(this page is work in progress)
A linear transform is a function from vector spaces V to W that satsifies
f[A ⊕ B] = f[A] ⊕ f[B] f[s ⊙ A] = s ⊙ f[A]
A and B are vectors. s is a real. ⊙ is the scalar product.
If the vectors are points in the plane, then, a linear transform means one of: shear, dilation, rotation, reflection around origin, reflection of a line assing the origin.
If a finite-dimensional abstract vector space V and W have chosen basis, then every linear trasform from V to W can be represented as a matrix.