Orthogonal complements
dim(V) + dim(orthogonal complement of V)=n Showing that if V is a subspace of Rn, then dim(V) + dim(V's orthogonal complement) = n
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- Let's say I've got some subspace of Rn called V.
- So V is a subspace of Rn.
- And let's say that I know its basis.
- Let's say the set.
- So I have a bunch of-- let me make that bracket a little
- nicer-- so let's say the set of the vectors v1, v2, all the
- way to vk, let's say that this is equal to-- this is the
- basis for V.
- And just as a reminder, that means that V's vectors both
- span V and they're linearly independent.
- You can kind of see there's a minimum set of vectors in Rn
- that span V.
- So, if I were to ask you what the dimension of V is, that's
- just the number of vectors you have in your
- basis for the subspace.
- So we have 1, 2 and we count to k vectors.
- So it is equal to k.
- Now let's think about, if we can somehow figure out what
- the dimension of the orthogonal
- complement of V can be.
- And to do that, let's construct a matrix.
- Let's construct a matrix whose column vectors
- are these basis vectors.
- So let's construct a matrix A, and let's say
- it looks like this.
- First column is v1.
- This first basis vector right there.
- v2 is the second one, and then you go all the way to vk.
- Just to make sure we remember the dimensions, we have k of
- these vectors, so we're going to have k columns.
- And then how many rows are we going to have?
- Well, as a member of Rn, so these are all going to have n
- entries in each of these vectors, there's going to be
- an n-- we're going to have n rows and k columns.
- It's an n by k matrix.
- Now, what's another way of expressing the subspace V?
- Well, the basis for V is-- or V is spanned by these basis
- vectors, which is the columns of these.
- So if I talk about the span-- so let me write out this-- V
- is equal to the span of these guys, v1, v2,
- all the way to vk.
- And that's just the same thing as the column space of A.
- Right?
- These are the column vectors, and the span of them, that's
- equal to the column space of A.
- Now, I said a little while ago, we want to somehow relate
- to the orthogonal complement of V.
- Well, what's the orthogonal complement of the
- column space of A?
- The orthogonal complement of the column space of A, I
- showed you-- I think it was two or three videos ago-- that
- the column space of A's orthogonal complement is equal
- to-- you could either view it as the null space of A
- transpose, or another way you call it is the left
- null space of A.
- This is equivalent to the orthogonal complement of the
- column space of A, which is also going to be equal to,
- which is also since this piece right here is the same thing
- as V, you take it's orthogonal complement, that's the same
- thing as V's orthogonal complement.
- So if we want to figure out there orthogonal complement
- of-- if we want to figure out the dimension-- if we want to
- figure out the dimensional of the orthogonal complement of
- V, we just need to figure out the dimension of the left null
- space of A, or the null space of A's transpose.
- Let me write that down.
- So the dimension-- get you tongue-tied sometimes-- the
- dimension of the orthogonal complement of V is going to be
- equal do the dimension of A transpose.
- Or another way to think of it is-- sorry, not just the
- dimension of A transpose, the dimension of the null space of
- A transpose.
- And if you have a good memory, I don't use the word a lot,
- this thing is the nullity-- this is the
- nullity of A transpose.
- The dimension of your null space is nullity, the
- dimension of your column space is your rank.
- Now let's see what we can do here.
- So let's just take A transpose, so you can just
- imagine A transpose for a second.
- I can just even draw it out.
- It's going to be a k by n matrix that looks like this.
- These columns are going to turn into rows.
- This is going to be v1 transpose, v2 transpose, all
- the way down to vk transpose vectors.
- These are all now row vectors.
- So we know one thing.
- We know one relationship between the rank and nullity
- of any matrix.
- We know that they're equal to the number of columns we have.
- We know that the rank of A transpose plus the nullity of
- A transpose is equal to the number of
- columns of A transpose.
- We have n columns.
- Each of these have n entries.
- It is equal to n.
- We saw this a while ago.
- And if you want just a bit of a reminder of where that comes
- from, when you take a-- if I wrote A transpose as a bunch
- of column vectors, which I can, or maybe let me take some
- other vector B, because I want to just remind you where this,
- why this made sense.
- If I take some vector B here, and it has got a bunch of
- column vectors, b1, b2, all the way to bn, and I put it
- into reduced row echelon form, you're going to have some
- pivot columns and some non-pivot columns.
- So let's say this is a pivot column.
- You know, I got a 1 and a bunch of 0's, let's say that
- this is one of them, and then let's say I got one other one
- that's out, and it would be a 0 there, it's a 1 down there,
- and everything else is a non-pivot column.
- I showed you in the last video that your basis for your
- column space is the number of pivot columns you have. So
- these guys are pivot columns.
- The corresponding column vectors form a basis for your
- column space.
- I showed you that in the last video.
- And so, if you want to know the dimension of your column
- space, you just have to count these things.
- You just count these things.
- This was equal to the number of, well, for this B's case,
- the rank of B is just equal to the number of pivot columns I
- have. Now the nullity is the dimension of your null space.
- We've done multiple problems where we found the null space
- of matrices.
- And every time, the dimension, it's a bit obvious, and I
- actually showed you this proof, it's related to the
- number of free columns you have, or non-pivot columns.
- So, if you have no pivot columns, then you are -- if
- all of your columns are pivot columns, and none of them have
- free variables or are associated with free
- variables, then you're null space is going to be trivial.
- It's just going to have the 0 vector.
- But the more free variables you have, the more
- dimensionality your null space has.
- So the free columns correspond to the null space, and they
- form actually a basis for your null space.
- And because of that, the basis for your null space vectors,
- plus the basis for your column space, is equal to the total
- number of columns you have. I showed that to you in the
- past, but it's always good to remind ourselves
- where things come from.
- But this was just a bit of a side.
- I did this with a separate vector B.
- Just to remind ourselves where this thing
- right here came from.
- Now, in the last video, I showed you that the rank of A
- transpose is the same thing is the rank of A.
- This is equal to, this part right here, is the same thing
- as the rank of A.
- I showed you that in the last video.
- When you transpose a matrix, it doesn't change its rank, or
- it doesn't change the dimension of its column space.
- So we can rewrite this statement, right here, as the
- rank of A plus the nullity of A transpose is equal to n, and
- the rank of A is the same thing as the dimension of the
- column space of A.
- And then the nullity of A transpose is the same thing as
- the dimension of the null space of A transpose-- that's
- just the definition of nullity-- they're going to be
- equal to n.
- Now what's the dimension-- what's the column space of A?
- The column space of A, that's what's spanned by these
- vectors right here, which were the basis for V.
- So this is the same thing as the dimension of V.
- The column space of A is the same thing as the dimension of
- my subspace V that I started this video with.
- And what is the null space of A transpose?
- The null space of A transpose, we saw already, that's the
- orthogonal complement of V.
- So I could write this as plus the dimension of the
- orthogonal complement of V is equal to n.
- And that's the result we wanted.
- If V is a subspace of Rn, that n is the same thing as that n,
- then the dimension of V plus the dimension of the
- orthogonal complement of V is going to be equal to n.
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