Introduction to Linear Transformations
Summary
Definition: transformation: a transformation (or function or mapping) T from to is a rule that assigns to each vector in a vector in . The set is the domain of T, and is the codomain.
For in , the vector is called the image of (under the action of T). The set of all images of vectors from the domain is called the range of the transformation T.
A transformation T is linear if it satisfies
The matrix product represents a linear transformation, as we have seen. If A is an m x n matrix, and are vectors in , and c is a scalar, then:
More generally, a linear transformation satisfies
also known as the principle of superposition.
In this section, several important examples of linear transformation representable by matrices are given, corresponding to