Page 236 - A Course in Linear Algebra with Applications

220 Chapter Seven: Orthogonality in Vector Spaces

Let / = ao+aix-\-a,2X 2 be an element of Ps(R). By 7.2.3,
1
a polynomial / belongs to S - if and only if it is orthogonal
to 1 and x; the conditions for this are

1
f 1 1
< , 1 > = / f{x)dx = a 0 + -a x + -a 2 = 0
/

and

1
f 1 1 1
< , x >= / xf(x)dx = -a 0 + -ai + -a 2 = 0.
/

Solving these equations, we find that ao = t/6, a\ = —t and
2
= t, where t is arbitrary. Hence / = t(x —£+|) is the most
a 2
1
1
general element of S -. It follows that S - is the 1-dimensional
subspace generated by the polynomial x 2 — x + .
|
,J
Notice in the last example that dim(S') + dim(5 -) = 3,
the dimension of Pa(R). This is no coincidence, as the follow-
ing fundamental theorem shows.
Theorem 7.2.4
Let S be a subspace of a finite-dimensional real inner product
space V; then

± ±
V = S®S and dim(V) = dim(S) + dim(5 ).

Proof
According to the definition in 5.3, we must prove that V =
1
S + S - and S D S x = 0. The second statement is true by
7.2.3, but the first one requires proof.
1
Certainly, if S = 0, then S - = V and the result is clear.
Having disposed of this case, we may assume that S is non-
v
zero and choose a basis ^,..., v m for S. Extend this basis of