Page 68 - 35Linear Algebra

68 Systems of Linear Equations

 
1
 −1 
Mx H = 0 says that   is a solution to the homogeneous equation.
2
 0 
1
Notice how adding any multiple of a homogeneous solution to the particular solution
yields another particular solution.

Reading homework: problem 4

2.6 Review Problems

Reading problems 4 , 5
Webwork: Solution sets 20, 21, 22
Geometry of solutions 23, 24, 25, 26

1. Write down examples of augmented matrices corresponding to each
of the five types of solution sets for systems of equations with three
unknowns.

2. Invent a simple linear system that has multiple solutions. Use the stan-
dard approach for solving linear systems and a non-standard approach
to obtain different descriptions of the solution set. Is the solution set
different with different approaches?

3. Let
 1 1 1   1 
a a · · · a x
1 2 k
 2 2 2   2 
a 1 a 2 · · · a   x 
k
M =  . .  and x =   .

.
 . . . . . .   . 
. 
 . 
a r a r · · · a r x k
1 2 k
2
Note: x does not denote the square of the column vector x. Instead
2
3
1
x , x , x , etc..., denote different variables (the components of x);
the superscript is an index. Although confusing at first, this nota-
tion was invented by Albert Einstein who noticed that quantities like
2 2
2 j
2 k
2 1
a x + a x · · · + a x =: P k a x , can be written unambiguously
k
1
2
j
j=1
2 j
as a x . This is called Einstein summation notation. The most im-
j
portant thing to remember is that the index j is a dummy variable,
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