Page 254 - DMTH502_LINEAR_ALGEBRA

Linear Algebra Sachin Kaushal, Lovely Professional University

Notes Unit 24: Inner Product and Inner Product Spaces

CONTENTS
Objectives
Introduction

24.1 Inner Product
24.2 Inner Product Space
24.3 Summary

24.4 Keywords
24.5 Review Questions
24.6 Further Readings

Objectives

After studying this unit, you will be able to:

 See that there is some similarity between the scalar product in vector analysis and the
concept of inner product.
 Understand that an inner product on a vector space V is a function which assigns to each
ordered pair of vectors , in V a scalar ( / ) in the field F such a way that for all , ,
in V and all scalars C

( /c ) c ( | ) ( | )

 Know the importance of the construction known as Gram–Schmidt orthogonalization
process to convert a set of independent vector ( , , … ) into an orthogonal set of
1 2 n
vectors ( , , … ).
1 2 n
 Understand orthogonal projection operators and their importance.

Introduction

In this unit the concept of inner product and inner product space is introduced and a similarity
is shown with the scalar product of dot product in vector analysis.
The Cauchy-Schwarz inequality is introduced.
With the help of examples it is shown how to obtain a set of orthogonal vectors ( , , … )
1 2 n
from a set of independent vectors ( , , … ) by means of a construction known as Gram-
1 2 n
Schmidt orthogonalization process.
By introducing orthogonal projection, E of V on W, it is seen that E is an idempotent linear
transformation of V onto W, W is the null space of F and V = W W .

24.1 Inner Product

In this unit we consider the vector space V over a field of real or complex numbers. In the first
case V is called a real vector space, in the second, a complex vector field. We have had some
experience of a real vector space in fact both analytic geometry and the subject matter of vector

248 LOVELY PROFESSIONAL UNIVERSITY