We’ve Been Taught Matrices Wrong: Relearning Matrices as Linear Functions

Before I had this representation, I would have had to make a full table where I write out f(x), and then apply g(x) on it per input like below:

But now that we have the polynomial representation, we can just do g(x)=10x g(f(x))=10⋅f(x) g(f(x))=20x

Using simple steps, we can figure out g(f(x)) and describe the composition function for any arbitrary input. Ok so we know we can represent a linear map as a matrix – but how do we apply this function on an input? So H, the matrix representing g(f(x)), is just G⋅F.

Thus, matrix multiplication is just a way to compose linear maps.

Source: www.dhruvonmath.com