Proof Regarding Projections
Suppose that we have an inner product space 
and a finite dimensional subspace 
.
Let  = proj_W \space \alpha)
. Prove that 
for all 
and 
in .
I've gone about various ways of trying to prove this; my main attack of this problem was to write out a basis for W, and then expand out the projection expression in terms of each basis vector, and then write out the inner products and see what I get (direct proof). This just leads me to a dead end.
I was able to sketch a quick proof in the Euclidean space with the dot product, but of course this is much too restrictive to be a proof (it's easier when you know exactly how the dot product is defined). This led me to the idea of representing the arbitrary dot product as a matrix, and then setting up the equations and trying another direct proof, but again it looks messy, and it seems like this kind of direct proof could get extremely long and bogged down, especially when you start expanding out individual linear combinations and distributing across inner products, etc. etc.
I need a new idea to approach this problem...any help would be appreciated! =)
Re: Proof Regarding Projections
Quote:
Originally Posted by
TaylorM0192 
Suppose that we have an inner product space
and a finite dimensional subspace
.
Let
. Prove that
for all
and
in
.
I've gone about various ways of trying to prove this; my main attack of this problem was to write out a basis for W, and then expand out the projection expression in terms of each basis vector, and then write out the inner products and see what I get (direct proof). This just leads me to a dead end.
I was able to sketch a quick proof in the Euclidean space with the dot product, but of course this is much too restrictive to be a proof (it's easier when you know exactly how the dot product is defined). This led me to the idea of representing the arbitrary dot product as a matrix, and then setting up the equations and trying another direct proof, but again it looks messy, and it seems like this kind of direct proof could get extremely long and bogged down, especially when you start expanding out individual linear combinations and distributing across inner products, etc. etc.
I need a new idea to approach this problem...any help would be appreciated! =)
Is this even true? You are stating that all projections onto finite dimensional subspaces are unitary. But, isn't this true if and only if 
? Am I misunderstanding?
Re: Proof Regarding Projections
Well, since I haven't proved anything yet, I suppose it could be false;)
In any case, the problem comes from Hoffman and Kunze #8.2.12:
"Let W be a finite dimensional subspace of an inner product space V, and let E be the orthogonal projection of V on W. Prove that (Ea|b) = (a|Eb) for all a and b in V."
Re: Proof Regarding Projections
Quote:
Originally Posted by
TaylorM0192 Well, since I haven't proved anything yet, I suppose it could be false;)
In any case, the problem comes from Hoffman and Kunze #8.2.12:
"Let W be a finite dimensional subspace of an inner product space V, and let E be the orthogonal projection of V on W. Prove that (Ea|b) = (a|Eb) for all a and b in V."
Oh, ok, fine, so it is an orthogonal projection! In this case you can actually find a nice discussion of this on wiki.
Re: Proof Regarding Projections
Hmm...is my notation incorrect?
I thought 
meant "the orthogonal projection of onto " and 
meant "the orthogonal component of onto ."
Anyway, this is how I remember the notation from my calculus courses, although I admit I don't see it used much in linear algebra texts.
And in any case, what would be the difference between a projection and an orthogonal projection?
Re: Proof Regarding Projections
Quote:
Originally Posted by
TaylorM0192 Hmm...is my notation incorrect?
I thought
meant "the orthogonal projection of
onto
" and
meant "the orthogonal component of
onto
."
Anyway, this is how I remember the notation from my calculus courses, although I admit I don't see it used much in linear algebra texts.
And in any case, what would be the difference between a projection and an orthogonal projection?
You know that every projection 
naturally decomposes as 
. In an inner product space whenever we have complements (i.e. subspaces whose direct sum is [implicitly assuming they are disjoint]) we'd love for them to be orthogonal complements. The fact that 
and 
are complements does not imply that they are orthogonal complements. When this nice condition is satisfied we call 
an orthogonal projection.
Re: Proof Regarding Projections
I really dislike my linear algebra professor this quarter...it has been a journey of self-discovery from day one!
Thanks man~
