Page 498 - Elementary_Linear_Algebra_with_Applications_Anton__9_edition

6.4 In this section we shall show how orthogonal projections can be used to
solve certain approximation problems. The results obtained in this section
BEST APPROXIMATION; have a wide variety of applications in both mathematics and science.
LEAST SQUARES

Orthogonal Projections Viewed as Approximations

If P is a point in ordinary 3-space and W is a plane through the origin, then the point Q in W that is closest to P can be

obtained by dropping a perpendicular from P to W (Figure 6.4.1a). Therefore, if we let , then the distance between P

and W is given by

In other words, among all vectors w in W, the vector minimizes the distance (Figure 6.4.1b).

Figure 6.4.1

There is another way of thinking about this idea. View u as a fixed vector that we would like to approximate by a vector in
W. Any such approximation w will result in an “error vector,”

that, unless u is in W, cannot be made equal to . However, by choosing

we can make the length of the error vector

as small as possible. Thus we can describe as the “best approximation” to u by vectors in W. The following theorem
will make these intuitive ideas precise.

THEOREM 6.4.1

Best Approximation Theorem is the best

If W is a finite-dimensional subspace of an inner product space V, and if u is a vector in V, then
approximation to u from W in the sense that

for every vector w in W that is different from .

Proof For every vector w in W, we can write