Matrices as transformations
Matrices are typically applied to vectors and other matrices through the process of matrix multiplication. However, the dry mechanics of matrix multiplication tend to hide what a matrix really represents and what matrix multiplication does. We aim to shed light on what matrices really are in this chapter. We’ll start by covering the basics of matrix multiplication and then show how matrices represent transformations.
Matrix multiplication
If we have a matrix 
of size 
, and a matrix 
of size 
, then we can multiply those two matrices together to get a new matrix 
, which is of size 
. The matrix element 
is calculated as follows:
Eq. 10
In this example, we are multiplying a 
matrix by an 
matrix. Schematically, we can write this as follows:
Eq. 11
From this, it is clear that the “inner” dimensions in this example match, both being 
. To multiply two matrices together, the inner dimensions must match, when we write...
