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Current time:0:00Total duration:12:18

Orthogonal complement of the orthogonal complement

Video transcript

let's have some subspace of RN called V let me draw it like this so that is RN that is RN at some subspace of it that will call V right here so that is my subspace V we know that the orthogonal complement of V the orthogonal complement of V is equal to the the set of all of the members of RN all of the members of RN so X is a member of RN such that X dot V is equal to 0 for every for every V that is a member of our subspace so our orthogonal complement of our subspaces only 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 and that's the zero vector let's 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 we want the orthogonal complement so that's the orthogonal complement that's the whole thing in pink but 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 such that X dot let me just say 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 now we can think 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 there's maybe this is just a subset or 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 yet we don't know whether those 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 so let's see if we can think about that a little bit better so let's say 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 in the last video that any vector in RN can be represented by a sum of some vector in a subspace and the subspaces complement so if I have so we know that X can be represented we can say that X can be represented as a sum of two vectors wanted 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 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 you know it could be 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 up but we're not sure whether V equals that thing but we say look it any anything that's in are in the Christoffel complement of the orthogonal complement is going to be a member of RN and anything in RN can be represented as the 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 dot if I take the dot product of X with W what is this going to be equal to well this is the orthogonal complement of the orthogonal complement so if 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 zero by definition these are all of the vectors this vector is definitely orthogonal to anything in just V perp write anything in V perp perp is orthogonal to anything in V perp so this thing is going to be equal to zero but what's another way of writing XW 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 is an original subspace and if you take the dot product of anything with our in our original subspace with this or with 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 W XW X is a member of the orthogonal complement of the orthogonal complement so u 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 0 vector that's the only vector in RN then 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 implies that our original vector X that our original vector X is equal to is equal to V is equal to V and V is a member of our subspace V right so that tells us that X X is a member of our subspace B so what we just show that if it's something is a member of the 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 all of this has to be has to be inside of this right there so this 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 this video said hey anything in our subspace is going to be a member of our orthogonal complement that you can kind of reason that in your head but let's use the same argument to just be a little bit more rigorous about it just be a little bit more rigorous about it so let's say let's say right now we say look if anything is in the orthogonal complement of the orthogonal complement then it's going to be in the original subspace let's go the other way let's say that something is in the original subspace is in the original subspace just like that if anything 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 start we have the orthogonal complement let me just draw that first so V perp and then you have your thought you have the complement you have the orthogonal complement of the orthogonal complement which could be this set right here right this is V perp 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 a sum of a vector that's here an S and a vector that's here so if I say if I say that dub you let me do it in purple if I say that the vector W or let me write it this way the vector V the vector V can be represented as the sum of the vector W and the vector and the vector X where 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 it's orthogonal complement and we learned that anything in RN can be represented in the sum of something in a subspace and the subspaces 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 what happens if I dot V if I dot V with W I'm doing the exact same argument that I did before well V anything 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 zero what what is what else is that going to be equal to well if we take if we write V in this way V dot W is the same thing as this thing dot W so W plus X dot W and this is going to be equal to wwww plus X dot W and then what's X dot W X is in the orthogonal complement of the orthogonal complement and W is in the orthogonal complement so if you take their dot product you're going to get 0 they're orthogonal to each other so this is just equal to W W or the length of W squared and since that has to equal 0 right we just have a bunch of equals here that tells us that once again the vector W has to be equal to 0 and that V 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 so we've just shown that if 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 of the orthogonal complement so we've proven it boat we've proved proved both ways if you look at the original statement we learned we wrote here that if you're a member of the orthogonal complement of the orthogonal complement your original member of the original subspace so we proved 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 ad and V perp perp is a member of our subspace so our subspace and V perp perp are the same set and of course it overlaps this equals this and of course it overlaps with V perp it's orthogonal complement only at the zero vector right there