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.