Unit 5
Linear Transformations I

Introduction

Unit 5 further develops the idea of vector valued functions through a detailed discussion of the special class of such functions known as linear transformations. Linear transformations are illustrated through examples based on operations that you have already encountered: contractions, dilations, rotations, and the identity and zero transformations.

We then consider two spaces that can be identified for every linear transformation, the kernel and the range. Finally, we consider the problem of finding the inverse of a linear transformation.

Objectives

After completing this unit, you should be able to

1. define the terms “vector valued function” and “linear transformation,” and determine whether a given vector valued function is a linear transformation.
2. solve linear transformation problems involving the following operations in two-dimensional space: rotation about the origin, mapping all vectors to the zero vector, mapping a space identically to itself, dilation of vectors, contraction of vectors, and shears.
3. prove that every non-singular linear transformation of R² is composed of a combination of shears, dilations (including contractions) and reflections.
4. solve linear transformation problems involving orthogonal projections from a vector space to one of its subspaces, and those involving the mapping of any n-dimensional vector space to Rⁿ.
5. define the terms “kernel,” “range,” “rank” and “nullity,” and explain their relation to linear transformations.
6. define what it means for a linear transformation to be “one-to-one.”
7. define the inverse of a linear transformation, and determine the standard matrix for the inverse of a one-to-one linear transformation.