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# Orthogonal complement of the orthogonal complement

Video transcript

Let's say I have some subspace
of rn called v. Let me draw it like this. So that it is r n. And I have some subspace of it
we'll call v right here. So that is my subspace v. We know that the orthogonal
complement v is equal to the set of all of the
members of rn. So x is a member of rn. Such that x dot v is equal to 0
for every v that is a member of r subspace. So our orthogonal complement of
our subspace is going to be all of the vectors that
are orthogonal to all of these vectors. And we've seen before that they
only overlap-- there's only one vector that's
a member of both. That's the zero vector. It's right there. Let's take the orthogonal
complement. Let's say it's this set right
here in pink, so that's the orthogonal complement. Fair enough. Now, what if we were to think
about the orthogonal complement of the orthogonal
complement? So, that's the orthogonal
complement in pink. We want the orthogonal
complement of that. So this is going to be all
of the x's-- let's just write it like this. All of the x's that are members
of rn such that x dot w is equal to 0. For every w that is a member
of the orthogonal complement of v. That's what that thing
is saying. So it's all of the vectors in
rn that are orthogonal to everything here. Obviously, all of the things in
v are going to be a member of that because these guys are
orthogonal to everything in these guys. But maybe this is just a subset
of the orthogonal complement of the orthogonal
complement. So maybe this thing in blue
right here looks like this. Maybe it's a slightly
larger set than v. Maybe there are some things,
these things that I'm shading in blue, maybe there are some
vectors that are orthogonal to the orthogonal complement of v
but that are outside of v. We don't know that yet. We don't know whether this
area right here exists. Or maybe the orthogonal
complement of the orthogonal complement. Maybe that takes us back to v. Maybe it's like the transpose
or an inverse function where it just goes back to our
original subspace. Let's see if we can
think about that a little bit better. Let's say that I have some
member of the orthogonal complement of the orthogonal
complement. So let's say I have some vector
x that is a member of the orthogonal complement of
the orthogonal complement. Now, we saw on the last video
that any vector in rn can be represented by a sum of some
vector in a subspace and the subspace's complement. So we know that x can be
represented-- we can say that x can be represented as the
sum of two vectors. One that's in v and one that's
in the orthogonal complement of v. So one, let's call that the
vector that's in v and let's call w the vector that's
in the orthogonal complement of V. Let me write it like this. Where v is a member of the
subspace v and the vector w is a member of the orthogonal
complement of v. Right? So this is some member. It could be some guy out here. It could be some
guy over here. He's a member of the orthogonal
complement of the orthogonal complement. Which is this whole area here. Which v is a subset of, but
we're not sure whether v equals that thing. But we say, look, anything
that's in the orthogonal complement of your orthogonal
complement, is going to be a member of rn. And anything in rn can be
represented as a sum of a vector in v and a vector
in the orthogonal complement of v. So that's all I wrote
right there. Now, what happens if I take the
dot product of x with w? What is this going
to be equal to? This is the orthogonal
complement of the orthogonal complement. Would you take the dot product
of any vector in this with any vector in the orthogonal
complement, which this vector is. Right? It's a member of the orthogonal
complement. You're going to get
0 by definition. These are all of the vectors. This factor is definitely
orthogonal to anything in just v perp right? Anything in v perp perp
is orthogonal to anything in v perp. So, this thing is going
to be equal to 0. But what's another way
of writing x dot w? We could write it like this. This is the same thing
as v plus w dot w. Which is the same thing as
v dot w plus w dot w. Now, what is v dot w? v is a member of our
original subspace. And if you take the dot product
of anything in our original subspace anything in
its orthogonal complement, you're going to get 0. So this term right here is going
to be 0, and you're just going to get this term which
is the same thing as the length of our vector
w squared. Now, that has to equal 0. Remember we just
wrote x dot w. x is a member of the orthogonal
complement of your orthogonal complement. So, you dot that with anything
in the orthogonal complement, that's got to be equal 0. But, if we write it the other
way, if we write it as the sum of v plus w and distribute this
w, we say that's the same thing as the magnitude
of w squared. So the magnitude of w squared
has got to be equal to 0. The magnitude of w squared, or
the length of w squared, has got to be equal to 0. Which tells us that w
is the zero vector. That's the only factor in rn
when you take its length and, especially when you square
it, you get 0. But you could just
take its length. So what does that mean? That means that our original
vector x is equal to v plus w. But w is just equal to 0. So that implies that
our original vector x is equal to v. And v is a member of
our subspace v. Right? So that tells us that x is a
member of our subspace v. So we just showed that if
something is a member of the orthogonal complement of the
orthogonal complement then that same vector has
to be a member of the original subspace. So there is no such thing as
something being in the orthogonal complement of the
orthogonal complement and not being a member of our
original subspace. All of this has to be inside
of this right there. So there is no outside
blue space like that. All of that is our original
subspace if you want to view it that way. Now I just at the beginning of
the video, anything in our subspace is going to be a
member of our orthogonal complement. And then you can kind of reason
that in your head. Let's use the same argument to
just be a little bit more rigorous about it. Right now we say if anything
is in the orthogonal complement of the orthogonal
complement, then it's going to be the original subspace. Let's go the other way. Let's say that something
is in the original subspace just like that. Let me draw another graph
right here because this might be useful. Let me draw rn again. Let me draw all of
rn like that. Now, we have the orthogonal
complement. Let me just draw that first.
So v perp And then you have the orthogonal complement of
the orthogonal complement which could be this
set right here. Right? This is v perp. I haven't even drawn
the subspace v. All I've shown is, I have some
subspace here, which I happen to call v perp. And then I have the orthogonal
complement of that subspace. So this means that anything in
rn can be represented as the sum of a vector that's here
and a vector that's here. So, if I say that w-- let
me do it in purple. If I say the vector w-- let
me write it this way. The vector v can be represented
as the sum of the vector w and the vector x where
w is a member of the orthogonal complement of v or
v perp And x is a member of its orthogonal complement. Notice, all I'm saying, I could
have called this set s. And then this would have been
s and its orthogonal complement. And we learned that anything in
rn could be represented as the sum of something in a
subspace and the subspace's orthogonal complement. So it doesn't matter that v is
somehow related to this. It can be represented as
a sum of a vector here plus a vector there. Fair enough. Now, what happens if
I dot v with w? I'm doing the exact same
argument that I did before. Well, if you take anything
that's a member of our original subspace, and you dot
it with anything in its orthogonal complement, that's
going to give us 0. What else is that going
to be equal to? If we write v in this way, v
dot w is the same thing as this thing dot w. So w plus x dot-- and this is
going to be equal to w dot w plus x dot w. And what's x dot w? x is in the
orthogonal complement of your orthogonal complement. And w is in the orthogonal
complement. So if you take the dot
product, you're going to get 0. They're orthogonal
to each other. So this is just equal to w dot
w or the length of w squared. And since since has to equal
0, we just have a bunch of equals here, that tells us that
once again the vector w has to be equal to 0. So that tells us v is
equal to w plus x. But if w is equal to 0, then v
is going to be equal to x. So we've just shown that if v is
a member of the subspace v, then v is a member of the
orthogonal complement of the orthogonal complement. Right? v is equal to x, which is
a member of the orthogonal complement or the orthogonal
complement. So we've proven it both ways. If you look at the original
statement, we wrote here that if you're a member of the
orthogonal complement of the orthogonal complement,
you're a member of the original subspace. So, we've proven this and,
earlier in the video, we proved that if x is a member of
the orthogonal complement of the orthogonal complement,
then x is a member of our subspace. So these two things
are equivalent. Anything that's in
the subspace is a member of v perp perp. Anything in v perp perp is
a member of our subspace. So, our subspace in v perp
perp are the same set. And of course it overlaps. This equals this. And of course it overlaps with
V perp and its orthogonal complement only at the zero
vector right there.