Linear Algebra

8

Inner Product Spaces

8.1 INTRODUCTION

Bilinear form, as we had seen in the last chapter, is a natural generalization of the dot products of ℝ² and ℝ³, allowing us to introduce the important geometric concept of perpendicularity. The dot product of a vector x ∈ ℝ² with itself is related to another important geometric concept; in fact, it is the square of the length of x. However, we do face a major difficulty if we try to introduce the idea of the length of a vector v as the square root of f (v, v) relative to an arbitrary bilinear form f. Even for symmetric bilinear forms, there may be non-zero vectors which are self-orthogonal, so that their lengths will be zero. Thus, to have a workable definition of the length of a vector available relative ...