Page 25 - 35Linear Algebra

1.4 So, What is a Matrix? 25

We have come full circle; matrices are just examples of the kinds of linear
operators that appear in algebra problems like those in section 1.3. Any
equation of the form Mv = w with M a matrix, and v, w n-vectors is called
a matrix equation. Chapter 2 is about efficiently solving systems of linear
equations, or equivalently matrix equations.

1.4.1 Matrix Multiplication is Composition of Functions
What would happen if we placed two of our expensive machines end to end?

?

The output of the first machine would be fed into the second.

1.(2x + 6y) + 2.(4x + 8y)
x 2x + 6y
0.(2x + 6y) + 1.(4x + 8y)
y 4x + 8y

10x + 22y
=
4x + 8y
Notice that the same final result could be achieved with a single machine:

x 10x + 22y
.
y 4x + 8y

There is a simple matrix notation for this called matrix multiplication

1 2 2 6 10 22
= .
0 1 4 8 4 8
Try review problem 6 to learn more about matrix multiplication.
In the language 10 of functions, if

f : U −→ V and g : V −→ W

10
The notation h : A → B means that h is a function with domain A and codomain B.
See the webwork background set3 if you are unfamiliar with this notation or these terms.

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