Page 47 - 35Linear Algebra

2.1 Gaussian Elimination 47

 
1 0 7 0 4

 0 1 3 4 1  x + 7z = 4
  ⇔
0 0 0 0 0 y + 3z+4w = 1
 
0 0 0 0 0
         
 x = 4 − 7z x 4 −7 0
 

1
y
 y = 1 − 3z − 4w       −3   −4 
⇔ ⇔   =   + z   + w  
0
z
 z = z       1   0 
 
w = w w 0 0 1
 
so the solution set is

     
 4 −7 0
 

1 −3 −4
 
     
  + z   + w   : z, w ∈ R .
 0  1   0  
 
 
0 0 1
 
From RREF to a Solution Set
You can imagine having three, four, or fifty-six non-pivot columns and
the same number of free variables indexing your solutions set. In general a
solution set to a system of equations with n free variables will be of the form
H
H
P
H
{x + µ 1 x + µ 2 x + · · · + µ n x : µ 1 , . . . , µ n ∈ R}.
2
n
1
The parts of these solutions play special roles in the associated matrix
equation. This will come up again and again long after we complete this
discussion of basic calculation methods, so we will use the general language
of linear algebra to give names to these parts now.
Definition: A homogeneous solution to a linear equation Lx = v, with
H
H
L and v known is a vector x such that Lx = 0 where 0 is the zero vector.
P
If you have a particular solution x to a linear equation and add
a sum of multiples of homogeneous solutions to it you obtain another
particular solution.

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