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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 -- 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 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 as
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you can add and multiply by
scalars.
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So these new,
new vector spaces,
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the example I took was the
space M of all three by three
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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 and
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multiplying by numbers.
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And that's fine,
we stay within this space of
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three 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
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matrices, 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, is the product
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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,
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well, 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 by three
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matrix?
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Nine.
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Nine is the,
is the dimension I'm going to
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find.
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And the most obvious basis
would be the matrix that's that
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matrix and then this matrix with
a one there and that's two of
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them, shall I put in the third
one, and then onwards,
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and the last one maybe 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 as nine
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dimensional space.
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It's just the nine numbers are
written in a square instead of
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in a column.
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But somehow it's different and,
and ought to be thought 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
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subspace 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 by threes,
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well, how many of these
original basis members belong to
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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,
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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 -- but that's,
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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,
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we can, 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, and the three entries
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above the diagonal,
then I know what the three
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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,
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actually, maybe we have got a
basis here for,
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the upper triangulars.
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I guess six of these guys,
one, two, three,
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four, and a,
and a couple more,
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would be upper triangular.
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So there's a accidental case
where the big basis contains in
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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 -- a basis
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is the basis for the big space,
we generally need to think it
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all over again to get a basis
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
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matrices and the upper
triangular.
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This is symmetric and upper
triangular.
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OK.
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What's the, what's the
dimension of that space?
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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,
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right -- 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,
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what about putting -- so this
is like, this intersection --
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is taking all the vectors that
are in both, that are symmetric
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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 or upper
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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,
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or possibly both,
so they, the diagonals
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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 of
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a nine dimensional space,
there's -- ooh,
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sorry, six.
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There's a six dimensional
subspace of a nine dimensional
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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 in also,
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separately, things that are in
U.
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This is the sum of any element
of S, that is,
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any symmetric matrix,
plus any in U,
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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, take all symmetric
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matrices, and add them 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 get out of
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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,
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just a little,
to, to think of these subspaces
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and what their intersection is
and what their sum is.
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And now can I give you a little
-- oh, well, let's figure out
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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
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threes.
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So the original spaces had,
the original symmetric space
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had 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 -- if I have
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two subspaces,
the dimension of one plus the
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dimension of the other --
equals the dimension of their
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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, this is the set of
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natural things to do with,
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 that
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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 that we'll give
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just a few minutes to.
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Suppose I have a differential
equation like d^2y/dx^2+ y=0.
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OK.
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I look at the solutions to that
equation.
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So what are the solutions to
that equation?
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y=cos(x) is a solution.
y=sin(x) is a solution.
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y equals -- well,
e to the (ix) is a solution,
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if you want,
if you allow me to put that in.
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But why should I put that in?
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It's already there.
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You see, I'm really looking at
a null space here.
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I'm looking at the null space
of a differential equation.
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That's the solution space.
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And describe the solution
space, all solutions to this
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differential equation.
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So the equation is y''+y=0.
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Cosine's, cosine's a solution,
sine is a solution.
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Now tell me all the solutions.
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They're -- so I don't need
e^(ix).
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Forget that.
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What are all the complete
solutions?
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Is what?
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A combination of these.
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The complete solution is y
equals some multiple of the
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cosine plus some multiple of the
sine.
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That's a vector space.
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That's a vector space.
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What's the dimension of that
space?
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What's a basis for that space?
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OK, let me ask you a basis
first.
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If I take the set of solutions
to that second order
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differential equation --
there it is,
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those are the solutions.
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What's a basis for that space?
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Now remember,
what's the, what question I
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asking?
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Because if you know the
question I'm asking,
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you'll see the answer.
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A basis means all the guys in
the space are combinations of
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these basis vectors.
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Well, this is a basis.
sin x, cos x there is a basis.
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Those two -- they're like the
special solutions,
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right?
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We had special solutions to
Ax=b.
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Now we've got special solutions
to differential equations.
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Sorry, we had special solutions
to Ax=0, I misspoke.
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The special solutions were for
the null space just as here
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we're talking about the null
space.
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Do you see that here is a --
those two -- and what's the
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dimension of the solution space?
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How many vectors in this basis?
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Two, the sine and cosine.
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Are those the only basis for
this space?
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By no means.
e^(ix) and e^(-ix) would be
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another basis.
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Lots of bases.
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But do you see that really what
a course in differential --
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in linear differential
equations is about is finding a
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basis for the solution space.
285
00:18:52 --> 00:18:58
The dimension of the solution
space will always be -- will be
286
00:18:58 --> 00:19:03
two, because we have a second
order equation.
287
00:19:03 --> 00:19:07
So that's, like there's 18.03
in --
288
00:19:07 --> 00:19:13.2
five minutes of 18.06 is enough
to, to take care of 18.03.
289
00:19:13.2 --> 00:19:13
OK.
290
00:19:13 --> 00:19:17.51
So there's a -- that's one more
example.
291
00:19:17.51 --> 00:19:23
And of course the point of the
example is these things don't
292
00:19:23 --> 00:19:25
look like vectors.
293
00:19:25 --> 00:19:28
They look like functions.
294
00:19:28 --> 00:19:34
But we can call them vectors,
because we can add them and we
295
00:19:34 --> 00:19:38
can multiply by constants,
so we can take linear
296
00:19:38 --> 00:19:39
combinations.
297
00:19:39 --> 00:19:43
That's all we have to be
allowed to do.
298
00:19:43 --> 00:19:48.92
So that's really why this idea
of linear algebra and basis and
299
00:19:48.92 --> 00:19:54
dimension and so on plays a
wider role than --
300
00:19:54 --> 00:19:58
our constant discussions of m
by n matrices.
301
00:19:58 --> 00:19:59
OK.
302
00:19:59 --> 00:20:03
That's what I wanted to say
about that topic.
303
00:20:03 --> 00:20:08
Now of course the key,
number associated with
304
00:20:08 --> 00:20:14
matrices, to go back to that
number, is the rank.
305
00:20:14 --> 00:20:20
And the rank,
what do we know about the rank?
306
00:20:20 --> 00:20:26
Well, we know it's not bigger
than m and it's not bigger than
307
00:20:26 --> 00:20:26.38
n.
308
00:20:26.38 --> 00:20:31
So but I'd like to have a
little discussion on the rank.
309
00:20:31 --> 00:20:34
Maybe I'll put that here.
310
00:20:34 --> 00:20:39
So I'm picking up this topic of
rank one matrices.
311
00:20:39 --> 00:20:44.92
And the reason I'm interested
in rank one matrices is that
312
00:20:44.92 --> 00:20:48
they ought to be simple.
313
00:20:48 --> 00:20:56
If the rank is only one,
the matrix can't get away from
314
00:20:56 --> 00:20:56
us.
315
00:20:56 --> 00:21:03
So for example,
let me take -- let me create a
316
00:21:03 --> 00:21:05
rank one matrix.
317
00:21:05 --> 00:21:05
OK.
318
00:21:05 --> 00:21:12.78
Suppose it's three -- suppose
it's two by three.
319
00:21:12.78 --> 00:21:19
And let me give you the first
row.
320
00:21:19 --> 00:21:22
What can the second row be?
321
00:21:22 --> 00:21:30
Tell me a possible second row
here, for, for this matrix to
322
00:21:30 --> 00:21:31
have rank one.
323
00:21:31 --> 00:21:35.14
A possible second row is?
324
00:21:35.14 --> 00:21:36
Two eight ten.
325
00:21:36 --> 00:21:42
The second row is a multiple of
the first row.
326
00:21:42 --> 00:21:46
It's not independent.
327
00:21:46 --> 00:21:51
So tell me a basis for the --
oh yeah, sorry to keep bringing
328
00:21:51 --> 00:21:53
up these same questions.
329
00:21:53 --> 00:21:56
After the quiz I'll stop,
but for now,
330
00:21:56 --> 00:21:59
tell me a basis for the row
space.
331
00:21:59 --> 00:22:04
A basis for the row space of
that matrix is the first row,
332
00:22:04 --> 00:22:04
right?
333
00:22:04 --> 00:22:07.2
The first row,
one four five.
334
00:22:07.2 --> 00:22:12
A basis for the column space of
this matrix is?
335
00:22:12 --> 00:22:15
What's the dimension of the
column space?
336
00:22:15 --> 00:22:19
The dimension of the column
space is also one,
337
00:22:19 --> 00:22:19
right?
338
00:22:19 --> 00:22:21
Because it's also the rank.
339
00:22:21 --> 00:22:26
The dimension -- you remember
the dimension of the column
340
00:22:26 --> 00:22:31
space equals the rank equals the
dimension of the column space of
341
00:22:31 --> 00:22:36
the transpose,
which is the row space of A.
342
00:22:36 --> 00:22:40
OK, and in this case it's one,
r is one.
343
00:22:40 --> 00:22:46
And sure enough,
all the columns are -- all the
344
00:22:46 --> 00:22:50
other columns are multiples of
that column.
345
00:22:50 --> 00:22:57.41
Now there's -- there ought to
be a nice way to see that,
346
00:22:57.41 --> 00:23:00
and here it is.
347
00:23:00 --> 00:23:04
I can write that matrix as its
pivot column,
348
00:23:04 --> 00:23:08.95
one two, times its -- times one
four five.
349
00:23:08.95 --> 00:23:14
A column times a row,
one column times one row gives
350
00:23:14 --> 00:23:16
me a matrix, right?
351
00:23:16 --> 00:23:22
If I multiply a column by a
row, that, g- that's a two by
352
00:23:22 --> 00:23:28
one matrix times a one by three
matrix, and the result of the
353
00:23:28 --> 00:23:32
multiplication is two by three.
354
00:23:32 --> 00:23:36
And it comes out right.
355
00:23:36 --> 00:23:45
So what I want to -- my point
is the rank one matrices that
356
00:23:45 --> 00:23:54
every rank one matrix has the
form some column times some row.
357
00:23:54 --> 00:24:02
So U is a column vector,
V is a column vector --
358
00:24:02 --> 00:24:06
but I make it into a row by
putting in V transpose.
359
00:24:06 --> 00:24:10
So that's the -- complete
picture of rank one matrices.
360
00:24:10 --> 00:24:13
We'll be interested in rank one
matrices.
361
00:24:13 --> 00:24:16
Later we'll find,
oh, their determinant,
362
00:24:16 --> 00:24:19.31
that'll be easy,
their eigenvalues,
363
00:24:19.31 --> 00:24:21
that'll be interesting.
364
00:24:21 --> 00:24:30
Rank one matrices are like the
building blocks for all
365
00:24:30 --> 00:24:31
matrices.
366
00:24:31 --> 00:24:36
And actually maybe you can
guess.
367
00:24:36 --> 00:24:44
If I took any matrix,
a five by seventeen matrix of
368
00:24:44 --> 00:24:52.5
rank four, then it seems pretty
likely --
369
00:24:52.5 --> 00:24:56
and it's true,
that I could break that five by
370
00:24:56 --> 00:25:01
seventeen matrix down as a
combination of rank one
371
00:25:01 --> 00:25:02
matrices.
372
00:25:02 --> 00:25:06.31
And probably how many of those
would I need?
373
00:25:06.31 --> 00:25:10
If I have a five by seventeen
matrix of rank four,
374
00:25:10 --> 00:25:13
I'll need four of them,
right.
375
00:25:13 --> 00:25:17
Four rank one matrices.
376
00:25:17 --> 00:25:21
So the rank one matrices are
the, are the building blocks.
377
00:25:21 --> 00:25:26
And out -- I can produce every,
I can produce every five by --
378
00:25:26 --> 00:25:31.4
every rank four matrix out of
four rank one matrices.
379
00:25:31.4 --> 00:25:31
OK.
380
00:25:31 --> 00:25:34
That brings me to a question,
of course.
381
00:25:34 --> 00:25:39
Would the rank four matrices
form a subspace?
382
00:25:39 --> 00:25:44
Let me take all five by
seventeen matrices and think
383
00:25:44 --> 00:25:50
about rank four -- the subset of
rank four matrices.
384
00:25:50 --> 00:25:53
Let me -- I'll write this down.
385
00:25:53 --> 00:25:59.23
You seem I'm reviewing for the
quiz, because I'm asking the
386
00:25:59.23 --> 00:26:04
kind of questions that are short
enough but --
387
00:26:04 --> 00:26:12
that bring out do you know what
these words mean.
388
00:26:12 --> 00:26:21
So I take -- my matrix space M
now is all five by seventeen
389
00:26:21 --> 00:26:22
matrices.
390
00:26:22 --> 00:26:29.24
And now the question I ask is
the subset of,
391
00:26:29.24 --> 00:26:35
of rank four matrices,
is that a subspace?
392
00:26:35 --> 00:26:43
If I add a matrix of --
so if I multiply a matrix of
393
00:26:43 --> 00:26:47
rank four by -- of rank four or
less, let's say,
394
00:26:47 --> 00:26:52.48
because I have to let the zero
matrix in if it's going to be a
395
00:26:52.48 --> 00:26:53.26
subspace.
396
00:26:53.26 --> 00:26:58
But, but that doesn't just
because the zero matrix got in
397
00:26:58 --> 00:27:01.23
there doesn't mean I have a
subspace.
398
00:27:01.23 --> 00:27:05
So if I -- so the,
the question really comes down
399
00:27:05 --> 00:27:09
to --
if I add two rank four
400
00:27:09 --> 00:27:12
matrices, is the sum rank four?
401
00:27:12 --> 00:27:14
What do you think?
402
00:27:14 --> 00:27:16
If -- no, not usually.
403
00:27:16 --> 00:27:17
Not usually.
404
00:27:17 --> 00:27:23
If I add two rank four
matrices, the sum is probably --
405
00:27:23 --> 00:27:26
what could I say about the sum?
406
00:27:26 --> 00:27:32
Well, actually,
well, the rank could be five.
407
00:27:32 --> 00:27:35
It's a general fact,
actually, that the rank of A
408
00:27:35 --> 00:27:39
plus B can't be more than rank
of A plus the rank of B.
409
00:27:39 --> 00:27:42.82
So this would say if I added
two of those,
410
00:27:42.82 --> 00:27:47
the rank couldn't be larger
than eight, but I know actually
411
00:27:47 --> 00:27:51
the rank couldn't be as large as
eight anyway.
412
00:27:51 --> 00:27:55
What -- how big could the rank
be, for, for the rank of a
413
00:27:55 --> 00:27:56.8
matrix in M?
414
00:27:56.8 --> 00:28:00
Could be as large as five,
right, right.
415
00:28:00 --> 00:28:03
So they're all sort of natural
ideas.
416
00:28:03 --> 00:28:07
So it's rank four matrices or
rank one matrices -- let me,
417
00:28:07 --> 00:28:11
let me change that to rank one.
418
00:28:11 --> 00:28:16.17
Let me take the subset of rank
one matrices.
419
00:28:16.17 --> 00:28:18
Is that a vector space?
420
00:28:18 --> 00:28:23
If I add a rank one matrix to a
rank one matrix?
421
00:28:23 --> 00:28:24
No.
422
00:28:24 --> 00:28:28
It's most likely going to have
rank two.
423
00:28:28 --> 00:28:33
So this is -- So I'll just make
that point.
424
00:28:33 --> 00:28:36
Not a subspace.
425
00:28:36 --> 00:28:36
OK.
426
00:28:36 --> 00:28:36
OK.
427
00:28:36 --> 00:28:43
Those are topics that I wanted
to, just fill out the,
428
00:28:43 --> 00:28:45
the previous lectures.
429
00:28:45 --> 00:28:52
The I'll ask one more subspace
question, a, a more,
430
00:28:52 --> 00:28:54
a more, likely example.
431
00:28:54 --> 00:29:01
Suppose I'm in -- let me put,
put this example on a new
432
00:29:01 --> 00:29:03
board.
433
00:29:03 --> 00:29:07
Suppose I'm in R,
in R^4.
434
00:29:07 --> 00:29:15
So my typical vector in R^4 has
four components,
435
00:29:15 --> 00:29:18
v1, v2, v3, and v4.
436
00:29:18 --> 00:29:29
Suppose I take the subspace of
vectors whose components add to
437
00:29:29 --> 00:29:30
zero.
438
00:29:30 --> 00:29:36
So I let S be all v,
all vectors v in four
439
00:29:36 --> 00:29:45
dimensional space with
v1+v2+v3+v4=0.
440
00:29:45 --> 00:29:50
So I just want to consider that
bunch of vectors.
441
00:29:50 --> 00:29:53
Is it a subspace,
first of all?
442
00:29:53 --> 00:29:55.23
It is a subspace.
443
00:29:55.23 --> 00:29:57
It is a subspace.
444
00:29:57 --> 00:30:00
What's -- how do we see that?
445
00:30:00 --> 00:30:01
It is a subspace.
446
00:30:01 --> 00:30:05
I -- formally I should check.
447
00:30:05 --> 00:30:11
If I have one vector that with
whose components add to zero and
448
00:30:11 --> 00:30:18
I multiply that vector by six --
the components still add to
449
00:30:18 --> 00:30:21
zero, just six times as -- six
times zero.
450
00:30:21 --> 00:30:24
If I have a couple of v and a w
and I add them,
451
00:30:24 --> 00:30:27
the, the components still add
to zero.
452
00:30:27 --> 00:30:28
OK, it's a subspace.
453
00:30:28 --> 00:30:32
What's the dimension of that
space and what's a basis for
454
00:30:32 --> 00:30:33
that space?
455
00:30:33 --> 00:30:38
So you see how I can just
describe a space and we --
456
00:30:38 --> 00:30:43
we can ask for the dimension --
ask for the basis first and the
457
00:30:43 --> 00:30:44
dimension.
458
00:30:44 --> 00:30:50
Of course, the dimension's the
one that's easy to tell me in a
459
00:30:50 --> 00:30:51
single word.
460
00:30:51 --> 00:30:55
What's the dimension of our
subspace S here?
461
00:30:55 --> 00:31:00
And a basis tell me -- some
vectors in it.
462
00:31:00 --> 00:31:08
Well, I'm going to make ask you
again to guess the dimension.
463
00:31:08 --> 00:31:11.11
Again I think I heard it.
464
00:31:11.11 --> 00:31:11
Three.
465
00:31:11 --> 00:31:14
The dimension is three.
466
00:31:14 --> 00:31:19.18
Now how does this connect to
our Ax=0?
467
00:31:19.18 --> 00:31:23
Is this the null space of
something?
468
00:31:23 --> 00:31:28
Is that the null space of a
matrix?
469
00:31:28 --> 00:31:37
And then we can look at the
matrix and, and we know
470
00:31:37 --> 00:31:42
everything about those
subspaces.
471
00:31:42 --> 00:31:48
This is the null space of what
matrix?
472
00:31:48 --> 00:31:56
What's the matrix where the
null space is then Ab=0.
473
00:31:56 --> 00:32:01.94
So I want this equation to be
Ab=0.
474
00:32:01.94 --> 00:32:06
b is now the vector.
475
00:32:06 --> 00:32:12
And what's the matrix that,
that we're seeing there?
476
00:32:12 --> 00:32:15
It's the matrix of four ones.
477
00:32:15 --> 00:32:22
Do you see that that's -- that
if I look at Ab=0 for this
478
00:32:22 --> 00:32:28
matrix A, I multiply by b and I
get this requirement,
479
00:32:28 --> 00:32:31
that the components add to
zero.
480
00:32:31 --> 00:32:36
So I'm really when I speak
about S --
481
00:32:36 --> 00:32:42
I'm speaking about the null
space of that matrix.
482
00:32:42 --> 00:32:43
OK.
483
00:32:43 --> 00:32:49
Let's just say we've got a
matrix now, we want its null
484
00:32:49 --> 00:32:50
space.
485
00:32:50 --> 00:32:54
Well, we -- tell me its rank
first.
486
00:32:54 --> 00:32:58
The rank of that matrix is one,
thanks.
487
00:32:58 --> 00:33:01.5
So r is one.
488
00:33:01.5 --> 00:33:09
What's the general formula for
the dimension of the null space?
489
00:33:09 --> 00:33:17
The dimension of the null space
of a matrix is -- in general,
490
00:33:17 --> 00:33:20
an m by n matrix of rank r?
491
00:33:20 --> 00:33:26
How many independent guys in
the null space?
492
00:33:26 --> 00:33:29
n-r, right?
n-r.
493
00:33:29 --> 00:33:32
In this case,
n is four, four columns.
494
00:33:32 --> 00:33:36
The rank is one,
so the null space is three
495
00:33:36 --> 00:33:37
dimensions.
496
00:33:37 --> 00:33:42
So of course y- you could see
it in this case,
497
00:33:42 --> 00:33:48
but you can also see it here in
our systematic way of dealing
498
00:33:48 --> 00:33:53
with the four fundamental
subspaces of a matrix.
499
00:33:53 --> 00:33:59
So what actually what,
what are all four subspaces
500
00:33:59 --> 00:33:59
then?
501
00:33:59 --> 00:34:02.34
The row space is clear.
502
00:34:02.34 --> 00:34:05
The row space is in R^4.
503
00:34:05 --> 00:34:11
Yeah, can we take the four
fundamental subspaces of this
504
00:34:11 --> 00:34:12
matrix?
505
00:34:12 --> 00:34:15
Let's just kill this example.
506
00:34:15 --> 00:34:20
The row space is one
dimensional.
507
00:34:20 --> 00:34:23
It's all multiples of that,
of that row.
508
00:34:23 --> 00:34:25
The null space is three
dimensional.
509
00:34:25 --> 00:34:29
Oh, you better give me a basis
for the null space.
510
00:34:29 --> 00:34:32.05
So what's a basis for the null
space?
511
00:34:32.05 --> 00:34:33
The special solutions.
512
00:34:33 --> 00:34:39
To find the special solutions,
I look for the free variables.
513
00:34:39 --> 00:34:44
The free variables here are --
there's the pivot.
514
00:34:44 --> 00:34:49
The free variables are two,
three, and four.
515
00:34:49 --> 00:34:53
So the basis,
basis for S,
516
00:34:53 --> 00:34:58
for S will be -- I'm expecting
three vectors,
517
00:34:58 --> 00:35:01
three special solutions.
518
00:35:01 --> 00:35:06
I give the value one to that
free variable,
519
00:35:06 --> 00:35:12
and what's the pivot variable
if the --
520
00:35:12 --> 00:35:15
this is going to be a vector in
S?
521
00:35:15 --> 00:35:16
Minus one.
522
00:35:16 --> 00:35:20
Now they're always added to --
the entries add to zero.
523
00:35:20 --> 00:35:25
The second special solution has
a one in the second free
524
00:35:25 --> 00:35:28
variable, and again a minus one
makes it right.
525
00:35:28 --> 00:35:33
The third one has a one in the
third free variable,
526
00:35:33 --> 00:35:37
and again a minus one makes it
right.
527
00:35:37 --> 00:35:38
That's my answer.
528
00:35:38 --> 00:35:42.11
That's the answer I would be
looking for.
529
00:35:42.11 --> 00:35:46.75
The -- a basis for this
subspace S, you would just list
530
00:35:46.75 --> 00:35:50
three vectors,
and those would be the natural
531
00:35:50 --> 00:35:51
three to list.
532
00:35:51 --> 00:35:56
Not the only possible three,
but those are the special
533
00:35:56 --> 00:35:57.75
three.
534
00:35:57.75 --> 00:36:04
OK, tell me about the column
space, What's the column space
535
00:36:04 --> 00:36:06
of this matrix A?
536
00:36:06 --> 00:36:10
So the column space is a
subspace of R^1,
537
00:36:10 --> 00:36:13
because m is only one.
538
00:36:13 --> 00:36:17
The columns only have one
component.
539
00:36:17 --> 00:36:23
So the column space of S,
the column space of A is
540
00:36:23 --> 00:36:30
somewhere in the space R^1,
because we only have --
541
00:36:30 --> 00:36:33
these columns are short.
542
00:36:33 --> 00:36:38
And what is the column space
actually?
543
00:36:38 --> 00:36:46
I just, it's just talking with
these words is what I'm doing.
544
00:36:46 --> 00:36:51
The column space for that
matrix is R^1.
545
00:36:51 --> 00:36:58
The column space for that
matrix is all multiples of that
546
00:36:58 --> 00:37:00
column.
547
00:37:00 --> 00:37:06
And all multiples give you all
of R^1.
548
00:37:06 --> 00:37:13
And what's the,
the remaining fourth space,
549
00:37:13 --> 00:37:18
the null space of A transpose
is what?
550
00:37:18 --> 00:37:21
So we transpose A.
551
00:37:21 --> 00:37:31
We look for combinations of the
columns now that give zero for A
552
00:37:31 --> 00:37:34
transpose.
553
00:37:34 --> 00:37:36
And there aren't any.
554
00:37:36 --> 00:37:41.68
The only thing,
the only combination of these
555
00:37:41.68 --> 00:37:47
rows to give the zero row is the
zero combination.
556
00:37:47 --> 00:37:47
OK.
557
00:37:47 --> 00:37:50
So let's just check dimensions.
558
00:37:50 --> 00:37:54
The null space has dimension
three.
559
00:37:54 --> 00:37:58.16
The row space has dimension
one.
560
00:37:58.16 --> 00:38:02
Three plus one is four.
561
00:38:02 --> 00:38:07
The column space has dimension
one, and what's the dimension of
562
00:38:07 --> 00:38:10.27
this, like, smallest possible
space?
563
00:38:10.27 --> 00:38:13
What's the dimension of the
zero space?
564
00:38:13 --> 00:38:14
It's a subspace.
565
00:38:14 --> 00:38:15
Zero.
566
00:38:15 --> 00:38:17
What else could it be?
567
00:38:17 --> 00:38:22
I mean, let's -- we have to
take a reasonable answer --
568
00:38:22 --> 00:38:25
and the only reasonable answer
is zero.
569
00:38:25 --> 00:38:30
So one plus zero gives -- this
was n, the number of columns,
570
00:38:30 --> 00:38:33
and this is m,
the number of rows.
571
00:38:33 --> 00:38:37
And let's just,
let me just say again then the,
572
00:38:37 --> 00:38:41
the, the subspace that has only
that one point,
573
00:38:41 --> 00:38:45
that point is zero dimensional,
of course.
574
00:38:45 --> 00:38:50
And the basis is empty,
because if the dimension is
575
00:38:50 --> 00:38:54
zero, there shouldn't be anybody
in the basis.
576
00:38:54 --> 00:38:58
So the basis of that smallest
subspace is the empty set.
577
00:38:58 --> 00:39:03
And the number of members in
the empty set is zero,
578
00:39:03 --> 00:39:06
so that's the dimension.
579
00:39:06 --> 00:39:06
OK.
580
00:39:06 --> 00:39:07
Good.
581
00:39:07 --> 00:39:13
Now I have just five minutes to
tell you about -- well,
582
00:39:13 --> 00:39:19
actually, about some,
some, some, this is now,
583
00:39:19 --> 00:39:25
this last topic of small world
graphs, and leads into,
584
00:39:25 --> 00:39:32
a lecture about graphs and
linear algebra.
585
00:39:32 --> 00:39:39
But let me tell you -- in these
last minutes the graph that I
586
00:39:39 --> 00:39:40
interested in.
587
00:39:40 --> 00:39:45
It's the graph where -- so what
is a graph?
588
00:39:45 --> 00:39:48
Better tell you that first.
589
00:39:48 --> 00:39:49
OK.
590
00:39:49 --> 00:39:50
What's a graph?
591
00:39:50 --> 00:39:51
OK.
592
00:39:51 --> 00:39:54
This isn't calculus.
593
00:39:54 --> 00:40:00.17
We're not, I'm not thinking of,
like, some sine curve.
594
00:40:00.17 --> 00:40:05
The word graph is used in a
completely different way.
595
00:40:05 --> 00:40:09
It's a set of,
a bunch of nodes and edges,
596
00:40:09 --> 00:40:12
edges connecting the nodes.
597
00:40:12 --> 00:40:18
So I have nodes like five nodes
and edges -- I'll put in some
598
00:40:18 --> 00:40:22
edges, I could put,
include them all.
599
00:40:22 --> 00:40:27
There's -- well,
let me put in a couple more.
600
00:40:27 --> 00:40:32
There's a graph with five nodes
and one two three four five six
601
00:40:32 --> 00:40:33
edges.
602
00:40:33 --> 00:40:38
And some five by six matrix is
going to tell us everything
603
00:40:38 --> 00:40:40
about that graph.
604
00:40:40 --> 00:40:46
Let me leave that matrix to
next time and tell you about the
605
00:40:46 --> 00:40:49
question I'm interested in.
606
00:40:49 --> 00:40:56
Suppose, suppose the graph
isn't just, just doesn't have
607
00:40:56 --> 00:41:00
just five nodes,
but suppose every,
608
00:41:00 --> 00:41:05
suppose every person in this
room is a node.
609
00:41:05 --> 00:41:12
And suppose there's an edge
between two nodes if those two
610
00:41:12 --> 00:41:14
people are friends.
611
00:41:14 --> 00:41:19
So have I described a graph?
612
00:41:19 --> 00:41:23
It's a pretty big graph,
hundred, hundred nodes.
613
00:41:23 --> 00:41:27.05
And I don't know how many edges
are in there.
614
00:41:27.05 --> 00:41:29
There's an edge if you're
friends.
615
00:41:29 --> 00:41:32
So that's the graph for this
class.
616
00:41:32 --> 00:41:37
A, a similar graph you could
take for the whole country,
617
00:41:37 --> 00:41:41
so two hundred and sixty
million nodes.
618
00:41:41 --> 00:41:45.03
And edges between friends.
619
00:41:45.03 --> 00:41:52
And the question for that graph
is how many steps does it take
620
00:41:52 --> 00:41:56
to get from anybody to anybody?
621
00:41:56 --> 00:42:03
What two people are furthest
apart in this friendship graph,
622
00:42:03 --> 00:42:04
say for the US?
623
00:42:04 --> 00:42:11
By furthest apart,
I mean the distance from --
624
00:42:11 --> 00:42:15
well, I'll tell you my distance
to Clinton.
625
00:42:15 --> 00:42:16
It's two.
626
00:42:16 --> 00:42:21
I happened to go to college
with somebody who knows Clinton.
627
00:42:21 --> 00:42:22
I don't know him.
628
00:42:22 --> 00:42:27
So my distance to Clinton is
not one, because I don't,
629
00:42:27 --> 00:42:31
happily or not,
don't know him.
630
00:42:31 --> 00:42:33
But I know somebody who does.
631
00:42:33 --> 00:42:37
He's a Senator and so I presume
he knows him.
632
00:42:37 --> 00:42:38
OK.
633
00:42:38 --> 00:42:42
I don't know what your -- well,
what's your distance to
634
00:42:42 --> 00:42:43
Clinton?
635
00:42:43 --> 00:42:46
Well, not more than three,
right.
636
00:42:46 --> 00:42:47
Actually, true.
637
00:42:47 --> 00:42:49
You know me.
638
00:42:49 --> 00:42:56
I take credit for reducing your
Clinton distance to three --
639
00:42:56 --> 00:43:00
what's your distance to Monica.
640
00:43:00 --> 00:43:06.99
Not, anybody below -- below
four is in trouble here.
641
00:43:06.99 --> 00:43:10
Or maybe three,
but, right.
642
00:43:10 --> 00:43:16
So -- and what's Hillary's
distance to Monica?
643
00:43:16 --> 00:43:22
I don't think we'd better put
that on tape here.
644
00:43:22 --> 00:43:25
That's one or two,
I guess.
645
00:43:25 --> 00:43:26
Is that right?
646
00:43:26 --> 00:43:31.79
I don't -- well,
we won't, think more about
647
00:43:31.79 --> 00:43:32
that.
648
00:43:32 --> 00:43:37
So actually,
the, the real question is what
649
00:43:37 --> 00:43:40.86
are large distances?
650
00:43:40.86 --> 00:43:44
How, how far apart could people
be separated?
651
00:43:44 --> 00:43:50.3
And roughly this number six
degrees of separation has kind
652
00:43:50.3 --> 00:43:54.88
of appeared as the movie title,
as the book title,
653
00:43:54.88 --> 00:43:57
and it's with this meaning.
654
00:43:57 --> 00:44:02
That roughly speaking --
six might be a fairly -- not
655
00:44:02 --> 00:44:03
too many people.
656
00:44:03 --> 00:44:08
If you sit next to somebody on
an airplane, you get talking to
657
00:44:08 --> 00:44:08.93
them.
658
00:44:08.93 --> 00:44:13
You begin to discuss mutual
friends to sort of find out,
659
00:44:13 --> 00:44:17
OK, what connections do you
have, and very often you'll find
660
00:44:17 --> 00:44:21
you're connected in,
like, two or three or four
661
00:44:21 --> 00:44:22
steps.
662
00:44:22 --> 00:44:25
And you remark,
it's a small world,
663
00:44:25 --> 00:44:28
and that's how this expression
small world came up.
664
00:44:28 --> 00:44:32.5
But six, I don't know if you
could find -- if it took six,
665
00:44:32.5 --> 00:44:36
I don't know if you would
successfully discover those six
666
00:44:36 --> 00:44:39
in a, in an airplane
conversation.
667
00:44:39 --> 00:44:44
But here's the math question,
and I'll leave it for next,
668
00:44:44 --> 00:44:49
for lecture twelve,
and do a lot of linear algebra
669
00:44:49 --> 00:44:51
in lecture twelve.
670
00:44:51 --> 00:44:56.35
But the interesting point is
that with a few shortcuts,
671
00:44:56.35 --> 00:45:00
the distances come down
dramatically.
672
00:45:00 --> 00:45:04
That, I mean,
all your distances to Clinton
673
00:45:04 --> 00:45:09
immediately drop to three by
taking linear algebra.
674
00:45:09 --> 00:45:13
That's, like,
an extra bonus for taking
675
00:45:13 --> 00:45:14
linear algebra.
676
00:45:14 --> 00:45:19
And to understand
mathematically what it is about
677
00:45:19 --> 00:45:24
these graphs --
or like the graphs of the World
678
00:45:24 --> 00:45:25
Wide Web.
679
00:45:25 --> 00:45:27
There's a fantastic graph.
680
00:45:27 --> 00:45:33
So many people would like to
understand and model the web.
681
00:45:33 --> 00:45:38
What the -- where the edges are
links and the nodes are,
682
00:45:38 --> 00:45:40
sites, websites.
683
00:45:40 --> 00:45:45
I'll leave you with that graph,
and I'll see you --
684
00:45:45 --> 00:45:48
have a good weekend,
and see you on Monday.