Page 480 - Elementary_Linear_Algebra_with_Applications_Anton__9_edition

in Theorem 6.3.1 are the coordinates of the vector u relative to the orthonormal basis , and
is the coordinate vector of u relative to this basis.

EXAMPLE 3 Coordinate Vector Relative to an Orthonormal Basis
Let

It is easy to check that is an orthonormal basis for with the Euclidean inner product. Express the vector

as a linear combination of the vectors in S, and find the coordinate vector .

Solution

Therefore, by Theorem 6.3.1 we have
that is,

The coordinate vector of u relative to S is

Remark The usefulness of Theorem 6.3.1 should be evident from this example if we remember that for nonorthonormal
bases, it is usually necessary to solve a system of equations in order to express a vector in terms of the basis.

Orthonormal bases for inner product spaces are convenient because, as the following theorem shows, many familiar formulas
hold for such bases.

THEOREM 6.3.2

If S is an orthonormal basis for an n-dimensional inner product space, and if

then
(a)