A matrix equationhas the same solution set as the vector equation

which has the same solution set as the linear system whose augmented matrix is

€

Ax = b

€

a1 a2 a3 K b[ ]

Therefore:

Ax = b has a solution if and only if

b is a linear combination of columns of A

€

x1a1 + x2a2 +L + xnan = b

REVIEW

Theorem 4:

The following statements are equivalent:

1. For each vector b, the equation has a solution.

2. Each vector b is a linear combination of the columns of A.

3. The columns of A span

4. A has a pivot position in every row.

Note: Theorem 4 is about a coefficient matrix A, not an augmented matrix.€

Rm

matrix. an be Let nmA

€

Ax = b

REVIEW

1.5 Solution Sets of Linear Systems

Definition of Homogeneous

A system of linear equations is said to be homogeneous

if it can be written in the form Ax = 0, where A is an matrix and 0 is the zero vector in Rm.nm

Example:

023

034

0452

321

321

321

xxx

xxx

xxx

Note: Every homogeneous linear system is consistent.

i.e. The homogeneous system Ax = 0 has at least one solution, namely the trivial solution, x = 0.

Important QuestionWhen does a homogenous system

have a non-trivial solution?

That is, when is there a non-zero vector x such that ?

€

Ax = 0

€

Ax = 0

Example 1: Determine if the following homogeneous systemhas a nontrivial solution:

086

0423

0453

321

321

321

xxx

xxx

xxx

Geometrically, what does the solution set represent?

The homogeneous equation Ax = 0 has a nontrivial solutionif and only if the equation has at least one free variable.

Basic variables: The variables corresponding to pivot columns

00000

31100

020101x 2x 3x 4x

Free variables: he others

free is

3

2

free is

4

43

42

1

x

xx

xx

x

Example 2: Describe all solutions of the homogeneous system

02310 321 xxx

Geometrically, what does the solution set represent?

Example 3: Describe all solutions for

486

1423

7453

321

321

321

xxx

xxx

xxx

Solutions of Nonhomogeneous Systems

i.e. Describe all solutions of where

€

Ax = b

816

423

453

A and

€

b =

7

−1

−4

⎡

⎣

⎢ ⎢ ⎢

⎤

⎦

⎥ ⎥ ⎥

Geometrically, what does the solution set represent?

Homogeneous

086

0423

0453

321

321

321

xxx

xxx

xxx

Nonhomogeneous

486

1423

7453

321

321

321

xxx

xxx

xxx

1 0-4

30

0 1 0 0

0 0 0 0

1 0-4

3-1

0 1 0 2

0 0 0 0

freex

x

xx

3

2

31

03

4

freex

x

xx

3

2

31

2

13

4

1

0

3/4

3

3

2

1

x

x

x

x

1

0

3/4

0

2

1

3

3

2

1

x

x

x

x

Homogeneous

086

0423

0453

321

321

321

xxx

xxx

xxx

Nonhomogeneous

486

1423

7453

321

321

321

xxx

xxx

xxx

1

0

3/4

3

3

2

1

x

x

x

x

1

0

3/4

0

2

1

3

3

2

1

x

x

x

x

x x

y y

z z

Theorem 6

Suppose is consistent for some given b, and let p be a solution. Then the solution set of is the set of all vectors of the form where is any solution of the homogeneous equation .

€

w = p + vh

€

vh

€

Ax = b

€

Ax = b

€

Ax = 0