Page 150 - DMTH502_LINEAR_ALGEBRA

Linear Algebra Richa Nandra, Lovely Professional University

Notes Unit 11: The Double Dual

CONTENTS

Objectives
Introduction

11.1 The Double Dual
11.2 The Transpose of Linear Transformation
11.3 Summary

11.4 Keywords
11.5 Review Questions

11.6 Further Readings
Objectives

After studying this unit, you will be able to:
 Understand the meanings of V and V * * and their corresponding basis * and * *.
*

 Know that the mapping L is an isomorphism of V onto V * * .

 See that if S is any subset of a finite dimensional vector space then S 0 0 is the subspace
spanned by S.
 Understand that the ',T the transpose of the linear transformation T is often called the
adjoint of T; however in this unit we use only the word transpose.

 See that if A be the matrix of T relative to basis , ' and be the matrix of ',T relative to
dual basis '* and then B ij A ji .

Introduction

In this unit the idea of dual and double dual finite dimensional spaces and their basis vectors are
explained.
Also the transpose T of the linear transformation is introduced. The alternate name of the
'
transpose transformation is word adjoint transformation.
11.1 The Double Dual

One question about dual bases which we did not answer in the last section was whether every
basis for V* is the dual of some basis for V. One way to answer that question is to consider V**,
the dual space of V*.

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