Elementary Linear Algebra, 11th Edition

CHAPTER 6Inner Product Spaces

CHAPTER CONTENTS

1. 6.1 Inner Products
2. 6.2 Angle and Orthogonality in Inner Product Spaces
3. 6.3 Gram–Schmidt Process; $QR$-Decomposition
4. 6.4 Best Approximation; Least Squares
5. 6.5 Mathematical Modeling Using Least Squares
6. 6.6 Function Approximation; Fourier Series

INTRODUCTION

In Chapter 3 we defined the dot product of vectors in $R^n$, and we used that concept to define notions of length, angle, distance, and orthogonality. In this chapter we will generalize those ideas so they are applicable in any vector space, not just $R^n$. We will also discuss various applications of these ideas.

6.1 Inner Products

In this section we will use the most important properties of the dot product on $R^n$ as axioms, which, if satisfied by the vectors in a vector space $V$, will enable us to extend the notions of length, distance, angle, and perpendicularity to general vector spaces.

General Inner Products

In Definition 4 of Section 3.2 we defined the dot product of two vectors in $R^n$, and in Theorem 3.2.2 we listed four fundamental properties of such products. Our first goal in this section is to extend the notion of a dot product to general real vector spaces by using those four properties as axioms. We make the following definition.

Note that Definition 1 applies only to real vector spaces. A definition of inner products on complex vector spaces is given in the exercises. ...