Suppose we have two composable linear maps, and . The composition is still linear, as you can check. There should be a connection between the matrix of with respect to some bases and the matrices for and .
Let and be linear maps. Let
be a basis of ,
be a basis of , and
be a basis of .
Then
Here is picture of this situation:
This theorem provides some justification for our definition of matrix multiplication: composition of linear maps corresponds to multiplication of matrices.
Let and . We will work out using the definition of the matrix of a linear map. For any ,
so the entry of is , which is the same as the entry of by the matrix multiplication formula. ∎