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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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00:12:27,520 --> 00:12:30,170
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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00:12:36,670 --> 00:12:42,200
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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00:12:50,300 --> 00:12:54,380
It's worth thinking about that.
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00:12:54,380 --> 00:12:58,330
It's just like stretch your
mind a little, just a little,
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00:12:58,330 --> 00:13:05,700
to, to think of these subspaces
and what their intersection is
201
00:13:05,700 --> 00:13:07,110
and what their sum is.
202
00:13:07,110 --> 00:13:08,950
And now can I give you
a little -- oh, well,
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00:13:08,950 --> 00:13:10,810
let's figure out the dimension.
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00:13:10,810 --> 00:13:17,220
So what's the
dimension of S plus U?
205
00:13:17,220 --> 00:13:25,470
In this example is nine, because
we got all three by threes.
206
00:13:25,470 --> 00:13:31,690
So the original spaces had, the
original symmetric space had
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00:13:31,690 --> 00:13:36,010
dimension six and the original
upper triangular space
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00:13:36,010 --> 00:13:38,010
had dimension six.
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00:13:38,010 --> 00:13:42,975
And actually I'm seeing
here a nice formula.
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00:13:48,360 --> 00:13:54,810
That the dimension of S
plus the dimension of U --
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00:13:54,810 --> 00:13:58,020
if I have two subspaces,
the dimension of one plus
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00:13:58,020 --> 00:14:03,750
the dimension of the other
-- equals the dimension
213
00:14:03,750 --> 00:14:07,850
of their intersection plus
the dimension of their sum.
214
00:14:07,850 --> 00:14:12,530
Six plus six is three plus nine.
215
00:14:16,550 --> 00:14:20,850
That's kind of satisfying, that
these natural operations --
216
00:14:20,850 --> 00:14:23,360
and we've -- this
is it, actually,
217
00:14:23,360 --> 00:14:27,650
this is the set of
natural things to do with,
218
00:14:27,650 --> 00:14:29,390
with subspaces.
219
00:14:29,390 --> 00:14:36,730
That, the dimensions
come out in a good way.
220
00:14:36,730 --> 00:14:37,640
OK.
221
00:14:37,640 --> 00:14:43,540
Maybe I'll take just one more
example of a vector space
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00:14:43,540 --> 00:14:48,270
that doesn't have vectors in it.
223
00:14:48,270 --> 00:14:51,296
It's come from
differential equations.
224
00:14:54,200 --> 00:14:57,120
So this is a one
more new vector space
225
00:14:57,120 --> 00:14:59,580
that we'll give just
a few minutes to.
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00:14:59,580 --> 00:15:06,380
Suppose I have a differential
equation like d^2y/dx^2+ y=0.
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00:15:09,590 --> 00:15:12,200
OK.
228
00:15:12,200 --> 00:15:14,175
I look at the solutions
to that equation.
229
00:15:16,790 --> 00:15:21,230
So what are the solutions
to that equation?
230
00:15:21,230 --> 00:15:24,815
y=cos(x) is a solution.
231
00:15:28,250 --> 00:15:31,960
y=sin(x) is a solution.
232
00:15:31,960 --> 00:15:37,730
y equals -- well, e to the (ix)
is a solution, if you want,
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00:15:37,730 --> 00:15:41,150
if you allow me to put that in.
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00:15:41,150 --> 00:15:42,520
But why should I put that in?
235
00:15:42,520 --> 00:15:45,890
It's already there.
236
00:15:45,890 --> 00:15:48,650
You see, I'm really looking
at a null space here.
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00:15:48,650 --> 00:15:52,730
I'm looking at the null space
of a differential equation.
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00:15:52,730 --> 00:15:55,210
That's the solution space.
239
00:15:55,210 --> 00:16:01,000
And describe the solution
space, all solutions
240
00:16:01,000 --> 00:16:03,290
to this differential equation.
241
00:16:03,290 --> 00:16:07,450
So the equation is y''+y=0.
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00:16:07,450 --> 00:16:11,630
Cosine's, cosine's a
solution, sine is a solution.
243
00:16:11,630 --> 00:16:14,430
Now tell me all the solutions.
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00:16:14,430 --> 00:16:18,930
They're -- so I
don't need e^(ix).
245
00:16:18,930 --> 00:16:21,220
Forget that.
246
00:16:21,220 --> 00:16:23,420
What are all the
complete solutions?
247
00:16:30,510 --> 00:16:32,860
Is what?
248
00:16:32,860 --> 00:16:34,840
A combination of these.
249
00:16:34,840 --> 00:16:38,070
The complete
solution is y equals
250
00:16:38,070 --> 00:16:48,350
some multiple of the cosine
plus some multiple of the sine.
251
00:16:48,350 --> 00:16:51,520
That's a vector space.
252
00:16:51,520 --> 00:16:52,660
That's a vector space.
253
00:16:52,660 --> 00:16:54,530
What's the dimension
of that space?
254
00:16:54,530 --> 00:16:57,500
What's a basis for that space?
255
00:16:57,500 --> 00:17:00,100
OK, let me ask
you a basis first.
256
00:17:00,100 --> 00:17:03,850
If I take the set of
solutions to that second order
257
00:17:03,850 --> 00:17:06,450
differential equation --
258
00:17:06,450 --> 00:17:10,520
there it is, those
are the solutions.
259
00:17:10,520 --> 00:17:12,010
What's a basis for that space?
260
00:17:14,680 --> 00:17:16,480
Now remember, what's
the, what question I
261
00:17:16,480 --> 00:17:16,980
asking?
262
00:17:16,980 --> 00:17:18,890
Because if you know the
question I'm asking,
263
00:17:18,890 --> 00:17:21,750
you'll see the answer.
264
00:17:21,750 --> 00:17:25,990
A basis means all
the guys in the space
265
00:17:25,990 --> 00:17:29,170
are combinations of
these basis vectors.
266
00:17:29,170 --> 00:17:31,320
Well, this is a basis.
267
00:17:31,320 --> 00:17:35,870
sin x, cos x there is a basis.
268
00:17:38,640 --> 00:17:44,090
Those two -- they're like
the special solutions, right?
269
00:17:44,090 --> 00:17:47,320
We had special
solutions to Ax=b.
270
00:17:47,320 --> 00:17:53,730
Now we've got special solutions
to differential equations.
271
00:17:53,730 --> 00:17:59,270
Sorry, we had special
solutions to Ax=0, I misspoke.
272
00:17:59,270 --> 00:18:01,760
The special solutions
were for the null space
273
00:18:01,760 --> 00:18:04,040
just as here we're talking
about the null space.
274
00:18:04,040 --> 00:18:07,360
Do you see that here
is a -- those two --
275
00:18:07,360 --> 00:18:13,170
and what's the dimension
of the solution space?
276
00:18:13,170 --> 00:18:23,360
How many vectors in this basis?
277
00:18:23,360 --> 00:18:27,430
Two, the sine and cosine.
278
00:18:27,430 --> 00:18:30,960
Are those the only
basis for this space?
279
00:18:30,960 --> 00:18:32,720
By no means.
280
00:18:32,720 --> 00:18:37,070
e^(ix) and e^(-ix)
would be another basis.
281
00:18:37,070 --> 00:18:38,220
Lots of bases.
282
00:18:38,220 --> 00:18:43,960
But do you see that really what
a course in differential --
283
00:18:43,960 --> 00:18:50,720
in linear differential equations
is about is finding a basis
284
00:18:50,720 --> 00:18:52,460
for the solution space.
285
00:18:52,460 --> 00:18:56,640
The dimension of the solution
space will always be --
286
00:18:56,640 --> 00:19:01,410
will be two, because we have
a second order equation.
287
00:19:01,410 --> 00:19:04,470
So that's, like
there's 18.03 in --
288
00:19:04,470 --> 00:19:10,310
five minutes of 18.06 is enough
to, to take care of 18.03.
289
00:19:10,310 --> 00:19:12,590
So there's a -- that's
one more example.
290
00:19:12,590 --> 00:19:13,090
OK.
291
00:19:13,090 --> 00:19:16,200
And of course the
point of the example
292
00:19:16,200 --> 00:19:24,020
is these things don't
look like vectors.
293
00:19:24,020 --> 00:19:26,080
They look like functions.
294
00:19:26,080 --> 00:19:31,520
But we can call them vectors,
because we can add them
295
00:19:31,520 --> 00:19:34,320
and we can multiply
by constants,
296
00:19:34,320 --> 00:19:36,470
so we can take
linear combinations.
297
00:19:36,470 --> 00:19:39,800
That's all we have
to be allowed to do.
298
00:19:39,800 --> 00:19:43,920
So that's really why this idea
of linear algebra and basis
299
00:19:43,920 --> 00:19:51,120
and dimension and so on
plays a wider role than --
300
00:19:51,120 --> 00:19:56,820
our constant discussions
of m by n matrices.
301
00:19:56,820 --> 00:19:57,520
OK.
302
00:19:57,520 --> 00:20:00,360
That's what I wanted to
say about that topic.
303
00:20:00,360 --> 00:20:09,740
Now of course the key, number
associated with matrices,
304
00:20:09,740 --> 00:20:13,510
to go back to that
number, is the rank.
305
00:20:13,510 --> 00:20:17,990
And the rank, what do
we know about the rank?
306
00:20:20,970 --> 00:20:23,360
Well, we know it's
not bigger than m
307
00:20:23,360 --> 00:20:25,310
and it's not bigger than n.
308
00:20:25,310 --> 00:20:29,060
So but I'd like to have a
little discussion on the rank.
309
00:20:29,060 --> 00:20:30,780
Maybe I'll put that here.
310
00:20:30,780 --> 00:20:33,895
So I'm picking up this
topic of rank one matrices.
311
00:20:38,360 --> 00:20:46,070
And the reason I'm interested
in rank one matrices
312
00:20:46,070 --> 00:20:48,900
is that they ought to be simple.
313
00:20:48,900 --> 00:20:55,450
If the rank is only one, the
matrix can't get away from
314
00:20:55,450 --> 00:20:55,950
us.
315
00:20:55,950 --> 00:20:59,840
So for example, let me take
-- let me create a rank one
316
00:20:59,840 --> 00:21:00,960
matrix.
317
00:21:00,960 --> 00:21:01,460
OK.
318
00:21:01,460 --> 00:21:04,840
Suppose it's three --
suppose it's two by three.
319
00:21:07,820 --> 00:21:10,943
And let me give
you the first row.
320
00:21:16,080 --> 00:21:19,750
What can the second row be?
321
00:21:19,750 --> 00:21:24,060
Tell me a possible second row
here, for, for this matrix
322
00:21:24,060 --> 00:21:27,570
to have rank one.
323
00:21:27,570 --> 00:21:30,380
A possible second row is?
324
00:21:30,380 --> 00:21:33,150
Two eight ten.
325
00:21:33,150 --> 00:21:41,390
The second row is a
multiple of the first row.
326
00:21:41,390 --> 00:21:43,380
It's not independent.
327
00:21:43,380 --> 00:21:45,580
So tell me a basis
for the -- oh yeah,
328
00:21:45,580 --> 00:21:50,250
sorry to keep bringing
up these same questions.
329
00:21:50,250 --> 00:21:53,750
After the quiz I'll
stop, but for now,
330
00:21:53,750 --> 00:21:57,200
tell me a basis
for the row space.
331
00:21:57,200 --> 00:22:03,470
A basis for the row space of
that matrix is the first row,
332
00:22:03,470 --> 00:22:03,980
right?
333
00:22:03,980 --> 00:22:06,860
The first row, one four five.
334
00:22:06,860 --> 00:22:11,020
A basis for the column
space of this matrix is?
335
00:22:11,020 --> 00:22:15,110
What's the dimension
of the column space?
336
00:22:15,110 --> 00:22:19,140
The dimension of the
column space is also one,
337
00:22:19,140 --> 00:22:19,640
right?
338
00:22:19,640 --> 00:22:21,130
Because it's also the rank.
339
00:22:21,130 --> 00:22:24,980
The dimension -- you remember
the dimension of the column
340
00:22:24,980 --> 00:22:33,090
space equals the rank equals the
dimension of the column space
341
00:22:33,090 --> 00:22:38,510
of the transpose, which
is the row space of A.
342
00:22:38,510 --> 00:22:44,620
OK, and in this case
it's one, r is one.
343
00:22:44,620 --> 00:22:48,080
And sure enough, all
the columns are --
344
00:22:48,080 --> 00:22:51,390
all the other columns are
multiples of that column.
345
00:22:51,390 --> 00:22:57,000
Now there's -- there ought
to be a nice way to see that,
346
00:22:57,000 --> 00:22:59,700
and here it is.
347
00:22:59,700 --> 00:23:06,530
I can write that matrix as
its pivot column, one two,
348
00:23:06,530 --> 00:23:10,520
times its --
349
00:23:10,520 --> 00:23:11,730
times one four five.
350
00:23:14,470 --> 00:23:18,570
A column times a row,
one column times one row
351
00:23:18,570 --> 00:23:21,340
gives me a matrix, right?
352
00:23:21,340 --> 00:23:25,370
If I multiply a
column by a row, that,
353
00:23:25,370 --> 00:23:30,500
g- that's a two by one matrix
times a one by three matrix,
354
00:23:30,500 --> 00:23:34,780
and the result of the
multiplication is two by three.
355
00:23:34,780 --> 00:23:36,630
And it comes out right.
356
00:23:36,630 --> 00:23:45,450
So what I want to -- my point
is the rank one matrices that
357
00:23:45,450 --> 00:23:52,830
every rank one matrix has the
form some column times some
358
00:23:52,830 --> 00:23:55,520
row.
359
00:23:55,520 --> 00:23:59,180
So U is a column vector,
V is a column vector --
360
00:23:59,180 --> 00:24:03,570
but I make it into a row
by putting in V transpose.
361
00:24:03,570 --> 00:24:12,330
So that's the -- complete
picture of rank one matrices.
362
00:24:12,330 --> 00:24:14,580
We'll be interested
in rank one matrices.
363
00:24:14,580 --> 00:24:19,270
Later we'll find, oh, their
determinant, that'll be easy,
364
00:24:19,270 --> 00:24:23,620
their eigenvalues,
that'll be interesting.
365
00:24:23,620 --> 00:24:26,450
Rank one matrices are
like the building blocks
366
00:24:26,450 --> 00:24:27,410
for all matrices.
367
00:24:27,410 --> 00:24:30,620
And actually maybe
you can guess.
368
00:24:37,220 --> 00:24:47,890
If I took any matrix, a five by
seventeen matrix of rank four,
369
00:24:47,890 --> 00:24:51,890
then it seems pretty
likely -- and it's true,
370
00:24:51,890 --> 00:24:56,060
that I could break that five
by seventeen matrix down
371
00:24:56,060 --> 00:24:59,840
as a combination of
rank one matrices.
372
00:24:59,840 --> 00:25:03,310
And probably how many
of those would I need?
373
00:25:03,310 --> 00:25:07,810
If I have a five by seventeen
matrix of rank four,
374
00:25:07,810 --> 00:25:11,300
I'll need four of them, right.
375
00:25:11,300 --> 00:25:13,490
Four rank one matrices.
376
00:25:13,490 --> 00:25:17,710
So the rank one matrices are
the, are the building blocks.
377
00:25:17,710 --> 00:25:23,680
And out -- I can produce every,
I can produce every five by --
378
00:25:23,680 --> 00:25:29,070
every rank four matrix out
of four rank one matrices.
379
00:25:29,070 --> 00:25:33,170
That brings me to a
question, of course.
380
00:25:33,170 --> 00:25:33,670
OK.
381
00:25:33,670 --> 00:25:36,406
Would the rank four
matrices form a subspace?
382
00:25:38,990 --> 00:25:43,450
Let me take all five by
seventeen matrices and think
383
00:25:43,450 --> 00:25:48,589
about rank four -- the
subset of rank four matrices.
384
00:25:48,589 --> 00:25:49,880
Let me -- I'll write this down.
385
00:25:53,670 --> 00:25:57,010
You seem I'm reviewing
for the quiz,
386
00:25:57,010 --> 00:26:00,800
because I'm asking the kind of
questions that are short enough
387
00:26:00,800 --> 00:26:04,780
but -- that bring out do you
know what these words mean.
388
00:26:04,780 --> 00:26:06,400
So I take --
389
00:26:06,400 --> 00:26:13,080
my matrix space M now is all
five by seventeen matrices.
390
00:26:13,080 --> 00:26:27,010
And now the question I ask is
the subset of, of rank four
391
00:26:27,010 --> 00:26:34,340
matrices, is that a subspace?
392
00:26:34,340 --> 00:26:38,650
If I add a matrix of -- so if I
multiply a matrix of rank four
393
00:26:38,650 --> 00:26:39,880
by --
394
00:26:39,880 --> 00:26:43,520
of rank four or less,
let's say, because I
395
00:26:43,520 --> 00:26:49,490
have to let the zero matrix in
if it's going to be a subspace.
396
00:26:49,490 --> 00:26:52,330
But, but that doesn't just
because the zero matrix
397
00:26:52,330 --> 00:26:56,610
got in there doesn't
mean I have a subspace.
398
00:26:56,610 --> 00:26:59,840
So if I -- so the, the question
really comes down to --
399
00:26:59,840 --> 00:27:05,605
if I add two rank four
matrices, is the sum rank four?
400
00:27:09,050 --> 00:27:09,800
What do you think?
401
00:27:12,730 --> 00:27:15,110
If -- no, not usually.
402
00:27:15,110 --> 00:27:16,540
Not usually.
403
00:27:16,540 --> 00:27:20,200
If I add two rank four
matrices, the sum is probably --
404
00:27:20,200 --> 00:27:22,750
what could I say about the sum?
405
00:27:22,750 --> 00:27:27,905
Well, actually, well,
the rank could be five.
406
00:27:30,790 --> 00:27:35,610
It's a general fact, actually,
that the rank of A plus B
407
00:27:35,610 --> 00:27:40,880
can't be more than rank
of A plus the rank of B.
408
00:27:40,880 --> 00:27:42,880
So this would say if
I added two of those,
409
00:27:42,880 --> 00:27:45,820
the rank couldn't be larger
than eight, but I know actually
410
00:27:45,820 --> 00:27:48,420
the rank couldn't be as
large as eight anyway.
411
00:27:48,420 --> 00:27:50,650
What -- how big
could the rank be,
412
00:27:50,650 --> 00:27:52,480
for, for the rank
of a matrix in M?
413
00:27:52,480 --> 00:27:57,450
Could be as large as
five, right, right.
414
00:27:57,450 --> 00:28:00,460
So they're all sort
of natural ideas.
415
00:28:00,460 --> 00:28:05,570
So it's rank four matrices
or rank one matrices --
416
00:28:05,570 --> 00:28:09,600
let me, let me change
that to rank one.
417
00:28:09,600 --> 00:28:12,670
Let me take the subset
of rank one matrices.
418
00:28:12,670 --> 00:28:15,660
Is that a vector space?
419
00:28:15,660 --> 00:28:19,860
If I add a rank one matrix
to a rank one matrix?
420
00:28:19,860 --> 00:28:20,460
No.
421
00:28:20,460 --> 00:28:23,050
It's most likely going
to have rank two.
422
00:28:23,050 --> 00:28:23,790
So this is --
423
00:28:23,790 --> 00:28:25,090
So I'll just make that point.
424
00:28:25,090 --> 00:28:27,690
Not a subspace.
425
00:28:32,030 --> 00:28:34,640
OK.
426
00:28:34,640 --> 00:28:35,140
OK.
427
00:28:35,140 --> 00:28:38,210
Those are topics that
I wanted to, just
428
00:28:38,210 --> 00:28:42,530
fill out the, the
previous lectures.
429
00:28:42,530 --> 00:28:46,730
The I'll ask one more
subspace question, a,
430
00:28:46,730 --> 00:28:50,570
a more, a more, likely example.
431
00:28:50,570 --> 00:28:53,751
Suppose I'm in -- let me put,
put this example on a new
432
00:28:53,751 --> 00:28:54,250
board.
433
00:28:56,850 --> 00:28:58,823
Suppose I'm in R, in R^4.
434
00:29:03,980 --> 00:29:10,850
So my typical vector in R^4 has
four components, v1, v2, v3,
435
00:29:10,850 --> 00:29:11,350
and v4.
436
00:29:15,560 --> 00:29:21,280
Suppose I take the
subspace of vectors
437
00:29:21,280 --> 00:29:23,850
whose components add to zero.
438
00:29:23,850 --> 00:29:34,070
So I let S be all v, all vectors
v in four dimensional space
439
00:29:34,070 --> 00:29:37,050
with v1+v2+v3+v4=0.
440
00:29:37,050 --> 00:29:39,320
So I just want to consider
that bunch of vectors.
441
00:29:39,320 --> 00:29:41,560
Is it a subspace, first of all?
442
00:29:41,560 --> 00:29:42,430
It is a subspace.
443
00:29:42,430 --> 00:29:57,880
It is a subspace.
444
00:29:57,880 --> 00:30:01,780
What's -- how do we see that?
445
00:30:01,780 --> 00:30:04,380
It is a subspace.
446
00:30:04,380 --> 00:30:06,890
I -- formally I should check.
447
00:30:06,890 --> 00:30:11,430
If I have one vector that with
whose components add to zero
448
00:30:11,430 --> 00:30:13,990
and I multiply that
vector by six --
449
00:30:13,990 --> 00:30:16,980
the components still add to
zero, just six times as --
450
00:30:16,980 --> 00:30:19,250
six times zero.
451
00:30:19,250 --> 00:30:22,840
If I have a couple of v
and a w and I add them,
452
00:30:22,840 --> 00:30:25,150
the, the components
still add to zero.
453
00:30:25,150 --> 00:30:27,170
OK, it's a subspace.
454
00:30:27,170 --> 00:30:29,500
What's the dimension
of that space
455
00:30:29,500 --> 00:30:32,730
and what's a basis
for that space?
456
00:30:32,730 --> 00:30:37,370
So you see how I can just
describe a space and we --
457
00:30:37,370 --> 00:30:41,530
we can ask for the dimension
-- ask for the basis first
458
00:30:41,530 --> 00:30:42,800
and the dimension.
459
00:30:42,800 --> 00:30:44,620
Of course, the
dimension's the one
460
00:30:44,620 --> 00:30:48,710
that's easy to tell
me in a single word.
461
00:30:48,710 --> 00:30:51,790
What's the dimension
of our subspace S here?
462
00:30:55,910 --> 00:30:57,940
And a basis tell me --
463
00:30:57,940 --> 00:31:01,250
some vectors in it.
464
00:31:01,250 --> 00:31:07,210
Well, I'm going to make ask you
again to guess the dimension.
465
00:31:07,210 --> 00:31:09,190
Again I think I heard it.
466
00:31:09,190 --> 00:31:11,270
The dimension is three.
467
00:31:11,270 --> 00:31:12,650
Three.
468
00:31:12,650 --> 00:31:18,380
Now how does this
connect to our Ax=0?
469
00:31:18,380 --> 00:31:21,370
Is this the null
space of something?
470
00:31:21,370 --> 00:31:24,500
Is that the null
space of a matrix?
471
00:31:24,500 --> 00:31:26,210
And then we can
look at the matrix
472
00:31:26,210 --> 00:31:30,240
and, and we know everything
about those subspaces.
473
00:31:30,240 --> 00:31:39,665
This is the null
space of what matrix?
474
00:31:47,010 --> 00:31:53,030
What's the matrix where the
null space is then Ab=0.
475
00:31:53,030 --> 00:31:57,195
So I want this
equation to be Ab=0.
476
00:31:59,920 --> 00:32:02,510
b is now the vector.
477
00:32:02,510 --> 00:32:07,850
And what's the matrix that,
that we're seeing there?
478
00:32:07,850 --> 00:32:14,580
It's the matrix of four ones.
479
00:32:14,580 --> 00:32:20,120
Do you see that that's -- that
if I look at Ab=0 for this
480
00:32:20,120 --> 00:32:27,180
matrix A, I multiply by b
and I get this requirement,
481
00:32:27,180 --> 00:32:29,280
that the components add to zero.
482
00:32:29,280 --> 00:32:33,670
So I'm really when
I speak about S --
483
00:32:33,670 --> 00:32:37,280
I'm speaking about the
null space of that matrix.
484
00:32:37,280 --> 00:32:38,050
OK.
485
00:32:38,050 --> 00:32:41,350
Let's just say we've
got a matrix now,
486
00:32:41,350 --> 00:32:44,000
we want its null space.
487
00:32:44,000 --> 00:32:47,610
Well, we -- tell
me its rank first.
488
00:32:47,610 --> 00:32:54,900
The rank of that
matrix is one, thanks.
489
00:32:54,900 --> 00:32:57,210
So r is one.
490
00:32:57,210 --> 00:32:59,610
What's the general
formula for the dimension
491
00:32:59,610 --> 00:33:01,530
of the null space?
492
00:33:01,530 --> 00:33:07,640
The dimension of the null
space of a matrix is --
493
00:33:07,640 --> 00:33:12,420
in general, an m by
n matrix of rank r?
494
00:33:12,420 --> 00:33:16,895
How many independent
guys in the null space?
495
00:33:21,140 --> 00:33:22,920
n-r, right?
496
00:33:22,920 --> 00:33:25,780
n-r.
497
00:33:25,780 --> 00:33:31,390
In this case, n is
four, four columns.
498
00:33:31,390 --> 00:33:35,330
The rank is one, so the null
space is three dimensions.
499
00:33:35,330 --> 00:33:39,440
So of course y- you could
see it in this case,
500
00:33:39,440 --> 00:33:44,160
but you can also see it
here in our systematic way
501
00:33:44,160 --> 00:33:49,450
of dealing with the four
fundamental subspaces
502
00:33:49,450 --> 00:33:51,730
of a matrix.
503
00:33:51,730 --> 00:33:55,410
So what actually what,
what are all four subspaces
504
00:33:55,410 --> 00:33:56,050
then?
505
00:33:56,050 --> 00:33:58,840
The row space is clear.
506
00:33:58,840 --> 00:34:01,420
The row space is in R^4.
507
00:34:01,420 --> 00:34:05,530
Yeah, can we take the
four fundamental subspaces
508
00:34:05,530 --> 00:34:06,910
of this matrix?
509
00:34:06,910 --> 00:34:08,815
Let's just kill this example.
510
00:34:11,860 --> 00:34:15,400
The row space is
one dimensional.
511
00:34:15,400 --> 00:34:19,830
It's all multiples
of that, of that row.
512
00:34:19,830 --> 00:34:22,780
The null space is
three dimensional.
513
00:34:22,780 --> 00:34:26,260
Oh, you better give me a
basis for the null space.
514
00:34:26,260 --> 00:34:28,080
So what's a basis
for the null space?
515
00:34:28,080 --> 00:34:30,489
The special solutions.
516
00:34:30,489 --> 00:34:34,830
To find the special solutions,
I look for the free variables.
517
00:34:34,830 --> 00:34:39,650
The free variables here
are -- there's the pivot.
518
00:34:39,650 --> 00:34:44,010
The free variables are
two, three, and four.
519
00:34:44,010 --> 00:34:52,730
So the basis, basis
for S, for S will be --
520
00:34:52,730 --> 00:35:01,710
I'm expecting three vectors,
three special solutions.
521
00:35:01,710 --> 00:35:06,470
I give the value one
to that free variable,
522
00:35:06,470 --> 00:35:12,000
and what's the pivot
variable if the --
523
00:35:12,000 --> 00:35:15,130
this is going to
be a vector in S?
524
00:35:15,130 --> 00:35:16,390
Minus one.
525
00:35:16,390 --> 00:35:20,250
Now they're always added to
-- the entries add to zero.
526
00:35:20,250 --> 00:35:22,870
The second special
solution has a one
527
00:35:22,870 --> 00:35:26,540
in the second free variable,
and again a minus one
528
00:35:26,540 --> 00:35:27,730
makes it right.
529
00:35:27,730 --> 00:35:30,940
The third one has a one in
the third free variable,
530
00:35:30,940 --> 00:35:34,910
and again a minus
one makes it right.
531
00:35:34,910 --> 00:35:35,780
That's my answer.
532
00:35:35,780 --> 00:35:39,770
That's the answer I
would be looking for.
533
00:35:39,770 --> 00:35:43,300
The -- a basis for
this subspace S,
534
00:35:43,300 --> 00:35:45,530
you would just
list three vectors,
535
00:35:45,530 --> 00:35:48,350
and those would be the
natural three to list.
536
00:35:48,350 --> 00:35:55,240
Not the only possible three,
but those are the special three.
537
00:35:55,240 --> 00:35:58,960
OK, tell me about
the column space,
538
00:35:58,960 --> 00:36:02,560
What's the column
space of this matrix A?
539
00:36:07,390 --> 00:36:13,500
So the column space
is a subspace of R^1,
540
00:36:13,500 --> 00:36:15,470
because m is only one.
541
00:36:15,470 --> 00:36:17,920
The columns only
have one component.
542
00:36:17,920 --> 00:36:23,360
So the column space of S, the
column space of A is somewhere
543
00:36:23,360 --> 00:36:26,410
in the space R^1,
because we only have --
544
00:36:26,410 --> 00:36:30,240
these columns are short.
545
00:36:30,240 --> 00:36:32,940
And what is the
column space actually?
546
00:36:36,330 --> 00:36:42,280
I just, it's just talking with
these words is what I'm doing.
547
00:36:42,280 --> 00:36:48,360
The column space for
that matrix is R^1.
548
00:36:48,360 --> 00:36:52,780
The column space
for that matrix is
549
00:36:52,780 --> 00:36:55,463
all multiples of that column.
550
00:36:58,410 --> 00:37:00,760
And all multiples
give you all of R^1.
551
00:37:03,630 --> 00:37:07,140
And what's the, the
remaining fourth space,
552
00:37:07,140 --> 00:37:12,680
the null space of A
transpose is what?
553
00:37:17,560 --> 00:37:21,880
So we transpose A.
554
00:37:21,880 --> 00:37:26,290
We look for combinations
of the columns
555
00:37:26,290 --> 00:37:30,310
now that give zero
for A transpose.
556
00:37:30,310 --> 00:37:32,190
And there aren't any.
557
00:37:32,190 --> 00:37:36,620
The only thing, the only
combination of these rows
558
00:37:36,620 --> 00:37:41,500
to give the zero row is
the zero combination.
559
00:37:41,500 --> 00:37:42,380
OK.
560
00:37:42,380 --> 00:37:44,415
So let's just check dimensions.
561
00:37:47,180 --> 00:37:51,320
The null space has
dimension three.
562
00:37:51,320 --> 00:37:53,420
The row space has dimension one.
563
00:37:53,420 --> 00:37:54,630
Three plus one is four.
564
00:37:57,180 --> 00:37:59,680
The column space
has dimension one,
565
00:37:59,680 --> 00:38:02,790
and what's the
dimension of this, like,
566
00:38:02,790 --> 00:38:05,510
smallest possible space?
567
00:38:05,510 --> 00:38:08,790
What's the dimension
of the zero space?
568
00:38:08,790 --> 00:38:09,675
It's a subspace.
569
00:38:13,520 --> 00:38:14,190
Zero.
570
00:38:14,190 --> 00:38:15,190
What else could it be?
571
00:38:15,190 --> 00:38:17,970
I mean, let's -- we have to
take a reasonable answer --
572
00:38:17,970 --> 00:38:20,190
and the only reasonable
answer is zero.
573
00:38:20,190 --> 00:38:25,900
So one plus zero gives -- this
was n, the number of columns,
574
00:38:25,900 --> 00:38:30,480
and this is m, the
number of rows.
575
00:38:30,480 --> 00:38:32,910
And let's just, let
me just say again
576
00:38:32,910 --> 00:38:37,030
then the, the, the subspace
that has only that one
577
00:38:37,030 --> 00:38:42,830
point, that point is zero
dimensional, of course.
578
00:38:42,830 --> 00:38:46,920
And the basis is empty, because
if the dimension is zero,
579
00:38:46,920 --> 00:38:49,480
there shouldn't be
anybody in the basis.
580
00:38:49,480 --> 00:38:56,160
So the basis of that smallest
subspace is the empty set.
581
00:38:56,160 --> 00:38:59,590
And the number of members
in the empty set is zero,
582
00:38:59,590 --> 00:39:01,400
so that's the dimension.
583
00:39:01,400 --> 00:39:02,610
OK.
584
00:39:02,610 --> 00:39:04,040
Good.
585
00:39:04,040 --> 00:39:10,740
Now I have just five
minutes to tell you about --
586
00:39:10,740 --> 00:39:16,010
well, actually, about some,
some, some, this is now,
587
00:39:16,010 --> 00:39:21,430
this last topic of small
world graphs, and leads into,
588
00:39:21,430 --> 00:39:27,140
a lecture about graphs
and linear algebra.
589
00:39:27,140 --> 00:39:29,700
But let me tell you --
590
00:39:29,700 --> 00:39:32,860
in these last minutes the
graph that I interested in.
591
00:39:32,860 --> 00:39:40,600
It's the graph where
-- so what is a graph?
592
00:39:40,600 --> 00:39:42,020
Better tell you that first.
593
00:39:42,020 --> 00:39:42,880
OK.
594
00:39:42,880 --> 00:39:43,516
What's a graph?
595
00:39:47,380 --> 00:39:48,180
OK.
596
00:39:48,180 --> 00:39:49,980
This isn't calculus.
597
00:39:49,980 --> 00:39:54,010
We're not, I'm not thinking
of, like, some sine curve.
598
00:39:54,010 --> 00:39:57,940
The word graph is used in
a completely different way.
599
00:39:57,940 --> 00:40:07,490
It's a set of, a bunch
of nodes and edges,
600
00:40:07,490 --> 00:40:10,060
edges connecting the nodes.
601
00:40:10,060 --> 00:40:17,280
So I have nodes like
five nodes and edges --
602
00:40:17,280 --> 00:40:21,880
I'll put in some edges, I
could put, include them all.
603
00:40:21,880 --> 00:40:23,970
There's -- well, let me
put in a couple more.
604
00:40:27,350 --> 00:40:31,380
There's a graph with five
nodes and one two three four
605
00:40:31,380 --> 00:40:34,690
five six edges.
606
00:40:34,690 --> 00:40:38,190
And some five by six
matrix is going to tell us
607
00:40:38,190 --> 00:40:41,100
everything about that graph.
608
00:40:41,100 --> 00:40:43,350
Let me leave that
matrix to next time
609
00:40:43,350 --> 00:40:46,350
and tell you about the
question I'm interested in.
610
00:40:46,350 --> 00:40:52,340
Suppose, suppose the
graph isn't just,
611
00:40:52,340 --> 00:40:56,230
just doesn't have just five
nodes, but suppose every,
612
00:40:56,230 --> 00:40:59,780
suppose every person
in this room is a node.
613
00:41:03,200 --> 00:41:07,310
And suppose there's an
edge between two nodes
614
00:41:07,310 --> 00:41:11,300
if those two people are friends.
615
00:41:11,300 --> 00:41:14,070
So have I described a graph?
616
00:41:14,070 --> 00:41:18,980
It's a pretty big graph,
hundred, hundred nodes.
617
00:41:18,980 --> 00:41:21,190
And I don't know how
many edges are in there.
618
00:41:24,920 --> 00:41:27,730
There's an edge
if you're friends.
619
00:41:27,730 --> 00:41:29,950
So that's the graph
for this class.
620
00:41:29,950 --> 00:41:35,120
A, a similar graph you could
take for the whole country,
621
00:41:35,120 --> 00:41:38,680
so two hundred and
sixty million nodes.
622
00:41:38,680 --> 00:41:43,340
And edges between friends.
623
00:41:43,340 --> 00:41:50,110
And the question for that graph
is how many steps does it take
624
00:41:50,110 --> 00:41:52,210
to get from anybody to anybody?
625
00:41:56,780 --> 00:42:01,950
What two people are furthest
apart in this friendship graph,
626
00:42:01,950 --> 00:42:04,070
say for the US?
627
00:42:04,070 --> 00:42:08,820
By furthest apart, I
mean the distance from --
628
00:42:08,820 --> 00:42:12,640
well, I'll tell you my
distance to Clinton.
629
00:42:12,640 --> 00:42:14,040
It's two.
630
00:42:14,040 --> 00:42:18,440
I happened to go to college
with somebody who knows Clinton.
631
00:42:18,440 --> 00:42:19,190
I don't know him.
632
00:42:19,190 --> 00:42:24,950
So my distance to Clinton is not
one, because I don't, happily
633
00:42:24,950 --> 00:42:26,680
or not, don't know him.
634
00:42:26,680 --> 00:42:29,050
But I know somebody who does.
635
00:42:29,050 --> 00:42:32,541
He's a Senator and so
I presume he knows him.
636
00:42:32,541 --> 00:42:33,040
OK.
637
00:42:33,040 --> 00:42:35,311
I don't know what your --
well, what's your distance
638
00:42:35,311 --> 00:42:35,810
to Clinton?
639
00:42:39,100 --> 00:42:40,950
Well, not more
than three, right.
640
00:42:40,950 --> 00:42:43,000
Actually, true.
641
00:42:43,000 --> 00:42:44,430
You know me.
642
00:42:44,430 --> 00:42:50,880
I take credit for reducing your
Clinton distance to three --
643
00:42:50,880 --> 00:42:52,195
what's your distance to Monica.
644
00:42:54,780 --> 00:43:04,090
Not, anybody below -- below
four is in trouble here.
645
00:43:04,090 --> 00:43:07,010
Or maybe three, but, right.
646
00:43:07,010 --> 00:43:14,740
So -- and what's Hillary's
distance to Monica?
647
00:43:14,740 --> 00:43:18,020
I don't think we'd better
put that on tape here.
648
00:43:18,020 --> 00:43:22,360
That's one or two, I guess.
649
00:43:22,360 --> 00:43:24,490
Is that right?
650
00:43:24,490 --> 00:43:28,980
I don't -- well, we won't,
think more about that.
651
00:43:28,980 --> 00:43:32,490
So actually, the,
the real question
652
00:43:32,490 --> 00:43:38,030
is what are large distances?
653
00:43:38,030 --> 00:43:41,910
How, how far apart could
people be separated?
654
00:43:41,910 --> 00:43:46,520
And roughly this number
six degrees of separation
655
00:43:46,520 --> 00:43:49,940
has kind of appeared as the
movie title, as the book title,
656
00:43:49,940 --> 00:43:52,050
and it's with this meaning.
657
00:43:52,050 --> 00:43:56,050
That roughly speaking --
658
00:43:56,050 --> 00:43:59,230
six might be a fairly --
659
00:43:59,230 --> 00:44:01,860
not too many people.
660
00:44:01,860 --> 00:44:04,860
If you sit next to
somebody on an airplane,
661
00:44:04,860 --> 00:44:07,850
you get talking to them.
662
00:44:07,850 --> 00:44:12,310
You begin to discuss mutual
friends to sort of find out,
663
00:44:12,310 --> 00:44:15,220
OK, what connections
do you have,
664
00:44:15,220 --> 00:44:17,450
and very often
you'll find you're
665
00:44:17,450 --> 00:44:21,880
connected in, like, two
or three or four steps.
666
00:44:21,880 --> 00:44:23,970
And you remark,
it's a small world,
667
00:44:23,970 --> 00:44:27,850
and that's how this expression
small world came up.
668
00:44:27,850 --> 00:44:31,450
But six, I don't know if you
could find -- if it took six,
669
00:44:31,450 --> 00:44:34,810
I don't know if you would
successfully discover those six
670
00:44:34,810 --> 00:44:36,980
in a, in an airplane
conversation.
671
00:44:36,980 --> 00:44:40,150
But here's the math
question, and I'll
672
00:44:40,150 --> 00:44:42,650
leave it for next,
for lecture twelve,
673
00:44:42,650 --> 00:44:46,010
and do a lot of linear
algebra in lecture twelve.
674
00:44:46,010 --> 00:44:54,860
But the interesting point is
that with a few shortcuts,
675
00:44:54,860 --> 00:44:58,290
the distances come
down dramatically.
676
00:44:58,290 --> 00:45:03,900
That, I mean, all your distances
to Clinton immediately drop
677
00:45:03,900 --> 00:45:06,460
to three by taking
linear algebra.
678
00:45:06,460 --> 00:45:11,580
That's, like, an extra bonus
for taking linear algebra.
679
00:45:11,580 --> 00:45:17,550
And to understand mathematically
what it is about these graphs
680
00:45:17,550 --> 00:45:18,120
--
681
00:45:18,120 --> 00:45:21,710
or like the graphs of
the World Wide Web.
682
00:45:21,710 --> 00:45:23,170
There's a fantastic graph.
683
00:45:23,170 --> 00:45:27,790
So many people would like to
understand and model the web.
684
00:45:27,790 --> 00:45:34,650
What the -- where the edges are
links and the nodes are, sites,
685
00:45:34,650 --> 00:45:37,530
websites.
686
00:45:37,530 --> 00:45:39,960
I'll leave you with that
graph, and I'll see you --
687
00:45:39,960 --> 00:45:42,740
have a good weekend,
and see you on Monday.