Page 449 - Elementary_Linear_Algebra_with_Applications_Anton__9_edition

The Euclidean inner product and the weighted Euclidean inner products are special cases of a general class of inner products on
, which we shall now describe. Let

be vectors in (expressed as matrices), and let A be an invertible matrix. It can be shown (Exercise 30) that if is
the Euclidean inner product on , then the formula

defines an inner product; it is called the inner product on generated by A. (3)
[see 7 in Section 4.1], it follows that 3
Recalling that the Euclidean inner product can be written as the matrix product
can be written in the alternative form

or, equivalently,

(4)

EXAMPLE 6 Inner Product Generated by the Identity Matrix in 3 yields
The inner product on generated by the identity matrix is the Euclidean inner product, since substituting

The weighted Euclidean inner product discussed in Example 2 is the inner product on generated by

because substituting this matrix in 4 yields

In general, the weighted Euclidean inner product (5)
is the inner product on generated by

(verify).