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