MAT225 Section Summary: 1.8

Introduction to Linear Transformations

Summary

Definition: transformation: a transformation (or function or mapping) T from tex2html_wrap_inline240 to tex2html_wrap_inline242 is a rule that assigns to each vector tex2html_wrap_inline244 in tex2html_wrap_inline240 a vector tex2html_wrap_inline248 in tex2html_wrap_inline242 . The set tex2html_wrap_inline240 is the domain of T, and tex2html_wrap_inline242 is the codomain.

For tex2html_wrap_inline244 in tex2html_wrap_inline240 , the vector tex2html_wrap_inline248 is called the image of tex2html_wrap_inline244 (under the action of T). The set of all images tex2html_wrap_inline248 of vectors tex2html_wrap_inline244 from the domain is called the range of the transformation T.

A transformation T is linear if it satisfies

The matrix product tex2html_wrap_inline290 represents a linear transformation, as we have seen. If A is an m x n matrix, tex2html_wrap_inline278 and tex2html_wrap_inline280 are vectors in tex2html_wrap_inline240 , and c is a scalar, then:

  1. tex2html_wrap_inline306
  2. tex2html_wrap_inline308

More generally, a linear transformation satisfies

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also known as the principle of superposition.

In this section, several important examples of linear transformation representable by matrices are given, corresponding to

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!

LONG ANDREW E
Sat Jan 29 20:50:37 EST 2011