. Gauss-Jordan Reduction
Example 1.6From a previous example, we have
x − y = 1
2x − y− z = 3x + y+ z = 6
⇒
1 −1 0 22 −1 −1 31 1 1 6
Row echelon form (also strict triangular):
x − y = 2
y − z = −1x + y+ z = 2
⇒
1 −1 0 2
0 1 −1 −1
0 0 1 2
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 635: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/635.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Row echelon form. Gauss-Jordan Reduction
Example 1.6From a previous example, we have
x − y = 1
2x − y− z = 3x + y+ z = 6
⇒
1 −1 0 22 −1 −1 31 1 1 6
Row echelon form (also strict triangular):
x − y = 2
y − z = −1x + y+ z = 2
⇒
1 −1 0 2
0 1 −1 −1
0 0 1 2
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 636: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/636.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Row echelon form. Gauss-Jordan Reduction
Reduced row echelon form :x = 3
y = 1z = 2
⇒
1 0 0 3
0 1 0 1
0 0 1 2
Example 1.7
x + y− 2z = 1
y− z = 3−x + 4y− 3z = 6
⇒
1 1 −2 10 1 −1 3−1 4 −3 1
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 637: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/637.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Row echelon form. Gauss-Jordan Reduction
Reduced row echelon form :
x = 3
y = 1z = 2
⇒
1 0 0 3
0 1 0 1
0 0 1 2
Example 1.7
x + y− 2z = 1
y− z = 3−x + 4y− 3z = 6
⇒
1 1 −2 10 1 −1 3−1 4 −3 1
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 638: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/638.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Row echelon form. Gauss-Jordan Reduction
Reduced row echelon form :x = 3
y = 1z = 2
⇒
1 0 0 3
0 1 0 1
0 0 1 2
Example 1.7
x + y− 2z = 1
y− z = 3−x + 4y− 3z = 6
⇒
1 1 −2 10 1 −1 3−1 4 −3 1
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 639: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/639.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Row echelon form. Gauss-Jordan Reduction
Reduced row echelon form :x = 3
y = 1z = 2
⇒
1 0 0 3
0 1 0 1
0 0 1 2
Example 1.7
x + y− 2z = 1
y− z = 3−x + 4y− 3z = 6
⇒
1 1 −2 10 1 −1 3−1 4 −3 1
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 640: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/640.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Row echelon form. Gauss-Jordan Reduction
Reduced row echelon form :x = 3
y = 1z = 2
⇒
1 0 0 3
0 1 0 1
0 0 1 2
Example 1.7
x + y− 2z = 1
y− z = 3−x + 4y− 3z = 6
⇒
1 1 −2 10 1 −1 3−1 4 −3 1
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 641: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/641.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Row echelon form. Gauss-Jordan Reduction
Reduced row echelon form :x = 3
y = 1z = 2
⇒
1 0 0 3
0 1 0 1
0 0 1 2
Example 1.7
x + y− 2z = 1
y− z = 3−x + 4y− 3z = 6
⇒
1 1 −2 10 1 −1 3−1 4 −3 1
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 642: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/642.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Row echelon form. Gauss-Jordan Reduction
Reduced row echelon form :x = 3
y = 1z = 2
⇒
1 0 0 3
0 1 0 1
0 0 1 2
Example 1.7
x + y− 2z = 1
y− z = 3−x + 4y− 3z = 6
⇒
1 1 −2 10 1 −1 3−1 4 −3 1
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 643: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/643.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Row echelon form. Gauss-Jordan Reduction
Row echelon form:
x + y− 2z = 1
y− z = 30 = 1
⇒
1 1 −2 1
0 1 −1 3
0 0 0 1
Reduced row echelon form:x + − z = 0
y− z = 00 = 1
⇒
1 0 −1 0
0 1 −1 0
0 0 0 1
Inconsistent system
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 644: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/644.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Row echelon form. Gauss-Jordan Reduction
Row echelon form:
x + y− 2z = 1
y− z = 30 = 1
⇒
1 1 −2 1
0 1 −1 3
0 0 0 1
Reduced row echelon form:x + − z = 0
y− z = 00 = 1
⇒
1 0 −1 0
0 1 −1 0
0 0 0 1
Inconsistent system
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 645: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/645.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Row echelon form. Gauss-Jordan Reduction
Row echelon form:
x + y− 2z = 1
y− z = 30 = 1
⇒
1 1 −2 1
0 1 −1 3
0 0 0 1
Reduced row echelon form:x + − z = 0
y− z = 00 = 1
⇒
1 0 −1 0
0 1 −1 0
0 0 0 1
Inconsistent system
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 646: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/646.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Row echelon form. Gauss-Jordan Reduction
Row echelon form:
x + y− 2z = 1
y− z = 30 = 1
⇒
1 1 −2 1
0 1 −1 3
0 0 0 1
Reduced row echelon form:x + − z = 0
y− z = 00 = 1
⇒
1 0 −1 0
0 1 −1 0
0 0 0 1
Inconsistent system
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 647: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/647.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Row echelon form. Gauss-Jordan Reduction
Row echelon form:
x + y− 2z = 1
y− z = 30 = 1
⇒
1 1 −2 1
0 1 −1 3
0 0 0 1
Reduced row echelon form:x + − z = 0
y− z = 00 = 1
⇒
1 0 −1 0
0 1 −1 0
0 0 0 1
Inconsistent system
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 648: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/648.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Row echelon form. Gauss-Jordan Reduction
Row echelon form:
x + y− 2z = 1
y− z = 30 = 1
⇒
1 1 −2 1
0 1 −1 3
0 0 0 1
Reduced row echelon form:x + − z = 0
y− z = 00 = 1
⇒
1 0 −1 0
0 1 −1 0
0 0 0 1
Inconsistent system
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 649: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/649.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Row echelon form. Gauss-Jordan Reduction
Row echelon form:
x + y− 2z = 1
y− z = 30 = 1
⇒
1 1 −2 1
0 1 −1 3
0 0 0 1
Reduced row echelon form:x + − z = 0
y− z = 00 = 1
⇒
1 0 −1 0
0 1 −1 0
0 0 0 1
Inconsistent system
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 650: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/650.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Row echelon form. Gauss-Jordan Reduction
How to solve a system of linear equations
1) Order the variables.
2) Write down the augmented matrix of the system.
3) Convert the matrix to row echelon form
4) Check for consistency.
5) Convert the matrix to reduced row echelon form Example 7
6) Write down the system corresponding to the reduced rowechelon form.
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 651: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/651.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Row echelon form. Gauss-Jordan Reduction
How to solve a system of linear equations
1) Order the variables.
2) Write down the augmented matrix of the system.
3) Convert the matrix to row echelon form
4) Check for consistency.
5) Convert the matrix to reduced row echelon form Example 7
6) Write down the system corresponding to the reduced rowechelon form.
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 652: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/652.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Row echelon form. Gauss-Jordan Reduction
How to solve a system of linear equations
1) Order the variables.
2) Write down the augmented matrix of the system.
3) Convert the matrix to row echelon form
4) Check for consistency.
5) Convert the matrix to reduced row echelon form Example 7
6) Write down the system corresponding to the reduced rowechelon form.
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 653: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/653.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Row echelon form. Gauss-Jordan Reduction
How to solve a system of linear equations
1) Order the variables.
2) Write down
the augmented matrix of the system.
3) Convert the matrix to row echelon form
4) Check for consistency.
5) Convert the matrix to reduced row echelon form Example 7
6) Write down the system corresponding to the reduced rowechelon form.
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 654: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/654.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Row echelon form. Gauss-Jordan Reduction
How to solve a system of linear equations
1) Order the variables.
2) Write down the augmented matrix
of the system.
3) Convert the matrix to row echelon form
4) Check for consistency.
5) Convert the matrix to reduced row echelon form Example 7
6) Write down the system corresponding to the reduced rowechelon form.
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 655: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/655.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Row echelon form. Gauss-Jordan Reduction
How to solve a system of linear equations
1) Order the variables.
2) Write down the augmented matrix of the system.
3) Convert the matrix to row echelon form
4) Check for consistency.
5) Convert the matrix to reduced row echelon form Example 7
6) Write down the system corresponding to the reduced rowechelon form.
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 656: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/656.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Row echelon form. Gauss-Jordan Reduction
How to solve a system of linear equations
1) Order the variables.
2) Write down the augmented matrix of the system.
3) Convert the matrix
to row echelon form
4) Check for consistency.
5) Convert the matrix to reduced row echelon form Example 7
6) Write down the system corresponding to the reduced rowechelon form.
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 657: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/657.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Row echelon form. Gauss-Jordan Reduction
How to solve a system of linear equations
1) Order the variables.
2) Write down the augmented matrix of the system.
3) Convert the matrix to row echelon form
4) Check for consistency.
5) Convert the matrix to reduced row echelon form Example 7
6) Write down the system corresponding to the reduced rowechelon form.
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 658: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/658.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Row echelon form. Gauss-Jordan Reduction
How to solve a system of linear equations
1) Order the variables.
2) Write down the augmented matrix of the system.
3) Convert the matrix to row echelon form
4) Check for consistency.
5) Convert the matrix to reduced row echelon form Example 7
6) Write down the system corresponding to the reduced rowechelon form.
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 659: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/659.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Row echelon form. Gauss-Jordan Reduction
How to solve a system of linear equations
1) Order the variables.
2) Write down the augmented matrix of the system.
3) Convert the matrix to row echelon form
4) Check for consistency.
5) Convert the matrix
to reduced row echelon form Example 7
6) Write down the system corresponding to the reduced rowechelon form.
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 660: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/660.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Row echelon form. Gauss-Jordan Reduction
How to solve a system of linear equations
1) Order the variables.
2) Write down the augmented matrix of the system.
3) Convert the matrix to row echelon form
4) Check for consistency.
5) Convert the matrix to reduced row
echelon form Example 7
6) Write down the system corresponding to the reduced rowechelon form.
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 661: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/661.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Row echelon form. Gauss-Jordan Reduction
How to solve a system of linear equations
1) Order the variables.
2) Write down the augmented matrix of the system.
3) Convert the matrix to row echelon form
4) Check for consistency.
5) Convert the matrix to reduced row echelon form Example 7
6) Write down the system corresponding to the reduced rowechelon form.
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 662: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/662.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Row echelon form. Gauss-Jordan Reduction
How to solve a system of linear equations
1) Order the variables.
2) Write down the augmented matrix of the system.
3) Convert the matrix to row echelon form
4) Check for consistency.
5) Convert the matrix to reduced row echelon form Example 7
6) Write down
the system corresponding to the reduced rowechelon form.
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 663: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/663.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Row echelon form. Gauss-Jordan Reduction
How to solve a system of linear equations
1) Order the variables.
2) Write down the augmented matrix of the system.
3) Convert the matrix to row echelon form
4) Check for consistency.
5) Convert the matrix to reduced row echelon form Example 7
6) Write down the system corresponding
to the reduced rowechelon form.
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 664: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/664.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Row echelon form. Gauss-Jordan Reduction
How to solve a system of linear equations
1) Order the variables.
2) Write down the augmented matrix of the system.
3) Convert the matrix to row echelon form
4) Check for consistency.
5) Convert the matrix to reduced row echelon form Example 7
6) Write down the system corresponding to the reduced rowechelon form.
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 665: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/665.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Row echelon form. Gauss-Jordan Reduction
7) Determine leading and free variables.
8) Rewrite the system so that the leading variables are on the leftwhile everything else is on the right.
9) Assign parameters to the free variables and write down thegeneral solution in parametric form.
Example 1.7 {x2 + 2x3 + 3x4 = 6
x1 + 2x2 + 3x3 + 4x4 = 10
In this case the variables are x1, x2, x3, x4. and the Augmentedmatrix is :
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 666: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/666.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Row echelon form. Gauss-Jordan Reduction
7) Determine leading and free variables.
8) Rewrite the system so that the leading variables are on the leftwhile everything else is on the right.
9) Assign parameters to the free variables and write down thegeneral solution in parametric form.
Example 1.7 {x2 + 2x3 + 3x4 = 6
x1 + 2x2 + 3x3 + 4x4 = 10
In this case the variables are x1, x2, x3, x4. and the Augmentedmatrix is :
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 667: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/667.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Row echelon form. Gauss-Jordan Reduction
7) Determine leading and free variables.
8) Rewrite the system
so that the leading variables are on the leftwhile everything else is on the right.
9) Assign parameters to the free variables and write down thegeneral solution in parametric form.
Example 1.7 {x2 + 2x3 + 3x4 = 6
x1 + 2x2 + 3x3 + 4x4 = 10
In this case the variables are x1, x2, x3, x4. and the Augmentedmatrix is :
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 668: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/668.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Row echelon form. Gauss-Jordan Reduction
7) Determine leading and free variables.
8) Rewrite the system so that the leading variables
are on the leftwhile everything else is on the right.
9) Assign parameters to the free variables and write down thegeneral solution in parametric form.
Example 1.7 {x2 + 2x3 + 3x4 = 6
x1 + 2x2 + 3x3 + 4x4 = 10
In this case the variables are x1, x2, x3, x4. and the Augmentedmatrix is :
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 669: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/669.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Row echelon form. Gauss-Jordan Reduction
7) Determine leading and free variables.
8) Rewrite the system so that the leading variables are on the left
while everything else is on the right.
9) Assign parameters to the free variables and write down thegeneral solution in parametric form.
Example 1.7 {x2 + 2x3 + 3x4 = 6
x1 + 2x2 + 3x3 + 4x4 = 10
In this case the variables are x1, x2, x3, x4. and the Augmentedmatrix is :
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 670: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/670.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Row echelon form. Gauss-Jordan Reduction
7) Determine leading and free variables.
8) Rewrite the system so that the leading variables are on the leftwhile everything else
is on the right.
9) Assign parameters to the free variables and write down thegeneral solution in parametric form.
Example 1.7 {x2 + 2x3 + 3x4 = 6
x1 + 2x2 + 3x3 + 4x4 = 10
In this case the variables are x1, x2, x3, x4. and the Augmentedmatrix is :
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 671: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/671.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Row echelon form. Gauss-Jordan Reduction
7) Determine leading and free variables.
8) Rewrite the system so that the leading variables are on the leftwhile everything else is on the right.
9) Assign parameters to the free variables and write down thegeneral solution in parametric form.
Example 1.7 {x2 + 2x3 + 3x4 = 6
x1 + 2x2 + 3x3 + 4x4 = 10
In this case the variables are x1, x2, x3, x4. and the Augmentedmatrix is :
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 672: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/672.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Row echelon form. Gauss-Jordan Reduction
7) Determine leading and free variables.
8) Rewrite the system so that the leading variables are on the leftwhile everything else is on the right.
9) Assign parameters
to the free variables and write down thegeneral solution in parametric form.
Example 1.7 {x2 + 2x3 + 3x4 = 6
x1 + 2x2 + 3x3 + 4x4 = 10
In this case the variables are x1, x2, x3, x4. and the Augmentedmatrix is :
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 673: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/673.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Row echelon form. Gauss-Jordan Reduction
7) Determine leading and free variables.
8) Rewrite the system so that the leading variables are on the leftwhile everything else is on the right.
9) Assign parameters to the free variables and
write down thegeneral solution in parametric form.
Example 1.7 {x2 + 2x3 + 3x4 = 6
x1 + 2x2 + 3x3 + 4x4 = 10
In this case the variables are x1, x2, x3, x4. and the Augmentedmatrix is :
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 674: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/674.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Row echelon form. Gauss-Jordan Reduction
7) Determine leading and free variables.
8) Rewrite the system so that the leading variables are on the leftwhile everything else is on the right.
9) Assign parameters to the free variables and write down thegeneral solution
in parametric form.
Example 1.7 {x2 + 2x3 + 3x4 = 6
x1 + 2x2 + 3x3 + 4x4 = 10
In this case the variables are x1, x2, x3, x4. and the Augmentedmatrix is :
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 675: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/675.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Row echelon form. Gauss-Jordan Reduction
7) Determine leading and free variables.
8) Rewrite the system so that the leading variables are on the leftwhile everything else is on the right.
9) Assign parameters to the free variables and write down thegeneral solution in parametric form.
Example 1.7 {x2 + 2x3 + 3x4 = 6
x1 + 2x2 + 3x3 + 4x4 = 10
In this case the variables are x1, x2, x3, x4. and the Augmentedmatrix is :
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 676: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/676.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Row echelon form. Gauss-Jordan Reduction
7) Determine leading and free variables.
8) Rewrite the system so that the leading variables are on the leftwhile everything else is on the right.
9) Assign parameters to the free variables and write down thegeneral solution in parametric form.
Example 1.7
{x2 + 2x3 + 3x4 = 6
x1 + 2x2 + 3x3 + 4x4 = 10
In this case the variables are x1, x2, x3, x4. and the Augmentedmatrix is :
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 677: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/677.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Row echelon form. Gauss-Jordan Reduction
7) Determine leading and free variables.
8) Rewrite the system so that the leading variables are on the leftwhile everything else is on the right.
9) Assign parameters to the free variables and write down thegeneral solution in parametric form.
Example 1.7 {x2 + 2x3 + 3x4 = 6
x1 + 2x2 + 3x3 + 4x4 = 10
In this case the variables are x1, x2, x3, x4. and the Augmentedmatrix is :
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 678: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/678.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Row echelon form. Gauss-Jordan Reduction
7) Determine leading and free variables.
8) Rewrite the system so that the leading variables are on the leftwhile everything else is on the right.
9) Assign parameters to the free variables and write down thegeneral solution in parametric form.
Example 1.7 {x2 + 2x3 + 3x4 = 6
x1 + 2x2 + 3x3 + 4x4 = 10
In this case
the variables are x1, x2, x3, x4. and the Augmentedmatrix is :
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 679: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/679.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Row echelon form. Gauss-Jordan Reduction
7) Determine leading and free variables.
8) Rewrite the system so that the leading variables are on the leftwhile everything else is on the right.
9) Assign parameters to the free variables and write down thegeneral solution in parametric form.
Example 1.7 {x2 + 2x3 + 3x4 = 6
x1 + 2x2 + 3x3 + 4x4 = 10
In this case the variables are
x1, x2, x3, x4. and the Augmentedmatrix is :
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 680: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/680.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Row echelon form. Gauss-Jordan Reduction
7) Determine leading and free variables.
8) Rewrite the system so that the leading variables are on the leftwhile everything else is on the right.
9) Assign parameters to the free variables and write down thegeneral solution in parametric form.
Example 1.7 {x2 + 2x3 + 3x4 = 6
x1 + 2x2 + 3x3 + 4x4 = 10
In this case the variables are x1,
x2, x3, x4. and the Augmentedmatrix is :
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 681: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/681.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Row echelon form. Gauss-Jordan Reduction
7) Determine leading and free variables.
8) Rewrite the system so that the leading variables are on the leftwhile everything else is on the right.
9) Assign parameters to the free variables and write down thegeneral solution in parametric form.
Example 1.7 {x2 + 2x3 + 3x4 = 6
x1 + 2x2 + 3x3 + 4x4 = 10
In this case the variables are x1, x2,
x3, x4. and the Augmentedmatrix is :
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 682: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/682.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Row echelon form. Gauss-Jordan Reduction
7) Determine leading and free variables.
8) Rewrite the system so that the leading variables are on the leftwhile everything else is on the right.
9) Assign parameters to the free variables and write down thegeneral solution in parametric form.
Example 1.7 {x2 + 2x3 + 3x4 = 6
x1 + 2x2 + 3x3 + 4x4 = 10
In this case the variables are x1, x2, x3,
x4. and the Augmentedmatrix is :
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 683: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/683.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Row echelon form. Gauss-Jordan Reduction
7) Determine leading and free variables.
8) Rewrite the system so that the leading variables are on the leftwhile everything else is on the right.
9) Assign parameters to the free variables and write down thegeneral solution in parametric form.
Example 1.7 {x2 + 2x3 + 3x4 = 6
x1 + 2x2 + 3x3 + 4x4 = 10
In this case the variables are x1, x2, x3, x4.
and the Augmentedmatrix is :
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 684: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/684.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Row echelon form. Gauss-Jordan Reduction
7) Determine leading and free variables.
8) Rewrite the system so that the leading variables are on the leftwhile everything else is on the right.
9) Assign parameters to the free variables and write down thegeneral solution in parametric form.
Example 1.7 {x2 + 2x3 + 3x4 = 6
x1 + 2x2 + 3x3 + 4x4 = 10
In this case the variables are x1, x2, x3, x4. and the Augmentedmatrix is :
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 685: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/685.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Row echelon form. Gauss-Jordan Reduction
(0 1 2 3 61 2 3 4 10
)To get it into row echelon form, we exchange the two rows:(
1 2 3 4 100 1 2 3 6
)Consistency check is passed. To convert into reduced row echelonform, add −2 times the 2nd row to the 1st row:(
1 0 −1 −2 −2
0 1 2 3 6
)The leading variables are x1 and x2 ; hence x3 and x4 are freevariables
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 686: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/686.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Row echelon form. Gauss-Jordan Reduction
(0 1 2 3 61 2 3 4 10
)
To get it into row echelon form, we exchange the two rows:(1 2 3 4 100 1 2 3 6
)Consistency check is passed. To convert into reduced row echelonform, add −2 times the 2nd row to the 1st row:(
1 0 −1 −2 −2
0 1 2 3 6
)The leading variables are x1 and x2 ; hence x3 and x4 are freevariables
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 687: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/687.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Row echelon form. Gauss-Jordan Reduction
(0 1 2 3 61 2 3 4 10
)To get it
into row echelon form, we exchange the two rows:(1 2 3 4 100 1 2 3 6
)Consistency check is passed. To convert into reduced row echelonform, add −2 times the 2nd row to the 1st row:(
1 0 −1 −2 −2
0 1 2 3 6
)The leading variables are x1 and x2 ; hence x3 and x4 are freevariables
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 688: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/688.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Row echelon form. Gauss-Jordan Reduction
(0 1 2 3 61 2 3 4 10
)To get it into row echelon form,
we exchange the two rows:(1 2 3 4 100 1 2 3 6
)Consistency check is passed. To convert into reduced row echelonform, add −2 times the 2nd row to the 1st row:(
1 0 −1 −2 −2
0 1 2 3 6
)The leading variables are x1 and x2 ; hence x3 and x4 are freevariables
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 689: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/689.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Row echelon form. Gauss-Jordan Reduction
(0 1 2 3 61 2 3 4 10
)To get it into row echelon form, we exchange
the two rows:(1 2 3 4 100 1 2 3 6
)Consistency check is passed. To convert into reduced row echelonform, add −2 times the 2nd row to the 1st row:(
1 0 −1 −2 −2
0 1 2 3 6
)The leading variables are x1 and x2 ; hence x3 and x4 are freevariables
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 690: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/690.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Row echelon form. Gauss-Jordan Reduction
(0 1 2 3 61 2 3 4 10
)To get it into row echelon form, we exchange the two rows:
(1 2 3 4 100 1 2 3 6
)Consistency check is passed. To convert into reduced row echelonform, add −2 times the 2nd row to the 1st row:(
1 0 −1 −2 −2
0 1 2 3 6
)The leading variables are x1 and x2 ; hence x3 and x4 are freevariables
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 691: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/691.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Row echelon form. Gauss-Jordan Reduction
(0 1 2 3 61 2 3 4 10
)To get it into row echelon form, we exchange the two rows:(
1 2 3 4 100 1 2 3 6
)
Consistency check is passed. To convert into reduced row echelonform, add −2 times the 2nd row to the 1st row:(
1 0 −1 −2 −2
0 1 2 3 6
)The leading variables are x1 and x2 ; hence x3 and x4 are freevariables
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 692: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/692.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Row echelon form. Gauss-Jordan Reduction
(0 1 2 3 61 2 3 4 10
)To get it into row echelon form, we exchange the two rows:(
1 2 3 4 100 1 2 3 6
)Consistency check
is passed. To convert into reduced row echelonform, add −2 times the 2nd row to the 1st row:(
1 0 −1 −2 −2
0 1 2 3 6
)The leading variables are x1 and x2 ; hence x3 and x4 are freevariables
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 693: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/693.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Row echelon form. Gauss-Jordan Reduction
(0 1 2 3 61 2 3 4 10
)To get it into row echelon form, we exchange the two rows:(
1 2 3 4 100 1 2 3 6
)Consistency check is passed.
To convert into reduced row echelonform, add −2 times the 2nd row to the 1st row:(
1 0 −1 −2 −2
0 1 2 3 6
)The leading variables are x1 and x2 ; hence x3 and x4 are freevariables
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 694: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/694.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Row echelon form. Gauss-Jordan Reduction
(0 1 2 3 61 2 3 4 10
)To get it into row echelon form, we exchange the two rows:(
1 2 3 4 100 1 2 3 6
)Consistency check is passed. To convert
into reduced row echelonform, add −2 times the 2nd row to the 1st row:(
1 0 −1 −2 −2
0 1 2 3 6
)The leading variables are x1 and x2 ; hence x3 and x4 are freevariables
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 695: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/695.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Row echelon form. Gauss-Jordan Reduction
(0 1 2 3 61 2 3 4 10
)To get it into row echelon form, we exchange the two rows:(
1 2 3 4 100 1 2 3 6
)Consistency check is passed. To convert into reduced row echelonform,
add −2 times the 2nd row to the 1st row:(1 0 −1 −2 −2
0 1 2 3 6
)The leading variables are x1 and x2 ; hence x3 and x4 are freevariables
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 696: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/696.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Row echelon form. Gauss-Jordan Reduction
(0 1 2 3 61 2 3 4 10
)To get it into row echelon form, we exchange the two rows:(
1 2 3 4 100 1 2 3 6
)Consistency check is passed. To convert into reduced row echelonform, add −2 times
the 2nd row to the 1st row:(1 0 −1 −2 −2
0 1 2 3 6
)The leading variables are x1 and x2 ; hence x3 and x4 are freevariables
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 697: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/697.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Row echelon form. Gauss-Jordan Reduction
(0 1 2 3 61 2 3 4 10
)To get it into row echelon form, we exchange the two rows:(
1 2 3 4 100 1 2 3 6
)Consistency check is passed. To convert into reduced row echelonform, add −2 times the 2nd row
to the 1st row:(1 0 −1 −2 −2
0 1 2 3 6
)The leading variables are x1 and x2 ; hence x3 and x4 are freevariables
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 698: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/698.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Row echelon form. Gauss-Jordan Reduction
(0 1 2 3 61 2 3 4 10
)To get it into row echelon form, we exchange the two rows:(
1 2 3 4 100 1 2 3 6
)Consistency check is passed. To convert into reduced row echelonform, add −2 times the 2nd row to the 1st row:
(1 0 −1 −2 −2
0 1 2 3 6
)The leading variables are x1 and x2 ; hence x3 and x4 are freevariables
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 699: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/699.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Row echelon form. Gauss-Jordan Reduction
(0 1 2 3 61 2 3 4 10
)To get it into row echelon form, we exchange the two rows:(
1 2 3 4 100 1 2 3 6
)Consistency check is passed. To convert into reduced row echelonform, add −2 times the 2nd row to the 1st row:(
1 0 −1 −2 −2
0 1 2 3 6
)
The leading variables are x1 and x2 ; hence x3 and x4 are freevariables
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 700: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/700.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Row echelon form. Gauss-Jordan Reduction
(0 1 2 3 61 2 3 4 10
)To get it into row echelon form, we exchange the two rows:(
1 2 3 4 100 1 2 3 6
)Consistency check is passed. To convert into reduced row echelonform, add −2 times the 2nd row to the 1st row:(
1 0 −1 −2 −2
0 1 2 3 6
)The leading variables
are x1 and x2 ; hence x3 and x4 are freevariables
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 701: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/701.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Row echelon form. Gauss-Jordan Reduction
(0 1 2 3 61 2 3 4 10
)To get it into row echelon form, we exchange the two rows:(
1 2 3 4 100 1 2 3 6
)Consistency check is passed. To convert into reduced row echelonform, add −2 times the 2nd row to the 1st row:(
1 0 −1 −2 −2
0 1 2 3 6
)The leading variables are x1 and
x2 ; hence x3 and x4 are freevariables
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 702: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/702.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Row echelon form. Gauss-Jordan Reduction
(0 1 2 3 61 2 3 4 10
)To get it into row echelon form, we exchange the two rows:(
1 2 3 4 100 1 2 3 6
)Consistency check is passed. To convert into reduced row echelonform, add −2 times the 2nd row to the 1st row:(
1 0 −1 −2 −2
0 1 2 3 6
)The leading variables are x1 and x2 ; hence
x3 and x4 are freevariables
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 703: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/703.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Row echelon form. Gauss-Jordan Reduction
(0 1 2 3 61 2 3 4 10
)To get it into row echelon form, we exchange the two rows:(
1 2 3 4 100 1 2 3 6
)Consistency check is passed. To convert into reduced row echelonform, add −2 times the 2nd row to the 1st row:(
1 0 −1 −2 −2
0 1 2 3 6
)The leading variables are x1 and x2 ; hence x3 and
x4 are freevariables
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 704: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/704.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Row echelon form. Gauss-Jordan Reduction
(0 1 2 3 61 2 3 4 10
)To get it into row echelon form, we exchange the two rows:(
1 2 3 4 100 1 2 3 6
)Consistency check is passed. To convert into reduced row echelonform, add −2 times the 2nd row to the 1st row:(
1 0 −1 −2 −2
0 1 2 3 6
)The leading variables are x1 and x2 ; hence x3 and x4
are freevariables
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 705: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/705.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Row echelon form. Gauss-Jordan Reduction
(0 1 2 3 61 2 3 4 10
)To get it into row echelon form, we exchange the two rows:(
1 2 3 4 100 1 2 3 6
)Consistency check is passed. To convert into reduced row echelonform, add −2 times the 2nd row to the 1st row:(
1 0 −1 −2 −2
0 1 2 3 6
)The leading variables are x1 and x2 ; hence x3 and x4 are freevariables
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 706: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/706.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Row echelon form. Gauss-Jordan Reduction
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 707: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/707.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Row echelon form. Gauss-Jordan Reduction
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 708: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/708.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Row echelon form. Gauss-Jordan Reduction
Back to the system:{x1 − x3 − 2x4 = −1x2 + 2x3 + 3x4 = 6
⇒{
x1 = x3 + 2x4 − 2x2 = −2x3 − 3x4 + 6
and the general solution is given byx1 = t + 2s − 2x2 = −2t − 3s + 6x3 = tx4 = s
(t, s ∈ R)
In vector form,
(x1, x2, x3, x4) = (−2, 6, 0, 0) + t(1,−2, 1, 0) + s(2,−3, 0, 1)
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 709: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/709.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Row echelon form. Gauss-Jordan Reduction
Back to the system:
{x1 − x3 − 2x4 = −1x2 + 2x3 + 3x4 = 6
⇒{
x1 = x3 + 2x4 − 2x2 = −2x3 − 3x4 + 6
and the general solution is given byx1 = t + 2s − 2x2 = −2t − 3s + 6x3 = tx4 = s
(t, s ∈ R)
In vector form,
(x1, x2, x3, x4) = (−2, 6, 0, 0) + t(1,−2, 1, 0) + s(2,−3, 0, 1)
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 710: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/710.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Row echelon form. Gauss-Jordan Reduction
Back to the system:{x1 − x3 − 2x4 = −1x2 + 2x3 + 3x4 = 6
⇒
{x1 = x3 + 2x4 − 2
x2 = −2x3 − 3x4 + 6
and the general solution is given byx1 = t + 2s − 2x2 = −2t − 3s + 6x3 = tx4 = s
(t, s ∈ R)
In vector form,
(x1, x2, x3, x4) = (−2, 6, 0, 0) + t(1,−2, 1, 0) + s(2,−3, 0, 1)
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 711: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/711.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Row echelon form. Gauss-Jordan Reduction
Back to the system:{x1 − x3 − 2x4 = −1x2 + 2x3 + 3x4 = 6
⇒{
x1 = x3 + 2x4 − 2x2 = −2x3 − 3x4 + 6
and the general solution is given byx1 = t + 2s − 2x2 = −2t − 3s + 6x3 = tx4 = s
(t, s ∈ R)
In vector form,
(x1, x2, x3, x4) = (−2, 6, 0, 0) + t(1,−2, 1, 0) + s(2,−3, 0, 1)
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 712: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/712.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Row echelon form. Gauss-Jordan Reduction
Back to the system:{x1 − x3 − 2x4 = −1x2 + 2x3 + 3x4 = 6
⇒{
x1 = x3 + 2x4 − 2x2 = −2x3 − 3x4 + 6
and the general solution is given by
x1 = t + 2s − 2x2 = −2t − 3s + 6x3 = tx4 = s
(t, s ∈ R)
In vector form,
(x1, x2, x3, x4) = (−2, 6, 0, 0) + t(1,−2, 1, 0) + s(2,−3, 0, 1)
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 713: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/713.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Row echelon form. Gauss-Jordan Reduction
Back to the system:{x1 − x3 − 2x4 = −1x2 + 2x3 + 3x4 = 6
⇒{
x1 = x3 + 2x4 − 2x2 = −2x3 − 3x4 + 6
and the general solution is given byx1 = t + 2s − 2x2 = −2t − 3s + 6x3 = tx4 = s
(t, s ∈ R)
In vector form,
(x1, x2, x3, x4) = (−2, 6, 0, 0) + t(1,−2, 1, 0) + s(2,−3, 0, 1)
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 714: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/714.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Row echelon form. Gauss-Jordan Reduction
Back to the system:{x1 − x3 − 2x4 = −1x2 + 2x3 + 3x4 = 6
⇒{
x1 = x3 + 2x4 − 2x2 = −2x3 − 3x4 + 6
and the general solution is given byx1 = t + 2s − 2x2 = −2t − 3s + 6x3 = tx4 = s
(t, s ∈ R)
In vector form,
(x1, x2, x3, x4) = (−2, 6, 0, 0) + t(1,−2, 1, 0) + s(2,−3, 0, 1)
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 715: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/715.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Row echelon form. Gauss-Jordan Reduction
Back to the system:{x1 − x3 − 2x4 = −1x2 + 2x3 + 3x4 = 6
⇒{
x1 = x3 + 2x4 − 2x2 = −2x3 − 3x4 + 6
and the general solution is given byx1 = t + 2s − 2x2 = −2t − 3s + 6x3 = tx4 = s
(t, s ∈ R)
In vector form,
(x1, x2, x3, x4) = (−2, 6, 0, 0) + t(1,−2, 1, 0) + s(2,−3, 0, 1)
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 716: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/716.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Row echelon form. Gauss-Jordan Reduction
Example 1.8 y + 3z = 0
x + y − 2z = 0x + 2y + az = 0
a ∈ R
The system is homogeneous (all right-hand sides are zeros).Therefore it is consistent ( x = y = z = 0 is a solution )The augmented matrix is: 0 1 3 0
1 1 2 01 2 a 0
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 717: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/717.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Row echelon form. Gauss-Jordan Reduction
Example 1.8
y + 3z = 0
x + y − 2z = 0x + 2y + az = 0
a ∈ R
The system is homogeneous (all right-hand sides are zeros).Therefore it is consistent ( x = y = z = 0 is a solution )The augmented matrix is: 0 1 3 0
1 1 2 01 2 a 0
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 718: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/718.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Row echelon form. Gauss-Jordan Reduction
Example 1.8 y + 3z = 0
x + y − 2z = 0x + 2y + az = 0
a ∈ R
The system is homogeneous (all right-hand sides are zeros).Therefore it is consistent ( x = y = z = 0 is a solution )The augmented matrix is: 0 1 3 0
1 1 2 01 2 a 0
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 719: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/719.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Row echelon form. Gauss-Jordan Reduction
Example 1.8 y + 3z = 0
x + y − 2z = 0x + 2y + az = 0
a ∈ R
The system is homogeneous (all right-hand sides are zeros).Therefore it is consistent ( x = y = z = 0 is a solution )The augmented matrix is: 0 1 3 0
1 1 2 01 2 a 0
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 720: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/720.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Row echelon form. Gauss-Jordan Reduction
Example 1.8 y + 3z = 0
x + y − 2z = 0x + 2y + az = 0
a ∈ R
The system
is homogeneous (all right-hand sides are zeros).Therefore it is consistent ( x = y = z = 0 is a solution )The augmented matrix is: 0 1 3 0
1 1 2 01 2 a 0
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 721: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/721.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Row echelon form. Gauss-Jordan Reduction
Example 1.8 y + 3z = 0
x + y − 2z = 0x + 2y + az = 0
a ∈ R
The system is homogeneous
(all right-hand sides are zeros).Therefore it is consistent ( x = y = z = 0 is a solution )The augmented matrix is: 0 1 3 0
1 1 2 01 2 a 0
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 722: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/722.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Row echelon form. Gauss-Jordan Reduction
Example 1.8 y + 3z = 0
x + y − 2z = 0x + 2y + az = 0
a ∈ R
The system is homogeneous (all right-hand sides are zeros).
Therefore it is consistent ( x = y = z = 0 is a solution )The augmented matrix is: 0 1 3 0
1 1 2 01 2 a 0
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 723: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/723.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Row echelon form. Gauss-Jordan Reduction
Example 1.8 y + 3z = 0
x + y − 2z = 0x + 2y + az = 0
a ∈ R
The system is homogeneous (all right-hand sides are zeros).Therefore
it is consistent ( x = y = z = 0 is a solution )The augmented matrix is: 0 1 3 0
1 1 2 01 2 a 0
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 724: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/724.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Row echelon form. Gauss-Jordan Reduction
Example 1.8 y + 3z = 0
x + y − 2z = 0x + 2y + az = 0
a ∈ R
The system is homogeneous (all right-hand sides are zeros).Therefore it is consistent
( x = y = z = 0 is a solution )The augmented matrix is: 0 1 3 0
1 1 2 01 2 a 0
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 725: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/725.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Row echelon form. Gauss-Jordan Reduction
Example 1.8 y + 3z = 0
x + y − 2z = 0x + 2y + az = 0
a ∈ R
The system is homogeneous (all right-hand sides are zeros).Therefore it is consistent ( x = y = z = 0 is a solution )
The augmented matrix is: 0 1 3 01 1 2 01 2 a 0
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 726: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/726.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Row echelon form. Gauss-Jordan Reduction
Example 1.8 y + 3z = 0
x + y − 2z = 0x + 2y + az = 0
a ∈ R
The system is homogeneous (all right-hand sides are zeros).Therefore it is consistent ( x = y = z = 0 is a solution )The augmented matrix is:
0 1 3 01 1 2 01 2 a 0
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 727: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/727.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Row echelon form. Gauss-Jordan Reduction
Example 1.8 y + 3z = 0
x + y − 2z = 0x + 2y + az = 0
a ∈ R
The system is homogeneous (all right-hand sides are zeros).Therefore it is consistent ( x = y = z = 0 is a solution )The augmented matrix is: 0 1 3 0
1 1 2 01 2 a 0
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 728: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/728.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Row echelon form. Gauss-Jordan Reduction
Since the 1st row cannot serve as a pivotal one, we interchange itwith the 2nd row: 0 1 3 0
1 1 −2 01 2 a 0
⇒
1 1 −2 00 1 3 01 2 a 0
Now we can start the elimination. First subtract the 1st row fromthe 3rd row: 1 1 −2 0
0 1 3 01 2 a 0
⇒
1 1 −2 00 1 3 00 1 a + 2 0
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 729: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/729.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Row echelon form. Gauss-Jordan Reduction
Since
the 1st row cannot serve as a pivotal one, we interchange itwith the 2nd row: 0 1 3 0
1 1 −2 01 2 a 0
⇒
1 1 −2 00 1 3 01 2 a 0
Now we can start the elimination. First subtract the 1st row fromthe 3rd row: 1 1 −2 0
0 1 3 01 2 a 0
⇒
1 1 −2 00 1 3 00 1 a + 2 0
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 730: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/730.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Row echelon form. Gauss-Jordan Reduction
Since the 1st row
cannot serve as a pivotal one, we interchange itwith the 2nd row: 0 1 3 0
1 1 −2 01 2 a 0
⇒
1 1 −2 00 1 3 01 2 a 0
Now we can start the elimination. First subtract the 1st row fromthe 3rd row: 1 1 −2 0
0 1 3 01 2 a 0
⇒
1 1 −2 00 1 3 00 1 a + 2 0
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 731: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/731.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Row echelon form. Gauss-Jordan Reduction
Since the 1st row cannot serve
as a pivotal one, we interchange itwith the 2nd row: 0 1 3 0
1 1 −2 01 2 a 0
⇒
1 1 −2 00 1 3 01 2 a 0
Now we can start the elimination. First subtract the 1st row fromthe 3rd row: 1 1 −2 0
0 1 3 01 2 a 0
⇒
1 1 −2 00 1 3 00 1 a + 2 0
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 732: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/732.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Row echelon form. Gauss-Jordan Reduction
Since the 1st row cannot serve as a pivotal one,
we interchange itwith the 2nd row: 0 1 3 0
1 1 −2 01 2 a 0
⇒
1 1 −2 00 1 3 01 2 a 0
Now we can start the elimination. First subtract the 1st row fromthe 3rd row: 1 1 −2 0
0 1 3 01 2 a 0
⇒
1 1 −2 00 1 3 00 1 a + 2 0
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 733: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/733.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Row echelon form. Gauss-Jordan Reduction
Since the 1st row cannot serve as a pivotal one, we interchange it
with the 2nd row: 0 1 3 01 1 −2 01 2 a 0
⇒
1 1 −2 00 1 3 01 2 a 0
Now we can start the elimination. First subtract the 1st row fromthe 3rd row: 1 1 −2 0
0 1 3 01 2 a 0
⇒
1 1 −2 00 1 3 00 1 a + 2 0
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 734: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/734.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Row echelon form. Gauss-Jordan Reduction
Since the 1st row cannot serve as a pivotal one, we interchange it
with the 2nd row: 0 1 3 01 1 −2 01 2 a 0
⇒
1 1 −2 00 1 3 01 2 a 0
Now we can start the elimination. First subtract the 1st row fromthe 3rd row: 1 1 −2 0
0 1 3 01 2 a 0
⇒
1 1 −2 00 1 3 00 1 a + 2 0
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 735: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/735.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Row echelon form. Gauss-Jordan Reduction
Since the 1st row cannot serve as a pivotal one, we interchange itwith the 2nd row:
0 1 3 01 1 −2 01 2 a 0
⇒
1 1 −2 00 1 3 01 2 a 0
Now we can start the elimination. First subtract the 1st row fromthe 3rd row: 1 1 −2 0
0 1 3 01 2 a 0
⇒
1 1 −2 00 1 3 00 1 a + 2 0
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 736: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/736.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Row echelon form. Gauss-Jordan Reduction
Since the 1st row cannot serve as a pivotal one, we interchange itwith the 2nd row: 0 1 3 0
1 1 −2 01 2 a 0
⇒
1 1 −2 00 1 3 01 2 a 0
Now we can start the elimination. First subtract the 1st row fromthe 3rd row: 1 1 −2 0
0 1 3 01 2 a 0
⇒
1 1 −2 00 1 3 00 1 a + 2 0
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 737: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/737.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Row echelon form. Gauss-Jordan Reduction
Since the 1st row cannot serve as a pivotal one, we interchange itwith the 2nd row: 0 1 3 0
1 1 −2 01 2 a 0
⇒
1 1 −2 00 1 3 01 2 a 0
Now we can start the elimination. First subtract the 1st row fromthe 3rd row: 1 1 −2 0
0 1 3 01 2 a 0
⇒
1 1 −2 00 1 3 00 1 a + 2 0
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 738: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/738.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Row echelon form. Gauss-Jordan Reduction
Since the 1st row cannot serve as a pivotal one, we interchange itwith the 2nd row: 0 1 3 0
1 1 −2 01 2 a 0
⇒
1 1 −2 00 1 3 01 2 a 0
Now
we can start the elimination. First subtract the 1st row fromthe 3rd row: 1 1 −2 0
0 1 3 01 2 a 0
⇒
1 1 −2 00 1 3 00 1 a + 2 0
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 739: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/739.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Row echelon form. Gauss-Jordan Reduction
Since the 1st row cannot serve as a pivotal one, we interchange itwith the 2nd row: 0 1 3 0
1 1 −2 01 2 a 0
⇒
1 1 −2 00 1 3 01 2 a 0
Now we can start the elimination.
First subtract the 1st row fromthe 3rd row: 1 1 −2 0
0 1 3 01 2 a 0
⇒
1 1 −2 00 1 3 00 1 a + 2 0
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 740: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/740.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Row echelon form. Gauss-Jordan Reduction
Since the 1st row cannot serve as a pivotal one, we interchange itwith the 2nd row: 0 1 3 0
1 1 −2 01 2 a 0
⇒
1 1 −2 00 1 3 01 2 a 0
Now we can start the elimination. First
subtract the 1st row fromthe 3rd row: 1 1 −2 0
0 1 3 01 2 a 0
⇒
1 1 −2 00 1 3 00 1 a + 2 0
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 741: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/741.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Row echelon form. Gauss-Jordan Reduction
Since the 1st row cannot serve as a pivotal one, we interchange itwith the 2nd row: 0 1 3 0
1 1 −2 01 2 a 0
⇒
1 1 −2 00 1 3 01 2 a 0
Now we can start the elimination. First subtract the 1st row
fromthe 3rd row: 1 1 −2 0
0 1 3 01 2 a 0
⇒
1 1 −2 00 1 3 00 1 a + 2 0
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 742: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/742.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Row echelon form. Gauss-Jordan Reduction
Since the 1st row cannot serve as a pivotal one, we interchange itwith the 2nd row: 0 1 3 0
1 1 −2 01 2 a 0
⇒
1 1 −2 00 1 3 01 2 a 0
Now we can start the elimination. First subtract the 1st row fromthe 3rd row:
1 1 −2 00 1 3 01 2 a 0
⇒
1 1 −2 00 1 3 00 1 a + 2 0
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 743: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/743.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Row echelon form. Gauss-Jordan Reduction
Since the 1st row cannot serve as a pivotal one, we interchange itwith the 2nd row: 0 1 3 0
1 1 −2 01 2 a 0
⇒
1 1 −2 00 1 3 01 2 a 0
Now we can start the elimination. First subtract the 1st row fromthe 3rd row: 1 1 −2 0
0 1 3 01 2 a 0
⇒
1 1 −2 00 1 3 00 1 a + 2 0
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 744: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/744.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Row echelon form. Gauss-Jordan Reduction
Since the 1st row cannot serve as a pivotal one, we interchange itwith the 2nd row: 0 1 3 0
1 1 −2 01 2 a 0
⇒
1 1 −2 00 1 3 01 2 a 0
Now we can start the elimination. First subtract the 1st row fromthe 3rd row: 1 1 −2 0
0 1 3 01 2 a 0
⇒
1 1 −2 00 1 3 00 1 a + 2 0
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 745: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/745.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Row echelon form. Gauss-Jordan Reduction
The 2nd row is our new pivotal row. Subtract the 2nd row fromthe 3rd row: 1 1 −2 0
0 1 3 00 1 a + 2 0
⇒
1 1 −2 00 1 3 00 0 a− 1 0
At this point row reduction splits into two cases.Case 1: a 6= 1. In this case, multiply the 3rd row by (a− 1)−1 1 1 −2 0
0 1 3 00 0 a− 1 0
⇒
1 1 −2 00 1 3 00 0 1 0
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 746: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/746.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Row echelon form. Gauss-Jordan Reduction
The 2nd row
is our new pivotal row. Subtract the 2nd row fromthe 3rd row: 1 1 −2 0
0 1 3 00 1 a + 2 0
⇒
1 1 −2 00 1 3 00 0 a− 1 0
At this point row reduction splits into two cases.Case 1: a 6= 1. In this case, multiply the 3rd row by (a− 1)−1 1 1 −2 0
0 1 3 00 0 a− 1 0
⇒
1 1 −2 00 1 3 00 0 1 0
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 747: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/747.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Row echelon form. Gauss-Jordan Reduction
The 2nd row is our new pivotal row.
Subtract the 2nd row fromthe 3rd row: 1 1 −2 0
0 1 3 00 1 a + 2 0
⇒
1 1 −2 00 1 3 00 0 a− 1 0
At this point row reduction splits into two cases.Case 1: a 6= 1. In this case, multiply the 3rd row by (a− 1)−1 1 1 −2 0
0 1 3 00 0 a− 1 0
⇒
1 1 −2 00 1 3 00 0 1 0
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 748: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/748.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Row echelon form. Gauss-Jordan Reduction
The 2nd row is our new pivotal row. Subtract the 2nd row
fromthe 3rd row: 1 1 −2 0
0 1 3 00 1 a + 2 0
⇒
1 1 −2 00 1 3 00 0 a− 1 0
At this point row reduction splits into two cases.Case 1: a 6= 1. In this case, multiply the 3rd row by (a− 1)−1 1 1 −2 0
0 1 3 00 0 a− 1 0
⇒
1 1 −2 00 1 3 00 0 1 0
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 749: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/749.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Row echelon form. Gauss-Jordan Reduction
The 2nd row is our new pivotal row. Subtract the 2nd row fromthe 3rd row:
1 1 −2 00 1 3 00 1 a + 2 0
⇒
1 1 −2 00 1 3 00 0 a− 1 0
At this point row reduction splits into two cases.Case 1: a 6= 1. In this case, multiply the 3rd row by (a− 1)−1 1 1 −2 0
0 1 3 00 0 a− 1 0
⇒
1 1 −2 00 1 3 00 0 1 0
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 750: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/750.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Row echelon form. Gauss-Jordan Reduction
The 2nd row is our new pivotal row. Subtract the 2nd row fromthe 3rd row: 1 1 −2 0
0 1 3 00 1 a + 2 0
⇒
1 1 −2 00 1 3 00 0 a− 1 0
At this point row reduction splits into two cases.Case 1: a 6= 1. In this case, multiply the 3rd row by (a− 1)−1 1 1 −2 0
0 1 3 00 0 a− 1 0
⇒
1 1 −2 00 1 3 00 0 1 0
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 751: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/751.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Row echelon form. Gauss-Jordan Reduction
The 2nd row is our new pivotal row. Subtract the 2nd row fromthe 3rd row: 1 1 −2 0
0 1 3 00 1 a + 2 0
⇒
1 1 −2 00 1 3 00 0 a− 1 0
At this point row reduction splits into two cases.Case 1: a 6= 1. In this case, multiply the 3rd row by (a− 1)−1 1 1 −2 0
0 1 3 00 0 a− 1 0
⇒
1 1 −2 00 1 3 00 0 1 0
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 752: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/752.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Row echelon form. Gauss-Jordan Reduction
The 2nd row is our new pivotal row. Subtract the 2nd row fromthe 3rd row: 1 1 −2 0
0 1 3 00 1 a + 2 0
⇒
1 1 −2 00 1 3 00 0 a− 1 0
At this point
row reduction splits into two cases.Case 1: a 6= 1. In this case, multiply the 3rd row by (a− 1)−1 1 1 −2 0
0 1 3 00 0 a− 1 0
⇒
1 1 −2 00 1 3 00 0 1 0
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 753: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/753.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Row echelon form. Gauss-Jordan Reduction
The 2nd row is our new pivotal row. Subtract the 2nd row fromthe 3rd row: 1 1 −2 0
0 1 3 00 1 a + 2 0
⇒
1 1 −2 00 1 3 00 0 a− 1 0
At this point row reduction splits
into two cases.Case 1: a 6= 1. In this case, multiply the 3rd row by (a− 1)−1 1 1 −2 0
0 1 3 00 0 a− 1 0
⇒
1 1 −2 00 1 3 00 0 1 0
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 754: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/754.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Row echelon form. Gauss-Jordan Reduction
The 2nd row is our new pivotal row. Subtract the 2nd row fromthe 3rd row: 1 1 −2 0
0 1 3 00 1 a + 2 0
⇒
1 1 −2 00 1 3 00 0 a− 1 0
At this point row reduction splits into two cases.
Case 1: a 6= 1. In this case, multiply the 3rd row by (a− 1)−1 1 1 −2 00 1 3 00 0 a− 1 0
⇒
1 1 −2 00 1 3 00 0 1 0
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 755: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/755.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Row echelon form. Gauss-Jordan Reduction
The 2nd row is our new pivotal row. Subtract the 2nd row fromthe 3rd row: 1 1 −2 0
0 1 3 00 1 a + 2 0
⇒
1 1 −2 00 1 3 00 0 a− 1 0
At this point row reduction splits into two cases.Case 1: a 6= 1. In this case, multiply the 3rd row by (a− 1)−1
1 1 −2 00 1 3 00 0 a− 1 0
⇒
1 1 −2 00 1 3 00 0 1 0
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 756: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/756.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Row echelon form. Gauss-Jordan Reduction
The 2nd row is our new pivotal row. Subtract the 2nd row fromthe 3rd row: 1 1 −2 0
0 1 3 00 1 a + 2 0
⇒
1 1 −2 00 1 3 00 0 a− 1 0
At this point row reduction splits into two cases.Case 1: a 6= 1. In this case, multiply the 3rd row by (a− 1)−1 1 1 −2 0
0 1 3 00 0 a− 1 0
⇒
1 1 −2 00 1 3 00 0 1 0
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 757: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/757.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Row echelon form. Gauss-Jordan Reduction
The 2nd row is our new pivotal row. Subtract the 2nd row fromthe 3rd row: 1 1 −2 0
0 1 3 00 1 a + 2 0
⇒
1 1 −2 00 1 3 00 0 a− 1 0
At this point row reduction splits into two cases.Case 1: a 6= 1. In this case, multiply the 3rd row by (a− 1)−1 1 1 −2 0
0 1 3 00 0 a− 1 0
⇒
1 1 −2 00 1 3 00 0 1 0
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 758: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/758.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Row echelon form. Gauss-Jordan Reduction
The matrix is converted into row echelon form. We proceedtowards reduced row echelon form. Subtract 3 times the 3rd rowfrom the 2nd row: 1 1 −2 0
0 1 3 00 0 1 0
⇒
1 1 −2 00 1 0 00 0 1 0
Add 2 times the 3rd row to the 1st row: 1 1 −2 0
0 1 0 00 0 1 0
⇒
1 1 0 00 1 0 00 0 1 0
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 759: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/759.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Row echelon form. Gauss-Jordan Reduction
The matrix
is converted into row echelon form. We proceedtowards reduced row echelon form. Subtract 3 times the 3rd rowfrom the 2nd row: 1 1 −2 0
0 1 3 00 0 1 0
⇒
1 1 −2 00 1 0 00 0 1 0
Add 2 times the 3rd row to the 1st row: 1 1 −2 0
0 1 0 00 0 1 0
⇒
1 1 0 00 1 0 00 0 1 0
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 760: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/760.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Row echelon form. Gauss-Jordan Reduction
The matrix is converted
into row echelon form. We proceedtowards reduced row echelon form. Subtract 3 times the 3rd rowfrom the 2nd row: 1 1 −2 0
0 1 3 00 0 1 0
⇒
1 1 −2 00 1 0 00 0 1 0
Add 2 times the 3rd row to the 1st row: 1 1 −2 0
0 1 0 00 0 1 0
⇒
1 1 0 00 1 0 00 0 1 0
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 761: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/761.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Row echelon form. Gauss-Jordan Reduction
The matrix is converted into row echelon form.
We proceedtowards reduced row echelon form. Subtract 3 times the 3rd rowfrom the 2nd row: 1 1 −2 0
0 1 3 00 0 1 0
⇒
1 1 −2 00 1 0 00 0 1 0
Add 2 times the 3rd row to the 1st row: 1 1 −2 0
0 1 0 00 0 1 0
⇒
1 1 0 00 1 0 00 0 1 0
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 762: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/762.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Row echelon form. Gauss-Jordan Reduction
The matrix is converted into row echelon form. We proceedtowards
reduced row echelon form. Subtract 3 times the 3rd rowfrom the 2nd row: 1 1 −2 0
0 1 3 00 0 1 0
⇒
1 1 −2 00 1 0 00 0 1 0
Add 2 times the 3rd row to the 1st row: 1 1 −2 0
0 1 0 00 0 1 0
⇒
1 1 0 00 1 0 00 0 1 0
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 763: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/763.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Row echelon form. Gauss-Jordan Reduction
The matrix is converted into row echelon form. We proceedtowards reduced row echelon form.
Subtract 3 times the 3rd rowfrom the 2nd row: 1 1 −2 0
0 1 3 00 0 1 0
⇒
1 1 −2 00 1 0 00 0 1 0
Add 2 times the 3rd row to the 1st row: 1 1 −2 0
0 1 0 00 0 1 0
⇒
1 1 0 00 1 0 00 0 1 0
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 764: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/764.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Row echelon form. Gauss-Jordan Reduction
The matrix is converted into row echelon form. We proceedtowards reduced row echelon form. Subtract 3 times the 3rd rowfrom the 2nd row:
1 1 −2 00 1 3 00 0 1 0
⇒
1 1 −2 00 1 0 00 0 1 0
Add 2 times the 3rd row to the 1st row: 1 1 −2 0
0 1 0 00 0 1 0
⇒
1 1 0 00 1 0 00 0 1 0
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 765: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/765.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Row echelon form. Gauss-Jordan Reduction
The matrix is converted into row echelon form. We proceedtowards reduced row echelon form. Subtract 3 times the 3rd rowfrom the 2nd row: 1 1 −2 0
0 1 3 00 0 1 0
⇒
1 1 −2 00 1 0 00 0 1 0
Add 2 times the 3rd row to the 1st row: 1 1 −2 0
0 1 0 00 0 1 0
⇒
1 1 0 00 1 0 00 0 1 0
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 766: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/766.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Row echelon form. Gauss-Jordan Reduction
The matrix is converted into row echelon form. We proceedtowards reduced row echelon form. Subtract 3 times the 3rd rowfrom the 2nd row: 1 1 −2 0
0 1 3 00 0 1 0
⇒
1 1 −2 00 1 0 00 0 1 0
Add 2 times the 3rd row to the 1st row: 1 1 −2 00 1 0 00 0 1 0
⇒
1 1 0 00 1 0 00 0 1 0
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 767: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/767.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Row echelon form. Gauss-Jordan Reduction
The matrix is converted into row echelon form. We proceedtowards reduced row echelon form. Subtract 3 times the 3rd rowfrom the 2nd row: 1 1 −2 0
0 1 3 00 0 1 0
⇒
1 1 −2 00 1 0 00 0 1 0
Add 2 times
the 3rd row to the 1st row: 1 1 −2 00 1 0 00 0 1 0
⇒
1 1 0 00 1 0 00 0 1 0
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 768: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/768.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Row echelon form. Gauss-Jordan Reduction
The matrix is converted into row echelon form. We proceedtowards reduced row echelon form. Subtract 3 times the 3rd rowfrom the 2nd row: 1 1 −2 0
0 1 3 00 0 1 0
⇒
1 1 −2 00 1 0 00 0 1 0
Add 2 times the 3rd row
to the 1st row: 1 1 −2 00 1 0 00 0 1 0
⇒
1 1 0 00 1 0 00 0 1 0
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 769: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/769.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Row echelon form. Gauss-Jordan Reduction
The matrix is converted into row echelon form. We proceedtowards reduced row echelon form. Subtract 3 times the 3rd rowfrom the 2nd row: 1 1 −2 0
0 1 3 00 0 1 0
⇒
1 1 −2 00 1 0 00 0 1 0
Add 2 times the 3rd row to the 1st row:
1 1 −2 00 1 0 00 0 1 0
⇒
1 1 0 00 1 0 00 0 1 0
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 770: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/770.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Row echelon form. Gauss-Jordan Reduction
The matrix is converted into row echelon form. We proceedtowards reduced row echelon form. Subtract 3 times the 3rd rowfrom the 2nd row: 1 1 −2 0
0 1 3 00 0 1 0
⇒
1 1 −2 00 1 0 00 0 1 0
Add 2 times the 3rd row to the 1st row: 1 1 −2 0
0 1 0 00 0 1 0
⇒
1 1 0 00 1 0 00 0 1 0
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 771: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/771.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Row echelon form. Gauss-Jordan Reduction
The matrix is converted into row echelon form. We proceedtowards reduced row echelon form. Subtract 3 times the 3rd rowfrom the 2nd row: 1 1 −2 0
0 1 3 00 0 1 0
⇒
1 1 −2 00 1 0 00 0 1 0
Add 2 times the 3rd row to the 1st row: 1 1 −2 0
0 1 0 00 0 1 0
⇒
1 1 0 00 1 0 00 0 1 0
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 772: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/772.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Row echelon form. Gauss-Jordan Reduction
Finally,subtract the 2nd row from the 1st row: 1 1 0 00 1 0 00 0 1 0
⇒
1 9 0 00 1 0 00 0 1 0
Thus x = y = z = 0 is the only solution
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 773: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/773.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Row echelon form. Gauss-Jordan Reduction
Finally,
subtract the 2nd row from the 1st row: 1 1 0 00 1 0 00 0 1 0
⇒
1 9 0 00 1 0 00 0 1 0
Thus x = y = z = 0 is the only solution
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 774: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/774.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Row echelon form. Gauss-Jordan Reduction
Finally,subtract the 2nd row
from the 1st row: 1 1 0 00 1 0 00 0 1 0
⇒
1 9 0 00 1 0 00 0 1 0
Thus x = y = z = 0 is the only solution
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 775: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/775.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Row echelon form. Gauss-Jordan Reduction
Finally,subtract the 2nd row from the 1st row:
1 1 0 00 1 0 00 0 1 0
⇒
1 9 0 00 1 0 00 0 1 0
Thus x = y = z = 0 is the only solution
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 776: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/776.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Row echelon form. Gauss-Jordan Reduction
Finally,subtract the 2nd row from the 1st row: 1 1 0 00 1 0 00 0 1 0
⇒
1 9 0 00 1 0 00 0 1 0
Thus x = y = z = 0 is the only solution
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 777: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/777.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Row echelon form. Gauss-Jordan Reduction
Finally,subtract the 2nd row from the 1st row: 1 1 0 00 1 0 00 0 1 0
⇒
1 9 0 00 1 0 00 0 1 0
Thus x = y = z = 0 is the only solution
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 778: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/778.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Row echelon form. Gauss-Jordan Reduction
Finally,subtract the 2nd row from the 1st row: 1 1 0 00 1 0 00 0 1 0
⇒
1 9 0 00 1 0 00 0 1 0
Thus x = y = z = 0
is the only solution
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 779: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/779.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Row echelon form. Gauss-Jordan Reduction
Finally,subtract the 2nd row from the 1st row: 1 1 0 00 1 0 00 0 1 0
⇒
1 9 0 00 1 0 00 0 1 0
Thus x = y = z = 0 is the only solution
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 780: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/780.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Row echelon form. Gauss-Jordan Reduction
Case 2: a = 1. In this case, the matrix is already in row echelonform: 1 1 −2 0
0 1 3 00 0 a− 1 0
⇒
1 1 −2 00 1 3 00 0 0 0
To get reduced row echelon form, subtract the 2nd row from the1st row: 1 1 −2 0
0 1 3 00 0 0 0
⇒
1 0 −5 00 1 3 00 0 0 0
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 781: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/781.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Row echelon form. Gauss-Jordan Reduction
Case 2: a = 1. In this case, the matrix is already in row echelonform:
1 1 −2 00 1 3 00 0 a− 1 0
⇒
1 1 −2 00 1 3 00 0 0 0
To get reduced row echelon form, subtract the 2nd row from the1st row: 1 1 −2 0
0 1 3 00 0 0 0
⇒
1 0 −5 00 1 3 00 0 0 0
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 782: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/782.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Row echelon form. Gauss-Jordan Reduction
Case 2: a = 1. In this case, the matrix is already in row echelonform: 1 1 −2 0
0 1 3 00 0 a− 1 0
⇒
1 1 −2 00 1 3 00 0 0 0
To get reduced row echelon form, subtract the 2nd row from the1st row: 1 1 −2 0
0 1 3 00 0 0 0
⇒
1 0 −5 00 1 3 00 0 0 0
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 783: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/783.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Row echelon form. Gauss-Jordan Reduction
Case 2: a = 1. In this case, the matrix is already in row echelonform: 1 1 −2 0
0 1 3 00 0 a− 1 0
⇒
1 1 −2 00 1 3 00 0 0 0
To get reduced row echelon form, subtract the 2nd row from the1st row: 1 1 −2 0
0 1 3 00 0 0 0
⇒
1 0 −5 00 1 3 00 0 0 0
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 784: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/784.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Row echelon form. Gauss-Jordan Reduction
Case 2: a = 1. In this case, the matrix is already in row echelonform: 1 1 −2 0
0 1 3 00 0 a− 1 0
⇒
1 1 −2 00 1 3 00 0 0 0
To get
reduced row echelon form, subtract the 2nd row from the1st row: 1 1 −2 0
0 1 3 00 0 0 0
⇒
1 0 −5 00 1 3 00 0 0 0
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 785: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/785.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Row echelon form. Gauss-Jordan Reduction
Case 2: a = 1. In this case, the matrix is already in row echelonform: 1 1 −2 0
0 1 3 00 0 a− 1 0
⇒
1 1 −2 00 1 3 00 0 0 0
To get reduced row echelon form,
subtract the 2nd row from the1st row: 1 1 −2 0
0 1 3 00 0 0 0
⇒
1 0 −5 00 1 3 00 0 0 0
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 786: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/786.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Row echelon form. Gauss-Jordan Reduction
Case 2: a = 1. In this case, the matrix is already in row echelonform: 1 1 −2 0
0 1 3 00 0 a− 1 0
⇒
1 1 −2 00 1 3 00 0 0 0
To get reduced row echelon form, subtract the 2nd row
from the1st row: 1 1 −2 0
0 1 3 00 0 0 0
⇒
1 0 −5 00 1 3 00 0 0 0
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 787: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/787.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Row echelon form. Gauss-Jordan Reduction
Case 2: a = 1. In this case, the matrix is already in row echelonform: 1 1 −2 0
0 1 3 00 0 a− 1 0
⇒
1 1 −2 00 1 3 00 0 0 0
To get reduced row echelon form, subtract the 2nd row from the1st row:
1 1 −2 00 1 3 00 0 0 0
⇒
1 0 −5 00 1 3 00 0 0 0
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 788: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/788.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Row echelon form. Gauss-Jordan Reduction
Case 2: a = 1. In this case, the matrix is already in row echelonform: 1 1 −2 0
0 1 3 00 0 a− 1 0
⇒
1 1 −2 00 1 3 00 0 0 0
To get reduced row echelon form, subtract the 2nd row from the1st row: 1 1 −2 0
0 1 3 00 0 0 0
⇒
1 0 −5 00 1 3 00 0 0 0
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 789: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/789.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Row echelon form. Gauss-Jordan Reduction
Case 2: a = 1. In this case, the matrix is already in row echelonform: 1 1 −2 0
0 1 3 00 0 a− 1 0
⇒
1 1 −2 00 1 3 00 0 0 0
To get reduced row echelon form, subtract the 2nd row from the1st row: 1 1 −2 0
0 1 3 00 0 0 0
⇒
1 0 −5 00 1 3 00 0 0 0
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 790: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/790.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Row echelon form. Gauss-Jordan Reduction
Then, z is a free variable, and the solution is given by{x − 5z = 0y + 3z = 0
⇒{
x = 5zy = −3z
Thus, the System of linear equations:x + 3z = 0
x + y − 2z = 0x + 2y + az = 0
has as a solution:
if a 6= 1 then (x , y , z) = (0, 0, 0).if a = 1 then (x , y , z) = t(5,−3, 1) t ∈ R.
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 791: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/791.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Row echelon form. Gauss-Jordan Reduction
Then,
z is a free variable, and the solution is given by{x − 5z = 0y + 3z = 0
⇒{
x = 5zy = −3z
Thus, the System of linear equations:x + 3z = 0
x + y − 2z = 0x + 2y + az = 0
has as a solution:
if a 6= 1 then (x , y , z) = (0, 0, 0).if a = 1 then (x , y , z) = t(5,−3, 1) t ∈ R.
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 792: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/792.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Row echelon form. Gauss-Jordan Reduction
Then, z is a free variable, and
the solution is given by{x − 5z = 0y + 3z = 0
⇒{
x = 5zy = −3z
Thus, the System of linear equations:x + 3z = 0
x + y − 2z = 0x + 2y + az = 0
has as a solution:
if a 6= 1 then (x , y , z) = (0, 0, 0).if a = 1 then (x , y , z) = t(5,−3, 1) t ∈ R.
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 793: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/793.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Row echelon form. Gauss-Jordan Reduction
Then, z is a free variable, and the solution is given by
{x − 5z = 0y + 3z = 0
⇒{
x = 5zy = −3z
Thus, the System of linear equations:x + 3z = 0
x + y − 2z = 0x + 2y + az = 0
has as a solution:
if a 6= 1 then (x , y , z) = (0, 0, 0).if a = 1 then (x , y , z) = t(5,−3, 1) t ∈ R.
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 794: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/794.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Row echelon form. Gauss-Jordan Reduction
Then, z is a free variable, and the solution is given by{x − 5z = 0y + 3z = 0
⇒
{x = 5zy = −3z
Thus, the System of linear equations:x + 3z = 0
x + y − 2z = 0x + 2y + az = 0
has as a solution:
if a 6= 1 then (x , y , z) = (0, 0, 0).if a = 1 then (x , y , z) = t(5,−3, 1) t ∈ R.
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 795: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/795.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Row echelon form. Gauss-Jordan Reduction
Then, z is a free variable, and the solution is given by{x − 5z = 0y + 3z = 0
⇒{
x = 5zy = −3z
Thus, the System of linear equations:
x + 3z = 0
x + y − 2z = 0x + 2y + az = 0
has as a solution:
if a 6= 1 then (x , y , z) = (0, 0, 0).if a = 1 then (x , y , z) = t(5,−3, 1) t ∈ R.
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 796: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/796.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Row echelon form. Gauss-Jordan Reduction
Then, z is a free variable, and the solution is given by{x − 5z = 0y + 3z = 0
⇒{
x = 5zy = −3z
Thus, the System of linear equations:x + 3z = 0
x + y − 2z = 0x + 2y + az = 0
has as a solution:
if a 6= 1 then (x , y , z) = (0, 0, 0).if a = 1 then (x , y , z) = t(5,−3, 1) t ∈ R.
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 797: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/797.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Row echelon form. Gauss-Jordan Reduction
Then, z is a free variable, and the solution is given by{x − 5z = 0y + 3z = 0
⇒{
x = 5zy = −3z
Thus, the System of linear equations:x + 3z = 0
x + y − 2z = 0x + 2y + az = 0
has as a solution:
if a 6= 1 then (x , y , z) = (0, 0, 0).if a = 1 then (x , y , z) = t(5,−3, 1) t ∈ R.
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 798: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/798.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Row echelon form. Gauss-Jordan Reduction
Then, z is a free variable, and the solution is given by{x − 5z = 0y + 3z = 0
⇒{
x = 5zy = −3z
Thus, the System of linear equations:x + 3z = 0
x + y − 2z = 0x + 2y + az = 0
has as a solution:
if a 6= 1
then (x , y , z) = (0, 0, 0).if a = 1 then (x , y , z) = t(5,−3, 1) t ∈ R.
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 799: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/799.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Row echelon form. Gauss-Jordan Reduction
Then, z is a free variable, and the solution is given by{x − 5z = 0y + 3z = 0
⇒{
x = 5zy = −3z
Thus, the System of linear equations:x + 3z = 0
x + y − 2z = 0x + 2y + az = 0
has as a solution:
if a 6= 1 then
(x , y , z) = (0, 0, 0).if a = 1 then (x , y , z) = t(5,−3, 1) t ∈ R.
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 800: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/800.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Row echelon form. Gauss-Jordan Reduction
Then, z is a free variable, and the solution is given by{x − 5z = 0y + 3z = 0
⇒{
x = 5zy = −3z
Thus, the System of linear equations:x + 3z = 0
x + y − 2z = 0x + 2y + az = 0
has as a solution:
if a 6= 1 then (x , y , z) = (0, 0, 0).
if a = 1 then (x , y , z) = t(5,−3, 1) t ∈ R.
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 801: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/801.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Row echelon form. Gauss-Jordan Reduction
Then, z is a free variable, and the solution is given by{x − 5z = 0y + 3z = 0
⇒{
x = 5zy = −3z
Thus, the System of linear equations:x + 3z = 0
x + y − 2z = 0x + 2y + az = 0
has as a solution:
if a 6= 1 then (x , y , z) = (0, 0, 0).if a = 1
then (x , y , z) = t(5,−3, 1) t ∈ R.
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 802: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/802.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Row echelon form. Gauss-Jordan Reduction
Then, z is a free variable, and the solution is given by{x − 5z = 0y + 3z = 0
⇒{
x = 5zy = −3z
Thus, the System of linear equations:x + 3z = 0
x + y − 2z = 0x + 2y + az = 0
has as a solution:
if a 6= 1 then (x , y , z) = (0, 0, 0).if a = 1 then
(x , y , z) = t(5,−3, 1) t ∈ R.
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 803: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/803.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Row echelon form. Gauss-Jordan Reduction
Then, z is a free variable, and the solution is given by{x − 5z = 0y + 3z = 0
⇒{
x = 5zy = −3z
Thus, the System of linear equations:x + 3z = 0
x + y − 2z = 0x + 2y + az = 0
has as a solution:
if a 6= 1 then (x , y , z) = (0, 0, 0).if a = 1 then (x , y , z) = t(5,−3, 1)
t ∈ R.
Dr. Marco A Roque Sol Linear Algebra. Session 1
![Page 804: Linear Algebra. Session 1roquesol/Math_304_Spring_2018_Session_1.pdf · For most students, it is a rst contact with modern mathematics. ... A Survey Linear algebra is a cornerstone](https://reader031.fdocuments.in/reader031/viewer/2022011803/5b74b1d07f8b9ac94a8c6078/html5/thumbnails/804.jpg)
Introduction
Table of ContentsA SurveySistems of linear equationsGaussian Elimination. Gauss-Jordan Reduction
Row echelon form. Gauss-Jordan Reduction
Then, z is a free variable, and the solution is given by{x − 5z = 0y + 3z = 0
⇒{
x = 5zy = −3z
Thus, the System of linear equations:x + 3z = 0
x + y − 2z = 0x + 2y + az = 0
has as a solution:
if a 6= 1 then (x , y , z) = (0, 0, 0).if a = 1 then (x , y , z) = t(5,−3, 1) t ∈ R.
Dr. Marco A Roque Sol Linear Algebra. Session 1