Transformation matrix
A general linear transformation between two vector spaces V and W, T:V-->W, maps one onto the other. It can be one-to-one (meaning it is invertible), but this is not necessary. The only requirements are these:
- T(u+v) = T(u) + T(v)
- T(kv) = kT(v)
Note that this is similar to proving a vector space V is a subspace of V', but not exactly so.
In the special case that T maps from V to V it is called a linear operator. Linear transformations can be expressed as matrix multiplication. Some examples include the rotation operator in 2-space. It is ( (cos x, -sin x) (sin x, cos x) ).