Page 263 - Elementary_Linear_Algebra_with_Applications_Anton__9_edition
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Proof (d) We have . Further, equality holds if and only if —that is, if and
only if .
EXAMPLE 2 Length and Distance in
Theorem 4.1.2 allows us to perform computations with Euclidean inner products in much the same way as we perform them
with ordinary arithmetic products. For example,
The reader should determine which parts of Theorem 4.1.2 were used in each step.
Norm and Distance in Euclidean n-Space
By analogy with the familiar formulas in and , we define the Euclidean norm (or Euclidean length) of a vector
in by
[Compare this formula to Formulas 1 and 2 in Section 3.2.] and in is defined by (1)
Similarly, the Euclidean distance between the points (2)
[See Formulas 3 and 4 of Section 3.2.]
EXAMPLE 3 Finding Norm and Distance
If and , then in the Euclidean space ,
and
The following theorem provides one of the most important inequalities in linear algebra: the Cauchy–Schwarz inequality.
THEOREM 4.1.3
Cauchy–Schwarz Inequality in
