MAT225 Section Summary: 1.8

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

• for all , in the domain of T
• for all and all scalars c.

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:

1. 
2. 

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

• projections (Example 2),
• shears (Example 3),
• scalings (Example 4 - contractions and dilations), and
• rotations (Example 5).

As you can well imagine, these sorts of transformations are very useful to the computer scientist, among others: if you want to simulate motion in a computer game, for example, you will be constantly projecting, rotating, and scaling objects. But for translations, computer scientists have need of affine transformations, as described in your homework problem #30, p. 81. Have fun!