Vectors are written as columns. If I have a linear function, I can describe it to you by just telling you what it does to each basis element, so that's what a m

Vectors are written as columns. If I have a linear function, I can describe it to you by just telling you what it does to each basis element, so that's what a matrix is: a list of each of the output (column) vectors you get when you apply it to each basis vectors as input. Multiplication is actually function composition: if A is the matrix for g and B is the matrix for f, then AB is the matrix for the function `x -> g(f(x))`.