Page 484 - Elementary_Linear_Algebra_with_Applications_Anton__9_edition

The following theorem, whose proof is requested in the exercises, provides formulas for calculating orthogonal projections.
THEOREM 6.3.5

Let W be a finite-dimensional subspace of an inner product space V.

(a) If is an orthonormal basis for W, and u is any vector in V, then

(6)

(b) If is an orthogonal basis for W, and u is any vector in V, then

(7)

EXAMPLE 6 Calculating Projections

Let have the Euclidean inner product, and let W be the subspace spanned by the orthonormal vectors and

. From 6 the orthogonal projection of on W is

The component of u orthogonal to W is

Observe that is orthogonal to both and , so this vector is orthogonal to each vector in the space W spanned by

and , as it should be.

Finding Orthogonal and Orthonormal Bases

We have seen that orthonormal bases exhibit a variety of useful properties. Our next theorem, which is the main result in this
section, shows that every nonzero finite-dimensional vector space has an orthonormal basis. The proof of this result is
extremely important, since it provides an algorithm, or method, for converting an arbitrary basis into an orthonormal basis.

THEOREM 6.3.6

Every nonzero finite-dimensional inner product space has an orthonormal basis.