WEBVTT
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OK.
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This is linear algebra
lecture eleven.
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And at the end of lecture ten,
I was talking about some vector
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spaces, but they're --
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the things in
those vector spaces
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were not what we
usually call vectors.
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Nevertheless, you
could add them and you
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could multiply by numbers,
so we can call them vectors.
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I think the example
I was working
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with they were matrices.
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So the -- so we had
like a matrix space,
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the space of all three
by three matrices.
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And I'd like to just pick
up on that, because --
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we've been so specific about
n dimensional space here,
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and you really want to see that
the same ideas work as long
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as you can add and
multiply by scalars.
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So these new, new vector
spaces, the example I took
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was the space M of all
three by three matrices.
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OK.
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I can add them, I can
multiply by scalars.
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I can multiply two of them
together, but I don't do that.
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That's not part of the
vector space picture.
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The vector space part is
just adding the matrices
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and multiplying by numbers.
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And that's fine, we stay
within this space of three
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by three matrices.
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And I had some subspaces
that were interesting,
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like the symmetric, the
subspace of symmetric matrices,
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symmetric three by threes.
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Or the subspace of upper
triangular three by threes.
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Now I, I use the word subspace
because it follows the rule.
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If I add two symmetric
matrices, I'm still symmetric.
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If I multiply two
symmetric matrices,
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is the product
automatically symmetric?
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No.
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But I'm not
multiplying matrices.
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I'm just adding.
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So I'm fine.
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This is a subspace.
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Similarly, if I add two
upper triangular matrices,
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I'm still upper triangular.
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And, that's a subspace.
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Now I just want to take these
as example and ask, well,
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what's a basis
for that subspace?
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What's the dimension
of that subspace?
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And what's bd- dimension
of the whole space?
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So, there's a natural basis for
all three by three matrices,
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and why don't we
just write it down.
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So, so M, a basis for M.
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Again, all three by threes.
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OK.
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And then I'll just count how
many members are in that basis
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and I'll know the dimension.
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And OK, it's going to
take me a little time.
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In fact, what is the dimension?
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Any idea of what I'm
coming up with next?
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How many numbers does it
take to specify that three
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by three matrix?
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Nine.
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Nine is the, is the
dimension I'm going to find.
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And the most obvious basis
would be the matrix that's
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that matrix and then this
matrix with a one there
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and that's two of them,
shall I put in the third one,
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and then onwards, and
the last one maybe
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would end with the one.
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OK.
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That's like the standard basis.
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In fact, our space is
practically the same
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as nine dimensional space.
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It's just the nine numbers
are written in a square
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instead of in a column.
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But somehow it's different
and, and ought to be thought
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of as --
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natural for itself.
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Because now what about the
symmetric three by threes?
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So that's a subspace.
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Just let's just think, what's
the dimension of that subspace
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and what's a basis
for that subspace.
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OK.
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And I guess this
question occurs to me.
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If I look at this subspace
of symmetric three
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by threes, well, how many
of these original basis
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members belong to the subspace?
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I think only three of them do.
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This one is symmetric.
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This last one is symmetric.
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And the one in the middle with
a, with a one in that position
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-- in the two two position,
would be symmetric.
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But so I've got three of these
original nine are symmetric,
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but, so this is an
example where --
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but that's, that's
not all, right?
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What's the dimension?
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Let's put the dimensions down.
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Dimension of the,
of M, was nine.
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What's the dimension of --
shall we call this S -- is what?
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What's the dimension of this?
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I'm sort of taking simple
examples where we can, we can,
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spot the answer to
these questions.
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So how many -- if I
have a symmetric --
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think of all symmetric
matrices as a subspace,
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how many parameters do I choose
in three by three symmetric
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matrices?
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Six, right.
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If I choose the
diagonal that's three,
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and the three entries
above the diagonal,
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then I know what the
three entries below.
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So the dimension is six.
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I guess what's the
dimension of this here?
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Let's call this space
U for upper triangular.
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So what's the dimension of that
space of all upper triangular
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three by threes?
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Again six.
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Again six.
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And, but we haven't got a -- we
haven't seen -- well, actually,
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maybe we have got a basis here
for, the upper triangulars.
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I guess six of these guys,
one, two, three, four,
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and a, and a couple more,
would be upper triangular.
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So there's a accidental case
where the big basis contains
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in it a basis for the subspace.
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But with the symmetric
guy, it didn't have.
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The symmetric guy the,
basis -- so you see --
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a basis is the basis
for the big space,
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we generally need to think it
all over again to get a basis
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for the subspace.
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And then how do I
get other subspaces?
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Well, we spoke before about, the
subspace the symmetric matrices
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and the upper triangular.
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This is symmetric
and upper triangular.
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What's the, what's the
dimension of that space?
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OK.
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Well, what's in that space?
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So what's -- if a matrix
is symmetric and also upper
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triangular, that
makes it diagonal.
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So this is the same as
the diagonal matrices,
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diagonal three by threes.
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And the dimension of this,
of S intersect U, right --
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you're OK with that symbol?
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That's, that's the vectors
that are in both S and U,
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and that's D.
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So S intersect U
is the diagonals.
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And the dimension of the
diagonal matrices is three.
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And we've got a
basis, no problem.
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OK, as I write that, I think,
OK, what about putting --
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so this is like,
this intersection --
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is taking all the
vectors that are
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in both, that are symmetric
and also upper triangular.
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Now we looked at the union.
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Suppose I take the
matrices that are symmetric
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or upper triangular.
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What -- why was that no good?
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So why is it no -- why I
not interested in the union,
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putting together
those two subspaces?
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So this, these are matrices that
are in S or in U, or possibly
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both, so they, the
diagonals included.
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But what's bad about this?
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It's not a subspace.
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It's like having,
taking, you know,
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a couple of lines in the
plane and stopping there.
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A line -- this is -- so there's
a three dimensional subspace
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of a nine dimensional space,
there's -- ooh, sorry, six.
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There's a six
dimensional subspace
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of a nine dimensional space.
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There's another one.
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But they, they're headed
in different directions,
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so we, we can't just
put them together.
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We have to fill in.
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So that's what we do.
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To get this bigger space that
I'll write with a plus sign,
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this is combinations of
things in S and things in U.
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OK.
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So that's the final space
I'm going to introduce.
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I have a couple of subspaces.
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I can take their intersection.
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And now I'm interested in not
their union but their sum.
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So this would be the,
this is the intersection,
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and this will be their sum.
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So what do I need
for a subspace here?
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I take anything in S
plus anything in U.
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I don't just take things
that are in S and pop
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in also, separately,
things that are in U.
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This is the sum of
any element of S,
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that is, any symmetric
matrix, plus any
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in U, any element of U.
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OK.
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Now as long as we've
got an example here,
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tell me what we get.
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If I take every
symmetric matrix,
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take all symmetric
matrices, and add them
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to all upper
triangular matrices,
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then I've got a whole lot of
matrices and it is a subspace.
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And what's -- it's
a vector space,
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and what vector space
would I then have?
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Any idea what,
what matrices can I
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get out of a symmetric
plus an upper triangular?
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I can get anything.
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I get all matrices.
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I get all three by threes.
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It's worth thinking about that.
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It's just like stretch your
mind a little, just a little,
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to, to think of these subspaces
and what their intersection is
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and what their sum is.
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And now can I give you
a little -- oh, well,
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let's figure out the dimension.
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So what's the
dimension of S plus U?
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In this example is nine, because
we got all three by threes.
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So the original spaces had, the
original symmetric space had
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dimension six and the original
upper triangular space
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had dimension six.
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And actually I'm seeing
here a nice formula.
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That the dimension of S
plus the dimension of U --
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if I have two subspaces,
the dimension of one plus
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the dimension of the other
-- equals the dimension
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of their intersection plus
the dimension of their sum.
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Six plus six is three plus nine.
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That's kind of satisfying, that
these natural operations --
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and we've -- this
is it, actually,
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this is the set of
natural things to do with,
00:14:27.650 --> 00:14:29.390
with subspaces.
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That, the dimensions
come out in a good way.
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OK.
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Maybe I'll take just one more
example of a vector space
00:14:43.540 --> 00:14:48.270
that doesn't have vectors in it.
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It's come from
differential equations.
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So this is a one
more new vector space
00:14:57.120 --> 00:14:59.580
that we'll give just
a few minutes to.
00:14:59.580 --> 00:15:06.380
Suppose I have a differential
equation like d^2y/dx^2+ y=0.
00:15:09.590 --> 00:15:12.200
OK.
00:15:12.200 --> 00:15:14.175
I look at the solutions
to that equation.
00:15:16.790 --> 00:15:21.230
So what are the solutions
to that equation?
00:15:21.230 --> 00:15:24.815
y=cos(x) is a solution.
00:15:28.250 --> 00:15:31.960
y=sin(x) is a solution.
00:15:31.960 --> 00:15:37.730
y equals -- well, e to the (ix)
is a solution, if you want,
00:15:37.730 --> 00:15:41.150
if you allow me to put that in.
00:15:41.150 --> 00:15:42.520
But why should I put that in?
00:15:42.520 --> 00:15:45.890
It's already there.
00:15:45.890 --> 00:15:48.650
You see, I'm really looking
at a null space here.
00:15:48.650 --> 00:15:52.730
I'm looking at the null space
of a differential equation.
00:15:52.730 --> 00:15:55.210
That's the solution space.
00:15:55.210 --> 00:16:01.000
And describe the solution
space, all solutions
00:16:01.000 --> 00:16:03.290
to this differential equation.
00:16:03.290 --> 00:16:07.450
So the equation is y''+y=0.
00:16:07.450 --> 00:16:11.630
Cosine's, cosine's a
solution, sine is a solution.
00:16:11.630 --> 00:16:14.430
Now tell me all the solutions.
00:16:14.430 --> 00:16:18.930
They're -- so I
don't need e^(ix).
00:16:18.930 --> 00:16:21.220
Forget that.
00:16:21.220 --> 00:16:23.420
What are all the
complete solutions?
00:16:30.510 --> 00:16:32.860
Is what?
00:16:32.860 --> 00:16:34.840
A combination of these.
00:16:34.840 --> 00:16:38.070
The complete
solution is y equals
00:16:38.070 --> 00:16:48.350
some multiple of the cosine
plus some multiple of the sine.
00:16:48.350 --> 00:16:51.520
That's a vector space.
00:16:51.520 --> 00:16:52.660
That's a vector space.
00:16:52.660 --> 00:16:54.530
What's the dimension
of that space?
00:16:54.530 --> 00:16:57.500
What's a basis for that space?
00:16:57.500 --> 00:17:00.100
OK, let me ask
you a basis first.
00:17:00.100 --> 00:17:03.850
If I take the set of
solutions to that second order
00:17:03.850 --> 00:17:06.450
differential equation --
00:17:06.450 --> 00:17:10.520
there it is, those
are the solutions.
00:17:10.520 --> 00:17:12.010
What's a basis for that space?
00:17:14.680 --> 00:17:16.480
Now remember, what's
the, what question I
00:17:16.480 --> 00:17:16.980
asking?
00:17:16.980 --> 00:17:18.890
Because if you know the
question I'm asking,
00:17:18.890 --> 00:17:21.750
you'll see the answer.
00:17:21.750 --> 00:17:25.990
A basis means all
the guys in the space
00:17:25.990 --> 00:17:29.170
are combinations of
these basis vectors.
00:17:29.170 --> 00:17:31.320
Well, this is a basis.
00:17:31.320 --> 00:17:35.870
sin x, cos x there is a basis.
00:17:38.640 --> 00:17:44.090
Those two -- they're like
the special solutions, right?
00:17:44.090 --> 00:17:47.320
We had special
solutions to Ax=b.
00:17:47.320 --> 00:17:53.730
Now we've got special solutions
to differential equations.
00:17:53.730 --> 00:17:59.270
Sorry, we had special
solutions to Ax=0, I misspoke.
00:17:59.270 --> 00:18:01.760
The special solutions
were for the null space
00:18:01.760 --> 00:18:04.040
just as here we're talking
about the null space.
00:18:04.040 --> 00:18:07.360
Do you see that here
is a -- those two --
00:18:07.360 --> 00:18:13.170
and what's the dimension
of the solution space?
00:18:13.170 --> 00:18:23.360
How many vectors in this basis?
00:18:23.360 --> 00:18:27.430
Two, the sine and cosine.
00:18:27.430 --> 00:18:30.960
Are those the only
basis for this space?
00:18:30.960 --> 00:18:32.720
By no means.
00:18:32.720 --> 00:18:37.070
e^(ix) and e^(-ix)
would be another basis.
00:18:37.070 --> 00:18:38.220
Lots of bases.
00:18:38.220 --> 00:18:43.960
But do you see that really what
a course in differential --
00:18:43.960 --> 00:18:50.720
in linear differential equations
is about is finding a basis
00:18:50.720 --> 00:18:52.460
for the solution space.
00:18:52.460 --> 00:18:56.640
The dimension of the solution
space will always be --
00:18:56.640 --> 00:19:01.410
will be two, because we have
a second order equation.
00:19:01.410 --> 00:19:04.470
So that's, like
there's 18.03 in --
00:19:04.470 --> 00:19:10.310
five minutes of 18.06 is enough
to, to take care of 18.03.
00:19:10.310 --> 00:19:12.590
So there's a -- that's
one more example.
00:19:12.590 --> 00:19:13.090
OK.
00:19:13.090 --> 00:19:16.200
And of course the
point of the example
00:19:16.200 --> 00:19:24.020
is these things don't
look like vectors.
00:19:24.020 --> 00:19:26.080
They look like functions.
00:19:26.080 --> 00:19:31.520
But we can call them vectors,
because we can add them
00:19:31.520 --> 00:19:34.320
and we can multiply
by constants,
00:19:34.320 --> 00:19:36.470
so we can take
linear combinations.
00:19:36.470 --> 00:19:39.800
That's all we have
to be allowed to do.
00:19:39.800 --> 00:19:43.920
So that's really why this idea
of linear algebra and basis
00:19:43.920 --> 00:19:51.120
and dimension and so on
plays a wider role than --
00:19:51.120 --> 00:19:56.820
our constant discussions
of m by n matrices.
00:19:56.820 --> 00:19:57.520
OK.
00:19:57.520 --> 00:20:00.360
That's what I wanted to
say about that topic.
00:20:00.360 --> 00:20:09.740
Now of course the key, number
associated with matrices,
00:20:09.740 --> 00:20:13.510
to go back to that
number, is the rank.
00:20:13.510 --> 00:20:17.990
And the rank, what do
we know about the rank?
00:20:20.970 --> 00:20:23.360
Well, we know it's
not bigger than m
00:20:23.360 --> 00:20:25.310
and it's not bigger than n.
00:20:25.310 --> 00:20:29.060
So but I'd like to have a
little discussion on the rank.
00:20:29.060 --> 00:20:30.780
Maybe I'll put that here.
00:20:30.780 --> 00:20:33.895
So I'm picking up this
topic of rank one matrices.
00:20:38.360 --> 00:20:46.070
And the reason I'm interested
in rank one matrices
00:20:46.070 --> 00:20:48.900
is that they ought to be simple.
00:20:48.900 --> 00:20:55.450
If the rank is only one, the
matrix can't get away from
00:20:55.450 --> 00:20:55.950
us.
00:20:55.950 --> 00:20:59.840
So for example, let me take
-- let me create a rank one
00:20:59.840 --> 00:21:00.960
matrix.
00:21:00.960 --> 00:21:01.460
OK.
00:21:01.460 --> 00:21:04.840
Suppose it's three --
suppose it's two by three.
00:21:07.820 --> 00:21:10.943
And let me give
you the first row.
00:21:16.080 --> 00:21:19.750
What can the second row be?
00:21:19.750 --> 00:21:24.060
Tell me a possible second row
here, for, for this matrix
00:21:24.060 --> 00:21:27.570
to have rank one.
00:21:27.570 --> 00:21:30.380
A possible second row is?
00:21:30.380 --> 00:21:33.150
Two eight ten.
00:21:33.150 --> 00:21:41.390
The second row is a
multiple of the first row.
00:21:41.390 --> 00:21:43.380
It's not independent.
00:21:43.380 --> 00:21:45.580
So tell me a basis
for the -- oh yeah,
00:21:45.580 --> 00:21:50.250
sorry to keep bringing
up these same questions.
00:21:50.250 --> 00:21:53.750
After the quiz I'll
stop, but for now,
00:21:53.750 --> 00:21:57.200
tell me a basis
for the row space.
00:21:57.200 --> 00:22:03.470
A basis for the row space of
that matrix is the first row,
00:22:03.470 --> 00:22:03.980
right?
00:22:03.980 --> 00:22:06.860
The first row, one four five.
00:22:06.860 --> 00:22:11.020
A basis for the column
space of this matrix is?
00:22:11.020 --> 00:22:15.110
What's the dimension
of the column space?
00:22:15.110 --> 00:22:19.140
The dimension of the
column space is also one,
00:22:19.140 --> 00:22:19.640
right?
00:22:19.640 --> 00:22:21.130
Because it's also the rank.
00:22:21.130 --> 00:22:24.980
The dimension -- you remember
the dimension of the column
00:22:24.980 --> 00:22:33.090
space equals the rank equals the
dimension of the column space
00:22:33.090 --> 00:22:38.510
of the transpose, which
is the row space of A.
00:22:38.510 --> 00:22:44.620
OK, and in this case
it's one, r is one.
00:22:44.620 --> 00:22:48.080
And sure enough, all
the columns are --
00:22:48.080 --> 00:22:51.390
all the other columns are
multiples of that column.
00:22:51.390 --> 00:22:57.000
Now there's -- there ought
to be a nice way to see that,
00:22:57.000 --> 00:22:59.700
and here it is.
00:22:59.700 --> 00:23:06.530
I can write that matrix as
its pivot column, one two,
00:23:06.530 --> 00:23:10.520
times its --
00:23:10.520 --> 00:23:11.730
times one four five.
00:23:14.470 --> 00:23:18.570
A column times a row,
one column times one row
00:23:18.570 --> 00:23:21.340
gives me a matrix, right?
00:23:21.340 --> 00:23:25.370
If I multiply a
column by a row, that,
00:23:25.370 --> 00:23:30.500
g- that's a two by one matrix
times a one by three matrix,
00:23:30.500 --> 00:23:34.780
and the result of the
multiplication is two by three.
00:23:34.780 --> 00:23:36.630
And it comes out right.
00:23:36.630 --> 00:23:45.450
So what I want to -- my point
is the rank one matrices that
00:23:45.450 --> 00:23:52.830
every rank one matrix has the
form some column times some
00:23:52.830 --> 00:23:55.520
row.
00:23:55.520 --> 00:23:59.180
So U is a column vector,
V is a column vector --
00:23:59.180 --> 00:24:03.570
but I make it into a row
by putting in V transpose.
00:24:03.570 --> 00:24:12.330
So that's the -- complete
picture of rank one matrices.
00:24:12.330 --> 00:24:14.580
We'll be interested
in rank one matrices.
00:24:14.580 --> 00:24:19.270
Later we'll find, oh, their
determinant, that'll be easy,
00:24:19.270 --> 00:24:23.620
their eigenvalues,
that'll be interesting.
00:24:23.620 --> 00:24:26.450
Rank one matrices are
like the building blocks
00:24:26.450 --> 00:24:27.410
for all matrices.
00:24:27.410 --> 00:24:30.620
And actually maybe
you can guess.
00:24:37.220 --> 00:24:47.890
If I took any matrix, a five by
seventeen matrix of rank four,
00:24:47.890 --> 00:24:51.890
then it seems pretty
likely -- and it's true,
00:24:51.890 --> 00:24:56.060
that I could break that five
by seventeen matrix down
00:24:56.060 --> 00:24:59.840
as a combination of
rank one matrices.
00:24:59.840 --> 00:25:03.310
And probably how many
of those would I need?
00:25:03.310 --> 00:25:07.810
If I have a five by seventeen
matrix of rank four,
00:25:07.810 --> 00:25:11.300
I'll need four of them, right.
00:25:11.300 --> 00:25:13.490
Four rank one matrices.
00:25:13.490 --> 00:25:17.710
So the rank one matrices are
the, are the building blocks.
00:25:17.710 --> 00:25:23.680
And out -- I can produce every,
I can produce every five by --
00:25:23.680 --> 00:25:29.070
every rank four matrix out
of four rank one matrices.
00:25:29.070 --> 00:25:33.170
That brings me to a
question, of course.
00:25:33.170 --> 00:25:33.670
OK.
00:25:33.670 --> 00:25:36.406
Would the rank four
matrices form a subspace?
00:25:38.990 --> 00:25:43.450
Let me take all five by
seventeen matrices and think
00:25:43.450 --> 00:25:48.589
about rank four -- the
subset of rank four matrices.
00:25:48.589 --> 00:25:49.880
Let me -- I'll write this down.
00:25:53.670 --> 00:25:57.010
You seem I'm reviewing
for the quiz,
00:25:57.010 --> 00:26:00.800
because I'm asking the kind of
questions that are short enough
00:26:00.800 --> 00:26:04.780
but -- that bring out do you
know what these words mean.
00:26:04.780 --> 00:26:06.400
So I take --
00:26:06.400 --> 00:26:13.080
my matrix space M now is all
five by seventeen matrices.
00:26:13.080 --> 00:26:27.010
And now the question I ask is
the subset of, of rank four
00:26:27.010 --> 00:26:34.340
matrices, is that a subspace?
00:26:34.340 --> 00:26:38.650
If I add a matrix of -- so if I
multiply a matrix of rank four
00:26:38.650 --> 00:26:39.880
by --
00:26:39.880 --> 00:26:43.520
of rank four or less,
let's say, because I
00:26:43.520 --> 00:26:49.490
have to let the zero matrix in
if it's going to be a subspace.
00:26:49.490 --> 00:26:52.330
But, but that doesn't just
because the zero matrix
00:26:52.330 --> 00:26:56.610
got in there doesn't
mean I have a subspace.
00:26:56.610 --> 00:26:59.840
So if I -- so the, the question
really comes down to --
00:26:59.840 --> 00:27:05.605
if I add two rank four
matrices, is the sum rank four?
00:27:09.050 --> 00:27:09.800
What do you think?
00:27:12.730 --> 00:27:15.110
If -- no, not usually.
00:27:15.110 --> 00:27:16.540
Not usually.
00:27:16.540 --> 00:27:20.200
If I add two rank four
matrices, the sum is probably --
00:27:20.200 --> 00:27:22.750
what could I say about the sum?
00:27:22.750 --> 00:27:27.905
Well, actually, well,
the rank could be five.
00:27:30.790 --> 00:27:35.610
It's a general fact, actually,
that the rank of A plus B
00:27:35.610 --> 00:27:40.880
can't be more than rank
of A plus the rank of B.
00:27:40.880 --> 00:27:42.880
So this would say if
I added two of those,
00:27:42.880 --> 00:27:45.820
the rank couldn't be larger
than eight, but I know actually
00:27:45.820 --> 00:27:48.420
the rank couldn't be as
large as eight anyway.
00:27:48.420 --> 00:27:50.650
What -- how big
could the rank be,
00:27:50.650 --> 00:27:52.480
for, for the rank
of a matrix in M?
00:27:52.480 --> 00:27:57.450
Could be as large as
five, right, right.
00:27:57.450 --> 00:28:00.460
So they're all sort
of natural ideas.
00:28:00.460 --> 00:28:05.570
So it's rank four matrices
or rank one matrices --
00:28:05.570 --> 00:28:09.600
let me, let me change
that to rank one.
00:28:09.600 --> 00:28:12.670
Let me take the subset
of rank one matrices.
00:28:12.670 --> 00:28:15.660
Is that a vector space?
00:28:15.660 --> 00:28:19.860
If I add a rank one matrix
to a rank one matrix?
00:28:19.860 --> 00:28:20.460
No.
00:28:20.460 --> 00:28:23.050
It's most likely going
to have rank two.
00:28:23.050 --> 00:28:23.790
So this is --
00:28:23.790 --> 00:28:25.090
So I'll just make that point.
00:28:25.090 --> 00:28:27.690
Not a subspace.
00:28:32.030 --> 00:28:34.640
OK.
00:28:34.640 --> 00:28:35.140
OK.
00:28:35.140 --> 00:28:38.210
Those are topics that
I wanted to, just
00:28:38.210 --> 00:28:42.530
fill out the, the
previous lectures.
00:28:42.530 --> 00:28:46.730
The I'll ask one more
subspace question, a,
00:28:46.730 --> 00:28:50.570
a more, a more, likely example.
00:28:50.570 --> 00:28:53.751
Suppose I'm in -- let me put,
put this example on a new
00:28:53.751 --> 00:28:54.250
board.
00:28:56.850 --> 00:28:58.823
Suppose I'm in R, in R^4.
00:29:03.980 --> 00:29:10.850
So my typical vector in R^4 has
four components, v1, v2, v3,
00:29:10.850 --> 00:29:11.350
and v4.
00:29:15.560 --> 00:29:21.280
Suppose I take the
subspace of vectors
00:29:21.280 --> 00:29:23.850
whose components add to zero.
00:29:23.850 --> 00:29:34.070
So I let S be all v, all vectors
v in four dimensional space
00:29:34.070 --> 00:29:37.050
with v1+v2+v3+v4=0.
00:29:37.050 --> 00:29:39.320
So I just want to consider
that bunch of vectors.
00:29:39.320 --> 00:29:41.560
Is it a subspace, first of all?
00:29:41.560 --> 00:29:42.430
It is a subspace.
00:29:42.430 --> 00:29:57.880
It is a subspace.
00:29:57.880 --> 00:30:01.780
What's -- how do we see that?
00:30:01.780 --> 00:30:04.380
It is a subspace.
00:30:04.380 --> 00:30:06.890
I -- formally I should check.
00:30:06.890 --> 00:30:11.430
If I have one vector that with
whose components add to zero
00:30:11.430 --> 00:30:13.990
and I multiply that
vector by six --
00:30:13.990 --> 00:30:16.980
the components still add to
zero, just six times as --
00:30:16.980 --> 00:30:19.250
six times zero.
00:30:19.250 --> 00:30:22.840
If I have a couple of v
and a w and I add them,
00:30:22.840 --> 00:30:25.150
the, the components
still add to zero.
00:30:25.150 --> 00:30:27.170
OK, it's a subspace.
00:30:27.170 --> 00:30:29.500
What's the dimension
of that space
00:30:29.500 --> 00:30:32.730
and what's a basis
for that space?
00:30:32.730 --> 00:30:37.370
So you see how I can just
describe a space and we --
00:30:37.370 --> 00:30:41.530
we can ask for the dimension
-- ask for the basis first
00:30:41.530 --> 00:30:42.800
and the dimension.
00:30:42.800 --> 00:30:44.620
Of course, the
dimension's the one
00:30:44.620 --> 00:30:48.710
that's easy to tell
me in a single word.
00:30:48.710 --> 00:30:51.790
What's the dimension
of our subspace S here?
00:30:55.910 --> 00:30:57.940
And a basis tell me --
00:30:57.940 --> 00:31:01.250
some vectors in it.
00:31:01.250 --> 00:31:07.210
Well, I'm going to make ask you
again to guess the dimension.
00:31:07.210 --> 00:31:09.190
Again I think I heard it.
00:31:09.190 --> 00:31:11.270
The dimension is three.
00:31:11.270 --> 00:31:12.650
Three.
00:31:12.650 --> 00:31:18.380
Now how does this
connect to our Ax=0?
00:31:18.380 --> 00:31:21.370
Is this the null
space of something?
00:31:21.370 --> 00:31:24.500
Is that the null
space of a matrix?
00:31:24.500 --> 00:31:26.210
And then we can
look at the matrix
00:31:26.210 --> 00:31:30.240
and, and we know everything
about those subspaces.
00:31:30.240 --> 00:31:39.665
This is the null
space of what matrix?
00:31:47.010 --> 00:31:53.030
What's the matrix where the
null space is then Ab=0.
00:31:53.030 --> 00:31:57.195
So I want this
equation to be Ab=0.
00:31:59.920 --> 00:32:02.510
b is now the vector.
00:32:02.510 --> 00:32:07.850
And what's the matrix that,
that we're seeing there?
00:32:07.850 --> 00:32:14.580
It's the matrix of four ones.
00:32:14.580 --> 00:32:20.120
Do you see that that's -- that
if I look at Ab=0 for this
00:32:20.120 --> 00:32:27.180
matrix A, I multiply by b
and I get this requirement,
00:32:27.180 --> 00:32:29.280
that the components add to zero.
00:32:29.280 --> 00:32:33.670
So I'm really when
I speak about S --
00:32:33.670 --> 00:32:37.280
I'm speaking about the
null space of that matrix.
00:32:37.280 --> 00:32:38.050
OK.
00:32:38.050 --> 00:32:41.350
Let's just say we've
got a matrix now,
00:32:41.350 --> 00:32:44.000
we want its null space.
00:32:44.000 --> 00:32:47.610
Well, we -- tell
me its rank first.
00:32:47.610 --> 00:32:54.900
The rank of that
matrix is one, thanks.
00:32:54.900 --> 00:32:57.210
So r is one.
00:32:57.210 --> 00:32:59.610
What's the general
formula for the dimension
00:32:59.610 --> 00:33:01.530
of the null space?
00:33:01.530 --> 00:33:07.640
The dimension of the null
space of a matrix is --
00:33:07.640 --> 00:33:12.420
in general, an m by
n matrix of rank r?
00:33:12.420 --> 00:33:16.895
How many independent
guys in the null space?
00:33:21.140 --> 00:33:22.920
n-r, right?
00:33:22.920 --> 00:33:25.780
n-r.
00:33:25.780 --> 00:33:31.390
In this case, n is
four, four columns.
00:33:31.390 --> 00:33:35.330
The rank is one, so the null
space is three dimensions.
00:33:35.330 --> 00:33:39.440
So of course y- you could
see it in this case,
00:33:39.440 --> 00:33:44.160
but you can also see it
here in our systematic way
00:33:44.160 --> 00:33:49.450
of dealing with the four
fundamental subspaces
00:33:49.450 --> 00:33:51.730
of a matrix.
00:33:51.730 --> 00:33:55.410
So what actually what,
what are all four subspaces
00:33:55.410 --> 00:33:56.050
then?
00:33:56.050 --> 00:33:58.840
The row space is clear.
00:33:58.840 --> 00:34:01.420
The row space is in R^4.
00:34:01.420 --> 00:34:05.530
Yeah, can we take the
four fundamental subspaces
00:34:05.530 --> 00:34:06.910
of this matrix?
00:34:06.910 --> 00:34:08.815
Let's just kill this example.
00:34:11.860 --> 00:34:15.400
The row space is
one dimensional.
00:34:15.400 --> 00:34:19.830
It's all multiples
of that, of that row.
00:34:19.830 --> 00:34:22.780
The null space is
three dimensional.
00:34:22.780 --> 00:34:26.260
Oh, you better give me a
basis for the null space.
00:34:26.260 --> 00:34:28.080
So what's a basis
for the null space?
00:34:28.080 --> 00:34:30.489
The special solutions.
00:34:30.489 --> 00:34:34.830
To find the special solutions,
I look for the free variables.
00:34:34.830 --> 00:34:39.650
The free variables here
are -- there's the pivot.
00:34:39.650 --> 00:34:44.010
The free variables are
two, three, and four.
00:34:44.010 --> 00:34:52.730
So the basis, basis
for S, for S will be --
00:34:52.730 --> 00:35:01.710
I'm expecting three vectors,
three special solutions.
00:35:01.710 --> 00:35:06.470
I give the value one
to that free variable,
00:35:06.470 --> 00:35:12.000
and what's the pivot
variable if the --
00:35:12.000 --> 00:35:15.130
this is going to
be a vector in S?
00:35:15.130 --> 00:35:16.390
Minus one.
00:35:16.390 --> 00:35:20.250
Now they're always added to
-- the entries add to zero.
00:35:20.250 --> 00:35:22.870
The second special
solution has a one
00:35:22.870 --> 00:35:26.540
in the second free variable,
and again a minus one
00:35:26.540 --> 00:35:27.730
makes it right.
00:35:27.730 --> 00:35:30.940
The third one has a one in
the third free variable,
00:35:30.940 --> 00:35:34.910
and again a minus
one makes it right.
00:35:34.910 --> 00:35:35.780
That's my answer.
00:35:35.780 --> 00:35:39.770
That's the answer I
would be looking for.
00:35:39.770 --> 00:35:43.300
The -- a basis for
this subspace S,
00:35:43.300 --> 00:35:45.530
you would just
list three vectors,
00:35:45.530 --> 00:35:48.350
and those would be the
natural three to list.
00:35:48.350 --> 00:35:55.240
Not the only possible three,
but those are the special three.
00:35:55.240 --> 00:35:58.960
OK, tell me about
the column space,
00:35:58.960 --> 00:36:02.560
What's the column
space of this matrix A?
00:36:07.390 --> 00:36:13.500
So the column space
is a subspace of R^1,
00:36:13.500 --> 00:36:15.470
because m is only one.
00:36:15.470 --> 00:36:17.920
The columns only
have one component.
00:36:17.920 --> 00:36:23.360
So the column space of S, the
column space of A is somewhere
00:36:23.360 --> 00:36:26.410
in the space R^1,
because we only have --
00:36:26.410 --> 00:36:30.240
these columns are short.
00:36:30.240 --> 00:36:32.940
And what is the
column space actually?
00:36:36.330 --> 00:36:42.280
I just, it's just talking with
these words is what I'm doing.
00:36:42.280 --> 00:36:48.360
The column space for
that matrix is R^1.
00:36:48.360 --> 00:36:52.780
The column space
for that matrix is
00:36:52.780 --> 00:36:55.463
all multiples of that column.
00:36:58.410 --> 00:37:00.760
And all multiples
give you all of R^1.
00:37:03.630 --> 00:37:07.140
And what's the, the
remaining fourth space,
00:37:07.140 --> 00:37:12.680
the null space of A
transpose is what?
00:37:17.560 --> 00:37:21.880
So we transpose A.
00:37:21.880 --> 00:37:26.290
We look for combinations
of the columns
00:37:26.290 --> 00:37:30.310
now that give zero
for A transpose.
00:37:30.310 --> 00:37:32.190
And there aren't any.
00:37:32.190 --> 00:37:36.620
The only thing, the only
combination of these rows
00:37:36.620 --> 00:37:41.500
to give the zero row is
the zero combination.
00:37:41.500 --> 00:37:42.380
OK.
00:37:42.380 --> 00:37:44.415
So let's just check dimensions.
00:37:47.180 --> 00:37:51.320
The null space has
dimension three.
00:37:51.320 --> 00:37:53.420
The row space has dimension one.
00:37:53.420 --> 00:37:54.630
Three plus one is four.
00:37:57.180 --> 00:37:59.680
The column space
has dimension one,
00:37:59.680 --> 00:38:02.790
and what's the
dimension of this, like,
00:38:02.790 --> 00:38:05.510
smallest possible space?
00:38:05.510 --> 00:38:08.790
What's the dimension
of the zero space?
00:38:08.790 --> 00:38:09.675
It's a subspace.
00:38:13.520 --> 00:38:14.190
Zero.
00:38:14.190 --> 00:38:15.190
What else could it be?
00:38:15.190 --> 00:38:17.970
I mean, let's -- we have to
take a reasonable answer --
00:38:17.970 --> 00:38:20.190
and the only reasonable
answer is zero.
00:38:20.190 --> 00:38:25.900
So one plus zero gives -- this
was n, the number of columns,
00:38:25.900 --> 00:38:30.480
and this is m, the
number of rows.
00:38:30.480 --> 00:38:32.910
And let's just, let
me just say again
00:38:32.910 --> 00:38:37.030
then the, the, the subspace
that has only that one
00:38:37.030 --> 00:38:42.830
point, that point is zero
dimensional, of course.
00:38:42.830 --> 00:38:46.920
And the basis is empty, because
if the dimension is zero,
00:38:46.920 --> 00:38:49.480
there shouldn't be
anybody in the basis.
00:38:49.480 --> 00:38:56.160
So the basis of that smallest
subspace is the empty set.
00:38:56.160 --> 00:38:59.590
And the number of members
in the empty set is zero,
00:38:59.590 --> 00:39:01.400
so that's the dimension.
00:39:01.400 --> 00:39:02.610
OK.
00:39:02.610 --> 00:39:04.040
Good.
00:39:04.040 --> 00:39:10.740
Now I have just five
minutes to tell you about --
00:39:10.740 --> 00:39:16.010
well, actually, about some,
some, some, this is now,
00:39:16.010 --> 00:39:21.430
this last topic of small
world graphs, and leads into,
00:39:21.430 --> 00:39:27.140
a lecture about graphs
and linear algebra.
00:39:27.140 --> 00:39:29.700
But let me tell you --
00:39:29.700 --> 00:39:32.860
in these last minutes the
graph that I interested in.
00:39:32.860 --> 00:39:40.600
It's the graph where
-- so what is a graph?
00:39:40.600 --> 00:39:42.020
Better tell you that first.
00:39:42.020 --> 00:39:42.880
OK.
00:39:42.880 --> 00:39:43.516
What's a graph?
00:39:47.380 --> 00:39:48.180
OK.
00:39:48.180 --> 00:39:49.980
This isn't calculus.
00:39:49.980 --> 00:39:54.010
We're not, I'm not thinking
of, like, some sine curve.
00:39:54.010 --> 00:39:57.940
The word graph is used in
a completely different way.
00:39:57.940 --> 00:40:07.490
It's a set of, a bunch
of nodes and edges,
00:40:07.490 --> 00:40:10.060
edges connecting the nodes.
00:40:10.060 --> 00:40:17.280
So I have nodes like
five nodes and edges --
00:40:17.280 --> 00:40:21.880
I'll put in some edges, I
could put, include them all.
00:40:21.880 --> 00:40:23.970
There's -- well, let me
put in a couple more.
00:40:27.350 --> 00:40:31.380
There's a graph with five
nodes and one two three four
00:40:31.380 --> 00:40:34.690
five six edges.
00:40:34.690 --> 00:40:38.190
And some five by six
matrix is going to tell us
00:40:38.190 --> 00:40:41.100
everything about that graph.
00:40:41.100 --> 00:40:43.350
Let me leave that
matrix to next time
00:40:43.350 --> 00:40:46.350
and tell you about the
question I'm interested in.
00:40:46.350 --> 00:40:52.340
Suppose, suppose the
graph isn't just,
00:40:52.340 --> 00:40:56.230
just doesn't have just five
nodes, but suppose every,
00:40:56.230 --> 00:40:59.780
suppose every person
in this room is a node.
00:41:03.200 --> 00:41:07.310
And suppose there's an
edge between two nodes
00:41:07.310 --> 00:41:11.300
if those two people are friends.
00:41:11.300 --> 00:41:14.070
So have I described a graph?
00:41:14.070 --> 00:41:18.980
It's a pretty big graph,
hundred, hundred nodes.
00:41:18.980 --> 00:41:21.190
And I don't know how
many edges are in there.
00:41:24.920 --> 00:41:27.730
There's an edge
if you're friends.
00:41:27.730 --> 00:41:29.950
So that's the graph
for this class.
00:41:29.950 --> 00:41:35.120
A, a similar graph you could
take for the whole country,
00:41:35.120 --> 00:41:38.680
so two hundred and
sixty million nodes.
00:41:38.680 --> 00:41:43.340
And edges between friends.
00:41:43.340 --> 00:41:50.110
And the question for that graph
is how many steps does it take
00:41:50.110 --> 00:41:52.210
to get from anybody to anybody?
00:41:56.780 --> 00:42:01.950
What two people are furthest
apart in this friendship graph,
00:42:01.950 --> 00:42:04.070
say for the US?
00:42:04.070 --> 00:42:08.820
By furthest apart, I
mean the distance from --
00:42:08.820 --> 00:42:12.640
well, I'll tell you my
distance to Clinton.
00:42:12.640 --> 00:42:14.040
It's two.
00:42:14.040 --> 00:42:18.440
I happened to go to college
with somebody who knows Clinton.
00:42:18.440 --> 00:42:19.190
I don't know him.
00:42:19.190 --> 00:42:24.950
So my distance to Clinton is not
one, because I don't, happily
00:42:24.950 --> 00:42:26.680
or not, don't know him.
00:42:26.680 --> 00:42:29.050
But I know somebody who does.
00:42:29.050 --> 00:42:32.541
He's a Senator and so
I presume he knows him.
00:42:32.541 --> 00:42:33.040
OK.
00:42:33.040 --> 00:42:35.311
I don't know what your --
well, what's your distance
00:42:35.311 --> 00:42:35.810
to Clinton?
00:42:39.100 --> 00:42:40.950
Well, not more
than three, right.
00:42:40.950 --> 00:42:43.000
Actually, true.
00:42:43.000 --> 00:42:44.430
You know me.
00:42:44.430 --> 00:42:50.880
I take credit for reducing your
Clinton distance to three --
00:42:50.880 --> 00:42:52.195
what's your distance to Monica.
00:42:54.780 --> 00:43:04.090
Not, anybody below -- below
four is in trouble here.
00:43:04.090 --> 00:43:07.010
Or maybe three, but, right.
00:43:07.010 --> 00:43:14.740
So -- and what's Hillary's
distance to Monica?
00:43:14.740 --> 00:43:18.020
I don't think we'd better
put that on tape here.
00:43:18.020 --> 00:43:22.360
That's one or two, I guess.
00:43:22.360 --> 00:43:24.490
Is that right?
00:43:24.490 --> 00:43:28.980
I don't -- well, we won't,
think more about that.
00:43:28.980 --> 00:43:32.490
So actually, the,
the real question
00:43:32.490 --> 00:43:38.030
is what are large distances?
00:43:38.030 --> 00:43:41.910
How, how far apart could
people be separated?
00:43:41.910 --> 00:43:46.520
And roughly this number
six degrees of separation
00:43:46.520 --> 00:43:49.940
has kind of appeared as the
movie title, as the book title,
00:43:49.940 --> 00:43:52.050
and it's with this meaning.
00:43:52.050 --> 00:43:56.050
That roughly speaking --
00:43:56.050 --> 00:43:59.230
six might be a fairly --
00:43:59.230 --> 00:44:01.860
not too many people.
00:44:01.860 --> 00:44:04.860
If you sit next to
somebody on an airplane,
00:44:04.860 --> 00:44:07.850
you get talking to them.
00:44:07.850 --> 00:44:12.310
You begin to discuss mutual
friends to sort of find out,
00:44:12.310 --> 00:44:15.220
OK, what connections
do you have,
00:44:15.220 --> 00:44:17.450
and very often
you'll find you're
00:44:17.450 --> 00:44:21.880
connected in, like, two
or three or four steps.
00:44:21.880 --> 00:44:23.970
And you remark,
it's a small world,
00:44:23.970 --> 00:44:27.850
and that's how this expression
small world came up.
00:44:27.850 --> 00:44:31.450
But six, I don't know if you
could find -- if it took six,
00:44:31.450 --> 00:44:34.810
I don't know if you would
successfully discover those six
00:44:34.810 --> 00:44:36.980
in a, in an airplane
conversation.
00:44:36.980 --> 00:44:40.150
But here's the math
question, and I'll
00:44:40.150 --> 00:44:42.650
leave it for next,
for lecture twelve,
00:44:42.650 --> 00:44:46.010
and do a lot of linear
algebra in lecture twelve.
00:44:46.010 --> 00:44:54.860
But the interesting point is
that with a few shortcuts,
00:44:54.860 --> 00:44:58.290
the distances come
down dramatically.
00:44:58.290 --> 00:45:03.900
That, I mean, all your distances
to Clinton immediately drop
00:45:03.900 --> 00:45:06.460
to three by taking
linear algebra.
00:45:06.460 --> 00:45:11.580
That's, like, an extra bonus
for taking linear algebra.
00:45:11.580 --> 00:45:17.550
And to understand mathematically
what it is about these graphs
00:45:17.550 --> 00:45:18.120
--
00:45:18.120 --> 00:45:21.710
or like the graphs of
the World Wide Web.
00:45:21.710 --> 00:45:23.170
There's a fantastic graph.
00:45:23.170 --> 00:45:27.790
So many people would like to
understand and model the web.
00:45:27.790 --> 00:45:34.650
What the -- where the edges are
links and the nodes are, sites,
00:45:34.650 --> 00:45:37.530
websites.
00:45:37.530 --> 00:45:39.960
I'll leave you with that
graph, and I'll see you --
00:45:39.960 --> 00:45:42.740
have a good weekend,
and see you on Monday.