Page 128 - DMTH502_LINEAR_ALGEBRA

Linear Algebra Sachin Kaushal, Lovely Professional University

Notes Unit 9: Representation of Transformations by Matrices

CONTENTS

Objectives
Introduction

9.1 Representation of Transformations by Matrices
9.2 Illustrative Examples
9.3 Summary

9.4 Keywords
9.5 Review Questions

9.6 Further Readings
Objectives

After studying this unit, you will be able to:

 Know that the matrix of the linear transformation depends on the basis vectors of V as
well as basis vectors of W where T is a linear transformation from V to W.

 See that the matrix of T depends upon the ordered basis relative to and ' and the matrix
of T relative to ordered basis is different from the previous matrix.
 See that when T defines a transformation from V to V then the idea of similar matrices
does come up.
 Understand how to find the matrix of T with the help of detailed solved examples.

Introduction

With the help of linear transformation one can deduce the rules for addition of matrices and
multiplication of two matrices.

One can also understand geometrically the meaning of linear transformation clearly.

9.1 Representation of Transformations by Matrices

Although we have been discussing linear transformations for some time, it has always been in
a detached way; to us a linear transformation has been a symbol (very often T) which acts in a
certain way on a vector space. When one gets right down to it, outside of the few concrete
examples encountered in the problems, we have really never come face to face with specific
linear transformations. At the same time it is clear that if one were to pursue the subject further
there would often arise the need of making a thorough and detailed study of a given linear
transformation. To mention one precise problem, presented with a linear transformation; how
does go about, in a “practical” and computable way, finding its characteristic roots?
What we seek first is a simple notation, or, perhaps more accurately, representation, for linear
transformations. We shall accomplish this by use of a particular basis of the vector space and by

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