Page 450 - Elementary_Linear_Algebra_with_Applications_Anton__9_edition

In the following examples we shall describe some inner products on vector spaces other than .

EXAMPLE 7 An Inner Product on
If

are any two matrices, then the following formula defines an inner product on (verify):
(Refer to Section 1.3 for the definition of the trace.) For example, if

then
The norm of a matrix U relative to this inner product is

and the unit sphere in this space consists of all matrices U whose entries satisfy the equation , which on squaring
yields

EXAMPLE 8 An Inner Product on , which on
If

are any two vectors in , then the following formula defines an inner product on (verify):
The norm of the polynomial p relative to this inner product is

and the unit sphere in this space consists of all polynomials p in whose coefficients satisfy the equation
squaring yields

Calculus Required

EXAMPLE 9 An Inner Product on