Page 581 - Elementary_Linear_Algebra_with_Applications_Anton__9_edition

EXAMPLE 5 Orthogonal Projections

In Section 6.4 we defined the orthogonal projection of onto a subspace W. [See Formula 6 and the definition preceding it
in that section.] Orthogonal projections can also be defined in general inner product spaces as follows: Suppose that W is a
finite-dimensional subspace of an inner product space V; then the orthogonal projection of V onto W is the transformation
defined by

(Figure 8.1.2). It follows from Theorem 6.3.5 that if

is any orthonormal basis for W, then is given by the formula

The proof that T is a linear transformation follows from properties of the inner product. For example,

Similarly, .

Figure 8.1.2
The orthogonal projection of V onto W.

EXAMPLE 6 Computing an Orthogonal Projection

As a special case of the preceding example, let have the Euclidean inner product. The vectors and
form an orthonormal basis for the
-plane. Thus, if is any vector in , the orthogonal
projection of onto the -plane is given by

(See Figure 8.1.3.)