Page 462 - Elementary_Linear_Algebra_with_Applications_Anton__9_edition

(c)
(d)

THEOREM 6.2.3
Properties of Distance
If u, v, and w are vectors in an inner product space V, and if k is any scalar, then
(a)
(b)
(c)
(d)

We shall prove part (d) of Theorem 6.2.2 and leave the remaining parts of Theorems Theorem 6.2.2 and Theorem 6.2.3 as
exercises.

Proof of Theorem 6.2.2d By definition,

Taking square roots gives .

Angle Between Vectors

We shall now show how the Cauchy–Schwarz inequality can be used to define angles in general inner product spaces. Suppose

that u and v are nonzero vectors in an inner product space V. If we divide both sides of Formula 6 by , we obtain

or, equivalently,