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; $QR$
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$$R^n$

6.1 Inner Products

In this section we will use the most important properties of the dot product on $R^n$$V$

General Inner Products

In Definition 4 of Section 3.2 we defined the dot product of two vectors in $R^n$

Note that Definition 1 applies only to real vector spaces. A definition of inner products on complex vector spaces is given in the exercises. Since we will have little need for complex vector spaces from this point on, you can assume that all vector spaces under discussion are real, even though some of the theorems are also valid in complex vector spaces.

DEFINITION 1

An inner product on a real vector space $V$$⟨\mathbf{u},\mathbf{v}⟩$$V$$\mathbf{u},\mathbf{v}$$\mathbf{w}$$V$$k$

1. $\begin{array}{cc}⟨\mathbf{u},\mathbf{v}⟩=⟨\mathbf{v},\mathbf{u}⟩\hfill & \left[\mathbf{\text{Symmetry axiom}}\right]\hfill \end{array}$
2. $\langle \mathbf{\text{u + v, w}}\mathbf{\rangle }=\mathbf{\langle }\mathbf{\text{u, w}}\mathbf{\rangle }\mathbf{\text{ + }}\mathbf{\langle }\mathbf{\text{v, w}}\mathbf{\rangle }\mathbf{\text{ [Additivity axiom]}}$
3. $\begin{array}{cc}⟨k\mathbf{u},\mathbf{v}⟩=k⟨\mathbf{u},\mathbf{v}⟩\hfill & \left[\mathbf{\text{Homogeneity axiom}}\right]\hfill \end{array}$
4. $⟨\mathbf{v},\mathbf{v}⟩\ge 0$$⟨\mathbf{v},\mathbf{v}⟩=0$$\mathbf{v}=\mathbf{0}$[Positivity axiom]

A real vector space with an inner product is called a real inner product space.

Because the axioms for a real inner product space are based on properties of the dot product, these inner product space axioms will be satisfied automatically if we define the inner product ...