Chapter 3 Bilinear Functions

The inner product is an example of a bilinear function that likely is familiar to the reader. In this book, we are more interested in a less restrictive type of bilinear function called a scalar product, but we will not ignore the inner product altogether. In order to decide whether a bilinear function is a scalar product, we need to determine whether it is symmetric and nondegenerate. These properties of bilinear functions will be the focus of the present chapter.

3.1 Bilinear Functions

Let V be a vector space. A function

equation

is said to be bilinear (on V) if it is linear in both arguments; that is,

equation

and

equation

for all vectors u, v, w in V and all real numbers c. In the literature, a bilinear function is sometimes called a quadratic form. We often denote

equation

writing v, w in place of b(v, w).

We say that b is:

symmetric
if v, w〉 = 〈w, v for all v, w in V.
alternating
if v, w〉 =  − 〈w, v for all v, w in V.
nondegenerate
if for all v in V, v, w〉 = 0 for all w in V implies v = 0.
degenerate
if b is not nondegenerate.

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