Page 66 - 35Linear Algebra

66 Systems of Linear Equations

Example 32 (Non-pivot variables determine the gemometry of the solution set)

 
  x 1   
1 0 1 −1 1  1x 1 + 0x 2 + 1x 3 − 1x 4 = 1
x 2
 
0 1 −1 1    = 1 ⇔ 0x 1 + 1x 2 − 1x 3 + 1x 4 = 1
  
 x 3 
0 0 0 0 0  0x 1 + 0x 2 + 0x 3 + 0x 4 = 0
x 4
Following the standard approach, express the pivot variables in terms of the non-pivot
variables and add “empty equations”. Here x 3 and x 4 are non-pivot variables.
        
x 1 1 −1 1
x 1 = 1 − x 3 + x 4 

1

x 2 = 1 + x 3 − x 4  x 2     1   −1 
⇔   =   + x 3   + x 4  
0
x 3 = x 3  x 3     1   0 



x 4 = x 4 x 4 0 0 1
The preferred way to write a solution set S is with set notation;
        
 x 1 1 −1 1 
 
1
 
 x 2     1   −1 
S =   =   + µ 1   + µ 2   : µ 1 , µ 2 ∈ R .
0
 x 3     1   0  

 
 
x 4 0 0 1
Notice that the first two components of the second two terms come from the non-pivot
columns. Another way to write the solution set is
P H H
S = x + µ 1 x + µ 2 x : µ 1 , µ 2 ∈ R ,
1 2
where
     
1 −1 1
1
P
x =   , x H =  1  , x H =  −1  .


1


 
2
0
   1   0 
0 0 1
P
Here x is a particular solution while x H and x H are called homogeneous solutions.
1 2
The solution set forms a plane.
2.5.3 Solutions and Linearity
Motivated by example 32, we say that the matrix equation Mx = v has
P
H
H
solution set {x + µ 1 x + µ 2 x | µ 1 , µ 2 ∈ R}. Recall that matrices are linear
1 2
operators. Thus
P
H
H
H
H
P
M(x + µ 1 x + µ 2 x ) = Mx + µ 1 Mx + µ 2 Mx = v ,
1
2
1
2
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