Page 63 - 35Linear Algebra

2.5 Solution Sets for Systems of Linear Equations 63

(b) Give an interpretation of the columns of M −1 in (M|I) ∼ (I|M −1 )
in terms of solutions to certain systems of linear equations.

6. How can you convince your fellow students to never make this mistake?

0
R =R 1 +R 2
1
  0  
1 0 2 3 R =R 1 −R 2 1 1 4 6
2
0
R =R 1 +2R 2
3
0 1 2 3 ∼ 1 −1 0 0
   
2 0 1 4 1 2 6 9
7. Is LU factorization of a matrix unique? Justify your answer.
∞. If you randomly create a matrix by picking numbers out of the blue,
it will probably be difficult to perform elimination or factorization;
fractions and large numbers will probably be involved. To invent simple
problems it is better to start with a simple answer:

(a) Start with any augmented matrix in RREF. Perform EROs to
make most of the components non-zero. Write the result on a
separate piece of paper and give it to your friend. Ask that friend
to find RREF of the augmented matrix you gave them. Make sure
they get the same augmented matrix you started with.

(b) Create an upper triangular matrix U and a lower triangular ma-
trix L with only 1s on the diagonal. Give the result to a friend to
factor into LU form.

(c) Do the same with an LDU factorization.

2.5 Solution Sets for Systems of Linear Equa-

tions

Algebraic equations problems can have multiple solutions. For example x(x−
1) = 0 has two solutions: 0 and 1. By contrast, equations of the form Ax = b
with A a linear operator (with scalars the real numbers) have the following
property:

If A is a linear operator and b is known, then Ax = b has either

1. One solution

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