Page 823 - Elementary_Linear_Algebra_with_Applications_Anton__9_edition

This result, called the Cauchy–Schwarz inequality for complex inner product spaces, differs from its real analog
(Theorem 6.2.1) in that an absolute value sign must be included on the left side.

Hint Let in the inequality of Exercise 29(b).

Prove: If and are vectors in , then
31.

This is the complex version of Formula 4 in Theorem 4.1.3.
Hint Use Exercise 30.

Prove that equality holds in the Cauchy–Schwarz inequality for complex vector spaces if and only if u and v are linearly
32. dependent.

Prove that if is an inner product on a complex vector space, then
33.

Prove that if is an inner product on a complex vector space, then
34.

Theorems 6.2.2 and 6.2.3 remain true in complex inner product spaces. In each part, prove that this is so.
35.

(a) Theorem 6.2.2a

(b) Theorem 6.2.2b

(c) Theorem 6.2.2c

(d) Theorem 6.2.2d

(e) Theorem 6.2.3a

(f) Theorem 6.2.3b

(g) Theorem 6.2.3c

(h) Theorem 6.2.3d

In Example 7 it was shown that the vectors
36.