REVIEW
A matrix equation Ax  b
has the same solution set as the vector equation
x1a1  x 2a 2 L  x n a n  b
which 
has the same solution set as the linear system
whose augmented matrix is
a1

a 2 a 3 K b
Therefore:
 Ax = b has a solution if and only if
b is a linear combination of columns of A
REVIEW
Theorem 4:
Let A be an m  n matrix.
The following statements are equivalent:
1. For each vector b, the equation Ax  b has a solution.
2. Each vector b is a linear combination of the columns of A.
 m
3. The columns of A span R
4. A has a pivot position in every row.

Note: Theorem 4 is about a coefficient matrix A, not an augmented matrix.
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
m n matrix and 0 is the zero vector in Rm.
Example:
2 x1  5 x2  4 x3  0

 x1  4 x2  3 x3  0
 x  3x  2 x  0
2
3
 1
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 Question
 When does a homogenous system Ax  0 have a non-trivial
solution?
That is, when is there a non-zero vector x such that Ax  0 ?


Example 1: Determine if the following homogeneous system
has a nontrivial solution:
3 x1  5 x2  4 x3  0

 3 x1  2 x2  4 x3  0
6 x  x  8 x  0
3
 1 2
Geometrically, what does the solution set represent?
Basic variables: The variables corresponding to pivot columns
Free variables: he others
x1
0
0

0
x2
1
0
0
x3 x4
0 2 0
1  1 3 
0 0 0

 x1 is free
 x  2 x
 2
4

 x3  x4  3
 x4 is free
The homogeneous equation Ax = 0 has a nontrivial solution
if and only if the equation has at least one free variable.
Example 2: Describe all solutions of the homogeneous system
10x1  3x2  2 x3  0
Geometrically, what does the solution set represent?
Solutions of Nonhomogeneous Systems
Example 3: Describe all solutions for 3x1  5 x2  4 x3  7

 3x1  2 x2  4 x3  1
6 x  x  8 x  4
3
 1 2
i.e. Describe all solutions of Ax  b where
 3 5  4
A   3  2 4 
 6 1  8
and

 7 
 
b  1

4 


Geometrically, what does the solution set represent?
Homogeneous
3 x1  5 x2  4 x3  0

 3 x1  2 x2  4 x3  0
6 x  x  8 x  0
3
 1 2

 1



 0


 0
0
-4
3
1
0
0
0

0



0

0
4

x


x3
 1
3

 x2  0
 x  free
 3

 x1 
 4 / 3
x   x  0 
3
 2

 x3 
 1 
Nonhomogeneous
3x1  5 x2  4 x3  7

 3x1  2 x2  4 x3  1
6 x  x  8 x  4
3
 1 2

 1



 0


 0
0
-4
3
1
0
0
0

-1 



2

0
4

x


x3  1
 1
3

 x2  2
 x  free
 3

 x1   1
 4 / 3
x    2   x  0 
 2   3

 x3   0 
 1 
Homogeneous
Nonhomogeneous
3 x1  5 x2  4 x3  0

 3 x1  2 x2  4 x3  0
6 x  x  8 x  0
3
 1 2
3x1  5 x2  4 x3  7

 3x1  2 x2  4 x3  1
6 x  x  8 x  4
3
 1 2
 x1 
 4 / 3
x   x  0 
3
 2

 x3 
 1 
 x1   1
 4 / 3
x    2   x  0 
 2   3

 x3   0 
 1 
y
y
x
x
z
z
Theorem 6
Suppose Ax  b is consistent for some given b, and let p be
a solution. Then the solution set of Ax  b is the set
of all vectors of the form w  p  v h where v h is any
solution of the homogeneous equation Ax  0 .




