Page 296 - DMTH502_LINEAR_ALGEBRA

Linear Algebra Richa Nandra, Lovely Professional University

Notes Unit 27: Introduction and Forms on Inner Product Spaces

CONTENTS
Objectives
Introduction

27.1 Overview
27.2 Forms on Inner Product Spaces
27.3 Summary

27.4 Keywords
27.5 Review Questions
27.6 Further Readings

Objectives

After studying this unit, you will be able to:

 See that the material covered in this unit on inner product spaces is more sophisticated and
generally more involved technically
 Understand more clearly sesquilinear form as well as bilinear forms

 See that the map f T isomorphism of the space of forms onto L(V, V) is understood well
 Know how to obtain the matrix of f in the ordered basis .

Introduction

In this unit the topics covered in the units 24, 25 and unit 26 are reviewed.
It is seen that these ideas can further be elaborated on an advanced stage.

It is shown that the section devotes to the relation between forms and linear operators.
One can see that for every Hermitian form f on a finite dimensional inner product space V, there
is an orthonormal basis of V in which f is represented by a diagonal matrix with real entries.

27.1 Overview

In the units 24, 25, 26 we have covered topics which are quite fundamental in nature. It covered
basically a lot of topics like inner products, inner product spaces, adjoint operators, unitary
operators and linear functionals. However, in the next few units we shall deal with inner product
spaces and spectral theory, forms on inner product spaces, positive forms and properties of the
normal operators. Apart from the formulation of the principal axis theorem or the orthogonal
diagonalization of self-adjoint operators the material covered in these units is sophisticated and
generally more technically involved. In these units the arguments and proofs are written in a
more condensed forms. Units 27 and 28 are devoted to results concerning forms on inner product
spaces and the relations between forms and linear operators. Unit 2 deals with spectral theory,
i.e. with the implication of the ideas of units 24, 25 and 26 concerning the diagonalization of self-
adjoint and normal operators.

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