Page 804 - Elementary_Linear_Algebra_with_Applications_Anton__9_edition

Theorem 4.1.2 listed the four main properties of the Euclidean inner product on . The following theorem is the
corresponding result for the complex Euclidean inner product on .
THEOREM 10.4.1

Properties of the Complex Inner Product
If u, v, and w are vectors in , and k is any complex number, then

(a)

(b)

(c)

(d) . Further, if and only if .

Note the difference between part (a) of this theorem and part (a) of Theorem 4.1.2. We will prove parts (a) and (d) and leave
the rest as exercises.

Proof (a) Let and . Then

and
so

Proof (d)

Moreover, equality holds if and only if . But this is true if and only if
; that is, it is true if and only if .

Remark We leave it as an exercise to prove that