Page 238 - A Course in Linear Algebra with Applications

222 Chapter Seven: Orthogonality in Vector Spaces

1
Hence V = S + S -, as required.
An important consequence of the theorem is
Corollary 7.2.5
If S is a subspace of a finite-dimensional real inner product
space V, then
(S^ = S.

Proof
Every vector in S is certainly orthogonal to every vector in
±
1 1
S ; thus S is a subspace of (S- )- . On the other hand, a
computation with dimensions using 7.2.4 yields
1
± ±
dim((5 ) ) =dim(V) - d i m ^ )
= dim(V) - (dim(F) - dim(S))
= dim(S)

± ±
Therefore S={S ) .
Projection on a subspace
The direct decomposition of an inner product space into
a subspace and its orthogonal complement afforded by 7.2.4
leads to wide generalization of the elementary notion of pro-
jection of one vector on another, as described in 7.1. This
generalized projection will prove invaluable during the discus-
sion of least squares in 7.4.
Let V be a finite-dimensional real inner product space, let
1
S be a subspace and let v an element of V. Since V = StSS- ,
there is a unique expression for v of the form

1
v = s + s -

1 1
where s and s- belong to S and S - respectively. Call s the
projection ofv on the subspace S. Of course, s -1 is the projec-
3
1
tion of v on the subspace S . For example, if V is R , and