As was shown in the last chapter with systems of equations, linear transformations can also be written in matrix form. If we take the linear function
from the previous chapter as an example (we'll call it
we can write this function as the multiplication
It is easy to verify that
is equivalent to
through matrix multiplication. Thus, multiplying any matrix by a vector is equivalent to performing a linear transformation on that vector. Thus, the matrix form is a very convenient way of representing linear functions.
In addition to multiplying a transform matrix by a vector, matrices can be multiplied in order to carry out a function convolution. Consider another linear function
If we convolve the two functions, it is easy to show (although it will not be done here) that the convolution is equivalent to the following:
Keep in mind that since matrix multiplication is not commutative that the order of the multiplication is important. The grouping of the multiplication is not important since matrix multiplication is associative. In other words, you can perform
first and still get the same result. By doing the former you are essentially transforming the transform
first and then transforming the vector
. In the latter you are transforming the vector
and then passing it to
to be transformed again. These two operations are equivalent, but switching the multiplication so that
is transformed by
does not yield an equivalent result.