Linear Algebra, 5th Edition

6.8* Bilinear and Quadratic Forms

There is a certain class of scalar-valued functions of two variables defined on a vector space that arises in the study of such diverse subjects as geometry and multivariable calculus. This is the class of bilinear forms. We study the basic properties of this class with a special emphasis on symmetric bilinear forms, and we consider some of its applications to quadratic surfaces and multivariable calculus. In this section, F denotes any field that does not have characteristic two, as defined on page 549.

Bilinear Forms

Definition.

Let V be a vector space over a field F. A function H from the set $\text{V}\times \text{V}$ of ordered pairs of vectors to F is called a bilinear form on V if H is linear in each variable when the other ...