Orthogonal complements
Orthogonal Complement of the Orthogonal Complement Finding that the orthogonal complement of the orthogonal complement of V is V
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- 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.
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