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
is said to be bilinear (on V) if it is linear in both arguments; that is,
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
writing 〈v, w〉 in place of b(v, w).
We say that b is:
- if 〈v, w〉 = 〈w, v〉 for all v, w in V.
- if 〈v, w〉 = − 〈w, v〉 for all v, w in V.
- if for all v in V, 〈v, w〉 = 0 for all w in V implies v = 0.
- if b is not nondegenerate.