1.1 Systems of Linear Equations1.2 Row Reduction and Echelon

Forms

Chapter 1: Linear Equations

Solve each system of equations

12

5 #1.

yx

yx

363

52 #2.

yx

yx

€

# 3. x + 2y + z = 5

−3x + y − z = 3

x + 3y =1

⎧ ⎨ ⎪

⎩ ⎪

Linear Equation:

System of linear equations: A collection of one or more linear equations involving the same variables

52 :Example xy

bxaxaxa nn 2211

0

5172 :Example

321

31

xxx

xx

numberscomplex or real are ,,, where 1 baa n

•Solution Set The set of all possible solutions to a

linear system.

•Two linear systems are called equivalent, if they have the same solution set.

.equivalent are 02

32&

042

624 :Ex

21

21

21

21

xx

xx

xx

xx

Matrix NotationAny linear system can be written as a

matrix (rectangular array) whose entries are the coefficients.

The size of a matrix tells how many rows and columns it has. (Ex: This is a 2x4 matrix.)

0111

51702

0111

51702

For the linear system:

The matrix

is called thecoefficient matrix, and the matrix

is called the augmented matrix.

111

1702

0

5172

321

31

xxx

xx

Algorithmic Procedure for Solving a Linear System

Basic idea: Replace one system with an equivalent system that is easier to solve.

Example: Solve the following system:

042

624

21

21

xx

xx

042

624

21

21

xx

xx

02

32

21

21

xx

xx

32

02

21

21

xx

xx

33

02

2

21

x

xx

1

02

2

21

x

xx

1

2

2

1

x

x

042

624

021

312

312

021

330

021

110

021

110

201

R1*(1/2) → R1R2*(-1/2) → R2

R1 ↔ R2

-2*R1+R2 → R2

R2*(1/3) → R2

2*R2+R1 → R1

/ 2/(-2)

*(-2)

+)

/3

Elementary Row Operations (Theorem 1.1.1)

360

341

452

360

452

341

360

230

341

360

3/210

3411

2

3

R1 R2 (-2)R1+R2 R2 R2/(-3) R2

Each of the following operations, performed on any linear system, produce a new linear system that is equivalent to the original.

1. Interchange two rows.2. Add a multiple of one row to another row.3. Multiply a row by any nonzero constant.

Gauss-Jordan elimination method to solve a system of linear equations

523

134

4452

Solve

321

321

321

xxx

xxx

xxx

5231

1341

4452Elementary RowOperations

100

010

001

2

2

3

200

200

300

321

321

321

xxx

xxx

xxx

2

2

3

3

2

1

x

x

x

Augmented matrix Reduced row-echelon form

5231

1341

4452

5231

4452

1341

4570

2230

1341

4570

3/23/210

1341

Practice with Elementary Row Operations

5231

2230

1341

5231

2230

1341

3/23/100

3/23/210

1341

2100

2010

3/113/101

Practice with Elementary Row Operations

2100

3/23/210

3/113/101

4570

3/23/210

1341

2100

3/23/210

1341

2100

3/23/210

1341

100

010

001

2

2

3

2100

2010

3/113/101

Practice with Elementary Row Operations

Gauss-Jordan elimination method to solve a system of linear equations

523

134

4452

Solve

321

321

321

xxx

xxx

xxx

5231

1341

4452Elementary RowOperations

100

010

001

2

2

3

200

200

300

321

321

321

xxx

xxx

xxx

2

2

3

3

2

1

x

x

x

Augmented matrix Reduced row-echelon form

Definition. A matrix is in reduced row-echelon form if it has the following four properties:

1. The leftmost nonzero entry in each row is 1. (If the row is not all zeros, this entry is called the leading 1.)

2. Each leading 1 is the only nonzero entry in its column. 3. Each leading 1 of a row is in a column to the right of the

leading 1 of the row above it.4. All nonzero rows are above any rows of all zeros.

Word ProblemsExample. A certain crazed zoologist is raising

experimental animals on a reserve near a toxic waste dump. There are four kinds of animals:

Neon dogs have one head, two tails, four eyes and four legs.

Headless cats have no head, one tail, no eyes and four legs.

Killer ducks have two heads, no tail, four eyes and two legs.

Hopping cows have one head, one tail, two eyes and one leg.

An investigator finds in the inventory book that there are 25 heads, 24 tails, 60 eyes and 72 legs on the reserve. How many of each kind of animal are there?

Existence and Uniqueness

Is the system consistent; that is, does at least one solution exist?

If a solution exists, is it the only one; that is, is the solution unique?

Find the solution set of each of the following:

02

32

21

21

xx

xx

02

32

21

21

xx

xx

624

32

21

21

xx

xx

No solutions

Solution set: {(1, 2)}

Infinitely many solutions.

Consistent

Inconsistent

Unique

Solution Set:

€

x1,x2( ) : x1 =1

2x2 +

3

2,x2 is free

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⎫ ⎬ ⎭

Solution set: { }

A linear system has one of the following:

No solutions, orExactly one solution, orInfinitely many solutions.

Practice: Writing a solution set

32

1

022

dcb

ca

cba