Page 124 - DMTH502_LINEAR_ALGEBRA

Linear Algebra Sachin Kaushal, Lovely Professional University

Notes Unit 8: Isomorphism

CONTENTS
Objectives

Introduction
8.1 Isomorphism
8.2 Summary

8.3 Keywords
8.4 Review Questions
8.5 Further Readings

Objectives

After studying this unit, you will be able to:
 Understand the linear transformation T is such that T transforms a subspace S of
independent vectors of vector space into an independent subspace T(S) of W.

 See that isomorphism is a homomorphism if the linear transformation T on V onto W is
one-one.

 Know that for finite vector space the linear transformation T is non-singular if and only if
dim V = dim W and T is isomorphism of V onto W.

Introduction

In dealing with two vector spaces over the same field, a transformation T from V into W can be
homomorphism or isomorphism.
After studying this unit one can see that a fine n-dimensional vector space and a space of n-tuple
co-ordinate space over the same field are isomorphic and so studying of one space gives all
information about the other space.

8.1 Isomorphism

If V and W are vector spaces over the field F, any one-one linear transformation T of V onto W is
called an isomorphism of V onto W. If there exists an isomorphism of V onto W, we say that V
is isomorphic to W.
Note that V is trivially isomorphic to V, the identity transformation operator being an
isomorphism of V onto V. Also, if V is isomorphic to W via an isomorphism T, then W is
-1
isomorphic to V, because then T is invertible and so T is an isomorphism of W onto V. Thus it
is easily verified that if V is isomorphic to W and W is isomorphic to Z, then V is isomorphic
to Z. Briefly, isomorphism is an equivalence relation on the class of vector spaces. If there exists
an isomorphism of V onto W, we sometimes say that V and W are isomorphic.
n
Theorem 1: Every n-dimensional vector spaceV over the field F is isomorphic to the space F .
n
Proof: Let V be an n-dimensional space over the field F and let , ... n be the ordered
n 1 2
n
basis for V. We defined a function T from V into F , as follows:

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