Page 112 - DMTH502_LINEAR_ALGEBRA

Linear Algebra Richa Nandra, Lovely Professional University

Notes Unit 7: Algebra of Linear Transformation

CONTENTS
Objectives
Introduction

7.1 Homomorphism
7.2 Linear Transformation
7.3 Algebra of Linear Transformation

7.4 Summary
7.5 Keywords
7.6 Review Questions
7.7 Further Readings

Objectives

After studying this unit, you will be able to:

 Know that linear transformation on the space is quite important. It helps in understanding
the space under various transformations.

 See that the knowledge of the basis and dimension help us that the properties of linear
transformation on the basis vector is central to the ideas of matrix mechanics.

Introduction

It will be seen that in the development of the algebra linear transformation plays an important
part in understanding the properties of spaces. It is seen that the set of linear transformations
also satisfy the properties of vector spaces.

7.1 Homomorphism

Consider two vector spaces V and W over the same field F i.e.
V v , , , , 
F
W w , , , , 
F
The vectors of two different systems might have different names, and the vector operations of
two systems might be defined in different ways.
A mapping H of V into W is called a homomorphism provided that all , B V and all a F,

( ) B H H H ...(1)
and (   ) H . a H ...(2)
If every vector of W is in the range of H, H is said to be homomorphism of V onto W.

A one-to-one homomorphism H of V onto W is called an isomorphism. If such a mapping exists, V,
and W are said to be isomorphic.

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