Sorry. Let me re-phrase my entire thread:
Affine sums are additions of points scaled by a scalar.
The dot product in my matrix is the multiplication of the matrix elements by a point in the vector, however the vector is a literal vector and not a matrix.
[x,y] * [a b c] = (x*a, y*a) + (x*b, y*b) + (x*c, y*c)
= (x(a+b+c), y(a+b+c))
The thing is, this should be done using tensors.