Page 461 - Elementary_Linear_Algebra_with_Applications_Anton__9_edition

nonnegative. Therefore,

This inequality implies that the quadratic polynomial has either no real roots or a repeated real root. Therefore, its

discriminant must satisfy the inequality . Expressing the coefficients a, b, and c in terms of the vectors u and v

gives 4 , or, equivalently,

Taking square roots of both sides and using the fact that and are nonnegative yields
which completes the proof.
For reference, we note that the Cauchy–Schwarz inequality can be written in the following two alternative forms:

(5)

(6)

The first of these formulas was obtained in the proof of Theorem 6.2.1, and the second is derived from the first using the fact
that and .

EXAMPLE 1 Cauchy–Schwarz Inequality in to be the

The Cauchy–Schwarz inequality for (Theorem 4.1.3) follows as a special case of Theorem 6.2.1 by taking
Euclidean inner product .

The next two theorems show that the basic properties of length and distance that were established in Theorems 4.1.4 and 4.1.5
for vectors in Euclidean n-space continue to hold in general inner product spaces. This is strong evidence that our definitions of
inner product, length, and distance are well chosen.

THEOREM 6.2.2

Properties of Length
If u and v are vectors in an inner product space V, and if k is any scalar, then

(a)

(b)