Page 483 - Elementary_Linear_Algebra_with_Applications_Anton__9_edition

Figure 6.3.1

THEOREM 6.3.4

Projection Theorem

If W is a finite-dimensional subspace of an inner product space V, then every vector u in V can be expressed in exactly
one way as

(3)

where is in W and is in .

The vector in the preceding theorem is called the orthogonal projection of u on W and is denoted by . The vector

is called the component of u orthogonal to W and is denoted by . Thus Formula 3 in the Projection Theorem can

be expressed as

(4)

Since it follows that

so Formula 4 can also be written as (5)
(Figure 6.3.2).

Figure 6.3.2