Unit 6
Linear Transformations II

Introduction

In this unit, we discuss the method for finding the matrix representation of a linear transformation with respect to a basis B in the domain space, and a basis B' in the image space. We then consider the question of how this matrix representation changes when there is a change of basis. For simplicity, we restrict our attention to the case of linear operators.

Later in this unit, you will study examples of linear transformations from Rⁿ to R^m, and from R² to R².

Objectives

After completing this unit, you should be able to

1. determine the matrix representation of any linear transformation for which the action on a given basis is given.
2. define two ways of viewing a linear operator, and explain how the passive way is derived from the active way.
3. explain what is meant when two matrices are said to be “similar” to one another.
4. find the matrix of a linear transformation with respect to a basis B', given the matrix of this transformation with respect to a basis B, and the transition matrix between B and B'.