MATH1850U/2050U:
Chapter 1 cont.
..
1
LINEAR SYSTEMS cont…
More on Linear Systems and Invertible Matrices (1.6; pg. 60)
Recall:
Last day, we learned about the concept of inverting a matrix.
Theorem:
Every system of linear equations has either no solution, exactly one solution,
or infinitely many solutions.
Proof (that if more than one solution, then infinitely many):
Theorem:
If
A
is an invertible
n
n
matrix, then for each column vector
b
, the system
of equations
b
x
A
has exactly one solution, namely
b
x
1
A
Proof:
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Chapter 1 cont.
..
2
Example:
Consider the following system with the information provided.
..what is the
solution to the system?
Question:
Say you wanted to solve ALL the systems
,
,
,
3
2
1
b
x
b
x
b
x
A
A
A
where
A
is the same for all systems.
How could do this efficiently?
Idea:
If
A
is invertible, then
,
,
,
3
1
3
2
1
2
1
1
1
b
x
b
x
b
x
A
A
A
or set up the
augmented matrix system




3
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 Spring '11
 PaulaTu
 Linear Algebra, Linear Equations, Equations, Linear Systems, Matrices, ax, Invertible matrix, Invertible Matrices

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