Page 445 - Elementary_Linear_Algebra_with_Applications_Anton__9_edition

The Euclidean inner product is the most important inner product on . However, there are various applications in which it is
desirable to modify the Euclidean inner product by weighting its terms differently. More precisely, if

are positive real numbers, which we shall call weights, and if and are vectors in , then it
can be shown (Exercise 26) that the formula

(1)

defines an inner product on ; it is called the weighted Euclidean inner product with weights , , …, .

To illustrate one way in which a weighted Euclidean inner product can arise, suppose that some physical experiment can produce
any of n possible numerical values

and that a series of m repetitions of the experiment yields these values with various frequencies; that is, occurs times,
occurs times, and so forth. Since there are a total of m repetitions of the experiment,

Thus the arithmetic average, or mean, of the observed numerical values (denoted by ) is (2)
If we let

then 2 can be expressed as the weighted inner product

Remark It will always be assumed that has the Euclidean inner product unless some other inner product is explicitly specified.
As defined in Section 4.1, we refer to with the Euclidean inner product as Euclidean n-space.

EXAMPLE 2 Weighted Euclidean Inner Product
Let and be vectors in . Verify that the weighted Euclidean inner product
satisfies the four inner product axioms.

Solution

Note first that if u and v are interchanged in this equation, the right side remains the same. Therefore,
If , then

which establishes the second axiom.