Why is it that when 'transformation A' is followed by 'transformation B', that the combined transformation is BA and not AB?

Remember that with matrix multiplication, "A times B" does not usually equal "B times A". If we take our 'matrix X' and transform it by 'matrix A'. We perform this transformation by calculating AX. If we then were to transform this new matrix 'AX' by a transformation represented by a matrix B, we would calculate B(AX). This leaves us with BAX. If we write this as (BA)X, we can see that the matrix 'BA' represents the combined transformation of A and then B.