Page 814 - Elementary_Linear_Algebra_with_Applications_Anton__9_edition

EXAMPLE 1 Inner Product on

Let and be vectors in . The Euclidean inner product
satisfies all the inner product axioms by Theorem 10.4.1.

EXAMPLE 2 Inner Product on Complex
If

are any matrices with complex entries, then the following formula defines a complex inner product on complex
(verify):

For example, if

then

EXAMPLE 3 Inner Product on Complex

Calculus Required

If is a complex-valued function of the real variable x, and if and are continuous on
then we define

In words, the integral of is the integral of the real part of f plus i times the integral of the imaginary part of f.