Linear Algebra

7

Bilinear Forms

7.1 INTRODUCTION

In analytic geometry for ℝ² and ℝ³ (or in vector analysis), natural geometric concepts such as length, distance and perpendicularity are introduced through the dot product or inner product of two vectors. The idea of bilinear forms generalizes the dot product, and provides the proper setting for concepts of length, distance and perpendicularity even in abstract vector spaces. Not surprisingly, these ideas turn out to be powerful geometric tools in advanced mathematics as well as in solving diverse applied problems. This chapter is a brief introduction to the elementary theory of bilinear forms.

7.2 BASIC CONCEPTS

We begin with a brief review of the definition of the dot product in analytic geometry. In ℝ², for ...