Page 479 - Elementary_Linear_Algebra_with_Applications_Anton__9_edition

We leave it for you to verify that the set is orthonormal by showing that

In an inner product space, a basis consisting of orthonormal vectors is called an orthonormal basis, and a basis consisting of
orthogonal vectors is called an orthogonal basis. A familiar example of an orthonormal basis is the standard basis for
with the Euclidean inner product:

This is the basis that is associated with rectangular coordinate systems (see Figure 5.4.4). More generally, in with the
Euclidean inner product, the standard basis

is orthonormal.

Coordinates Relative to Orthonormal Bases

The interest in finding orthonormal bases for inner product spaces is motivated in part by the following theorem, which
shows that it is exceptionally simple to express a vector in terms of an orthonormal basis.

THEOREM 6.3.1

If is an orthonormal basis for an inner product space V, and u is any vector in V, then

Proof Since is a basis, a vector u can be expressed in the form

We shall complete the proof by showing that for , …, n. For each vector in S, we
have

Since is an orthonormal set, we have

Therefore, the above expression for simplifies to

Using the terminology and notation introduced in Section 5.4, the scalars