SUMMARY Linear Algebra and Its Applications 3ed

LINEAR ALGEBRA

AND ITS APPLICATIONS

THIRD EDITION UPDATE

David C. LayUniversity of Maryland – College Park

STUDY GUIDE

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Reproduced by Pearson Addison-Wesley from electronic files supplied by the author.

All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, ortransmitted, in any form or by any means, electronic, mechanical, photocopying, recording, orotherwise, without the prior written permission of the publisher. Printed in the United States ofAmerica.

ISBN 0-321-28066-0

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Brief ContentsINTRODUCTION

HOW TO STUDY LINEAR ALGEBRA

C H A P T E R 1 LINEAR EQUATIONS IN LINEAR ALGEBRA

C H A P T E R 2 MATRIX ALGEBRA

C H A P T E R 3 DETERMINANTS

C H A P T E R 4 VECTOR SPACES

C H A P T E R 5 EIGENVALUES AND EIGENVECTORS

C H A P T E R 6 ORTHOGONALITY AND LEAST SQUARES

C H A P T E R 7 SYMMETRIC MATRICES

A P P E N D I C E S TECHNOLOGY INDEX OF PROCEDURES AND TERMS

INTRODUCTION TO MATLAB

NOTES FOR THE MAPLE COMPUTER ALGEBRA SYSTEM

NOTES FOR THE MATHEMATICA COMPUTERALGEBRA SYSTEM

NOTES FOR THE TI-83+/86/89 GRAPHIC CALCULATORS

NOTES FOR THE HP-48G GRAPHIC CALCULATOR

ContentsINTRODUCTION viiTechnology Support vii

Review Materials on the Web viii

HOW TO STUDY LINEAR ALGEBRA ixStrategies for Success in Linear Algebra ix

C H A P T E R 1 LINEAR EQUATIONS IN LINEAR ALGEBRA1.1 Systems of Linear Systems 1-1

1.2 Row Reduction and Echelon Forms 1-7

1.3 Vector Equations 1-10

1.4 The Matrix Equation Ax = b 1-15

Mastering Linear Algebra Concepts: Span

1.5 Solution Sets of Linear Systems 1-20

1.6 Applications of Linear Systems 1-25

1.7 Linear Independence 1-29

Mastering Linear Algebra Concepts: Linear Independence

1.8 Introduction to Linear Transformations 1-33

Mastering Linear Algebra Concepts: Linear Transformation

1.9 Matrix of a Linear Transformation 1-37

Mastering Linear Algebra Concepts: Existence and Uniqueness

1.10 Linear Models in Business, Science, and Engineering 1-42

Supplementary Exercises 1-46

Glossary Checklist 1-47

C H A P T E R 2 MATRIX ALGEBRA2.1 Matrix Operations 2-1

2.2 The Inverse of a Matrix 2-4

2.3 Characterizations of Invertible Matrices 2-9

Expanded Table for the IMT 2-10

Mastering Linear Algebra Concepts: Reviewing and Reflecting

2.4 Partitioned Matrices 2-14

The Principles of Induction 2-20

2.5 Matrix Factorizations 2-20

Permuted LU Factorizations 2-24

2.6 The Leontief Input-Output Model 2-28

2.7 Applications to Computer Graphics 2-30

2.8 Subspaces of Rn 2-33

Mastering Linear Algebra Concepts: Subspace, Column Space, Null Space, Basis

2.9 Dimension and Rank 2-38

Expanded Table for the IMT 2-39

Mastering Linear Algebra Concepts: Dimension and Rank

Supplementary Exercises 2-42

Glossary Checklist 2-42

C H A P T E R 3 DETERMINANTS3.1 Introduction to Determinants 3-1

3.2 Properties of Determinants 3-5

3.3 Cramer’s Rule, Volume and Linear Transformations 3-8

A Geometric Proof 3-12

Glossary Checklist 3-13

C H A P T E R 4 VECTOR SPACES4.1 Vector Spaces and Subspaces 4-1

4.2 Null Spaces, Column Spaces, and Linear Transformations 4-4

Mastering Linear Algebra Concepts: Vector Space, Subspace, Col A and Nul A

4.3 Linearly Independent Sets; Bases 4-7

Mastering Linear Algebra Concepts: Basis

4.4 Coordinate Systems 4-11

Isomorphic Vector Spaces 4-12

4.5 The Dimension of a Vector Space 4-16

4.6 Rank 4-19

Expanded Table for the IMT 4-21

Mastering Linear Algebra Concepts: Major Review of Key Concepts

4.7 Change of Basis 4-24

4.8 Applications to Difference Equations 4-27

The Casorati Test 4-31

4.9 Applications to Markov Chains 4-32

Glossary Checklist 4-35

C H A P T E R 5 EIGENVALUES AND EIGENVECTORS5.1 Eigenvectors and Eigenvalues 5-1

5.2 The Characteristic Equation 5-5

Factoring a Polynomial 5-8

5.3 Diagonalization 5-9

Mastering Linear Algebra Concepts: Eigenvalue, Eigenvector, Eigenspace

5.4 Eigenvalues and Linear Transformations 5-16

5.5 Complex Eigenvalues 5-19

5.6 Discrete Dynamical Systems 5-22

5.7 Applications to Differential Equations 5-26

5.8 Iterative Estimates for Eigenvalues 5-29

Glossary Checklist 5-34

C H A P T E R 6 ORTHOGONALITY AND LEAST SQUARES6.1 Inner Product, Length, and Orthogonality 6-1

6.2 Orthogonal Sets 6-2

Mastering Linear Algebra Concepts: Orthogonal Basis

6.3 Orthogonal Projections 6-5

6.4 The Gram-Schmidt Process 6-9

6.5 Least-Squares Problems 6-12

6.6 Applications to Linear Models 6-16

The Geometrey of a Linear Model 6-19

6.7 Inner Product Spaces 6-20

6.8 Applications of Inner Product Spaces 6-22

The Linearity of an Orthoganal Projection 6-25

Glossary Checklist 6-26

C H A P T E R 7 SYMMETRIC MATRICES7.1 Diagonalization of Symmetric Matrices 7-1

7.2 Quadratic Forms 7-6

Mastering Linear Algebra Concepts: Diagonalization and Quadratic Forms

7.3 Constrained Optimization 7-8

7.4 The Singular Value Decomposition 7-10

Computing an SVD 7-11

7.5 Applications to Image Processing and Statistics 7-16

Supplementary Exercises 7-19

Glossary Checklist 7-19

A P P E N D I C E S TECHNOLOGY INDEX OF PROCEDURES AND TERMS

Technology Index of Procedures and Terms TECH-1

INTRODUCTION TO MATLABGetting Started with Matlab ML-1

Script M-Files ML-3

Index of Matlab Commands ML-5

NOTES FOR THE MAPLE COMPUTER ALGEBRA SYSTEMGetting Started with Maple MP-1

Study Guide Notes MP-4

Index of Maple Commands MP-22

NOTES FOR THE MATHEMATICA COMPUTER ALGEBRA SYSTEMGetting Started with Mathematica MM-1

Study Guide Notes MM-5

Index of Mathematica Commands MM-24

NOTES FOR THE TI-83+/86/89 GRAPHIC CALCULATORSGetting Started with a TI-83+ Calculator TI-1

Getting Started with a TI-86 Calculator TI-2

Getting Started with a TI-89 Calculator TI-3

Study Guide Notes TI-4

Index of TI Calculator Commands TI-29

NOTES FOR THE HP-48G GRAPHIC CALCULATORGetting Started with an HP-48G Calculator HP-1

Study Guide Notes HP-2

Index of HP Calculator Commands HP-18

SUMMARY Linear Algebra and Its Applications 3ed

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