Page 48 - 35Linear Algebra

48 Systems of Linear Equations

Particular and Homogeneous Solutions

Check now that the parts of the solutions with free variables as coefficients
from the previous examples are homogeneous solutions, and that by adding
a homogeneous solution to a particular solution one obtains a solution to the
matrix equation. This will come up over and over again. As an example
d
without matrices, consider the differential equation dx 2 2 f = 3. A particular
3 2
solution is x while x and 1 are homogeneous solutions. The solution set is
2
3 2
{ x + ax + c1 : a, b ∈ R}. You can imagine similar differential equations
2
with more homogeneous solutions.
You need to become very adept at reading off solutions sets of linear
systems from the RREF of their augmented matrix; it is a basic skill for
linear algebra, and we will continue using it up to the last page of the book!

Worked examples of Gaussian elimination

2.2 Review Problems

Reading problems 1 , 2
Augmented matrix 6
Webwork: 2 × 2 systems 7, 8, 9, 10, 11, 12
3 × 2 systems 13, 14
3 × 3 systems 15, 16, 17

1. State whether the following augmented matrices are in RREF and com-
pute their solution sets.
 
1 0 0 0 3 1
0 1 0 0 1 2
 
  ,
0 0 1 0 1 3
 
0 0 0 1 2 0

 
1 1 0 1 0 1 0
0 0 1 2 0 2 0
 
  ,
0 0 0 0 1 3 0
 
0 0 0 0 0 0 0
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