linear-guest

2.5 Solution Sets for Systems of Linear Equations 67
for any µ 1 , µ 2 ∈ R. Choosing µ 1 = µ 2 = 0, we obtain
Mx P = v .
This is why x P is an example of a particular solution.
Setting µ 1 = 1, µ 2 = 0, and subtracting Mx P = v we obtain
Mx H 1 = 0 .
Likewise, setting µ 1 = 0, µ 2 = 1, we obtain
Mx H 2 = 0 .
Here x H 1 and x H 2 are examples of what are called homogeneous solutions to
the system. They do not solve the original equation Mx = v, but instead its
associated homogeneous equation My = 0.
We have just learnt a fundamental lesson of linear algebra: the solution
set to Ax = b, where A is a linear operator, consists of a particular solution
plus homogeneous solutions.
{Solutions} = {Particular solution + Homogeneous solutions}
Example 33 Consider the matrix equation of example 32. It has solution set
⎧⎛
⎞ ⎛ ⎞ ⎛ ⎞
⎫
1 −1 1
⎪⎨
S = ⎜1
⎟
⎝0⎠ ⎪⎩
+ µ ⎜ 1
⎟
1 ⎝ 1⎠ + µ ⎜−1
⎪⎬
⎟
2 ⎝ 0⎠ : µ 1, µ 2 ∈ R .
⎪⎭
0 0 1
⎛ ⎞
1
Then Mx P = v says that ⎜1
⎟
⎝0⎠ is a solution to the original matrix equation, which is
0
certainly true, but this is not the only solution.
Mx H 1
⎛ ⎞
−1
= 0 says that ⎜ 1
⎟
⎝ 1⎠ is a solution to the homogeneous equation.
0
67