17.4. ORTHOGONAL COMPLEMENT 413
17.4 Orthogonal ComplementTheorem 17.4.1 Let V be a finite dimensional inner product space and let H be a sub-space. Then H⊥ defined by H⊥ = {y ∈V : ⟨y,h⟩= 0 for all h ∈ H} is also a subspace and
V = H⊕H⊥
where the symbol means that every vector in V can be obtained as a sum of two vectors,one in H and the other in H⊥ in exactly one way.
Proof: Let {u1, · · · ,ur} be an orthonormal basis for H. Define the projection map
Pv≡r
∑k=1⟨v,uk⟩uk
As shown in Theorem 17.3.2, Pv is the unique point of H closest to v and ⟨v−Pv,h⟩ = 0for all h ∈ H. Thus v = v−Pv+Pv which shows that V = H +H⊥. It remains to verifythat there is a unique way to represent v as such a sum. Suppose then that
v = h+y = ĥ+ ŷ
where the y vectors are in H⊥ and the h vectors in H. Then
h− ĥ = ŷ−y
Now H⊥ is a subspace and so, taking the inner product of both sides with h− ĥ, you get∣∣h− ĥ∣∣2 = 0
and so h = ĥ which then requires that y = ŷ. ■Note in the above that it is routine from the formula to see that P is linear and also
P2 = P. This is why it is called a projection.
17.5 Fourier SeriesOne of the most important applications of these ideas about approximation is to Fourierseries. Much more can be said about these than will be presented here. However, Theorem17.3.2 is a very useful framework for discussing these series.
For x ∈ R, define eix by the following formula
eix ≡ cosx+ isinx
The reason for defining it this way is that ei0 = 1, and(eix)′= ieix if you use this definition.
Also it follows from the trigonometry identities that ei(x+y) = eixeiy. This is because
eixeiy = (cosx+ isinx)(cosy+ isiny)
= cosxcosy− sinxsiny+ i(sinxcosy+ cosxsiny)
= cos(x+ y)+ isin(x+ y) = ei(x+y)