Page 23 - 35Linear Algebra

1.4 So, What is a Matrix? 23

This is of the same Lv = w form as our opening examples. The matrix
encodes fruit per container. The equation is roughly fruit per container
times number of containers equals fruit. To solve for number of containers
we want to somehow “divide” by the matrix.
Another way to think about the above example is to remember the rule
for multiplying a matrix times a vector. If you have forgotten this, you can
actually guess a good rule by making sure the matrix equation is the same
as the system of linear equations. This would require that

2 6 x 2x + 6y
:=
4 8 y 4x + 8y
Indeed this is an example of the general rule that you have probably seen
before

p q x px + qy p q
:= = x + y .
r s y rx + sy r s

Notice, that the second way of writing the output on the right hand side of
this equation is very useful because it tells us what all possible outputs a
matrix times a vector look like – they are just sums of the columns of the
matrix multiplied by scalars. The set of all possible outputs of a matrix
times a vector is called the column space (it is also the image of the linear
function defined by the matrix).

Reading homework: problem 2

Multiplication by a matrix is an example of a Linear Function, because it
takes one vector and turns it into another in a “linear” way. Of course, we
can have much larger matrices if our system has more variables.

Matrices in Space!

Thus matrices can be viewed as linear functions. The statement of this for
the matrix in our fruity example is as follows.

2 6 x 2 6 x
1. λ = λ and
4 8 y 4 8 y

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