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OK.
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Good.
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The final class in linear
algebra at MIT this Fall is to
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review the whole course.
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And, you know the best way I
know how to review is to take
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old exams and just think through
the problems.
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So it will be a three-hour exam
next Thursday.
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Nobody will be able to take an
exam before Thursday,
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anybody who needs to take it in
some different way after
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Thursday should see me next
Monday.
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I'll be in my office Monday.
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OK.
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May I just read out some
problems and,
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let me bring the board down,
and let's start.
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OK.
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Here's a question.
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This is about a 3-by-n matrix.
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And we're given -- so we're
given -- given -- A x equals 1 0
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0 has no solution.
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And we're also given A x equals
0 1 0 has exactly one solution.
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OK.
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So you can probably anticipate
my first question,
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what can you tell me about m?
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It's an m-by-n matrix of rank
r, as always,
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what can you tell me about
those three numbers?
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So what can you tell me about
m, the number of rows,
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n, the number of columns,
and r, the rank?
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OK.
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See, do you want to tell me
first what m is?
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How many rows in this matrix?
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Must be three,
right?
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We can't tell what n is,
but we can certainly tell that
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m is three.
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OK.
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And, what do these things tell
us?
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Let's take them one at a time.
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When I discover that some
equation has no solution,
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that there's some right-hand
side with no answer,
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what does that tell me about
the rank of the matrix?
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It's smaller m.
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Is that right?
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If there is no solution,
that tells me that some rows of
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the matrix are combinations of
other rows.
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Because if I had a pivot in
every row, then I would
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certainly be able to solve the
system.
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I would have particular
solutions and all the good
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stuff.
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So any time that there's a
system with no solutions,
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that tells me that r must be
below m.
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What about the fact that if,
when there is a solution,
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there's only one?
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What does that tell me?
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Well, normally there would be
one solution,
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and then we could add in
anything in the null space.
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So this is telling me the null
space only has the 0 vector in
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it.
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There's just one solution,
period, so what does that tell
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me?
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The null space has only the
zero vector in it?
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What does that tell me about
the relation of r to n?
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So this one solution only,
that means the null space of
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the matrix must be just the zero
vector, and what does that tell
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me about r and n?
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They're equal.
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The columns are independent.
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So I've got,
now, r equals n,
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and r less than m,
and now I also know m is three.
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So those are really the facts I
know.
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n=r and those numbers are
smaller than three.
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Sorry, yes, yes.
r is smaller than m,
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and n, of course,
is also.
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So I guess this summarizes what
we can tell.
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In fact, why not give me a
matrix -- because I would often
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ask for an example of such a
matrix -- can you give me a
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matrix A that's an example?
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That shows this possibility?
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Exactly, that there's no
solution with that right-hand
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side, but there's exactly one
solution with this right-hand
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side.
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Anybody want to suggest a
matrix that does that?
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Let's see.
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What do I -- what vector do I
want in the column space?
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I want zero,
one, zero, to be in the column
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space, because I'm able to solve
for that.
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So let's put zero,
one, zero in the column space.
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Actually, I could stop right
there.
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That would be a matrix with m
equal three, three rows,
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and n and r are both one,
rank one, one column,
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and, of course,
there's no solution to that
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one.
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So that's perfectly good as it
is.
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Or if you, kind of,
have a prejudice against
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matrices that only have one
column, I'll accept a second
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column.
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So what could I include as a
second column that would just be
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a different answer but equally
good?
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I could put this vector in the
column space,
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too, if I wanted.
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That would now be a case with
r=n=2, but, of course,
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three m eq- m is still three,
and this vector is not in the
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column space.
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So you're -- this is just like
prompting us to remember all
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those things,
column space,
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null space, all that stuff.
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Now, I probably asked a second
question about this type of
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thing.
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Ah.
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OK.
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Oh, I even asked,
write down an example of a
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matrix that fits the
description.
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Hm.
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I guess I haven't learned
anything in twenty-six years.
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CK.
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Cross out all statements that
are false about any matrix with
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these -- so again,
these are -- this is the
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preliminary sta- these are the
facts about my matrix,
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this is one example.
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But, of course,
by having an example,
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it will be easy to check some
of these facts,
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or non-facts.
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Let me, let me write down some,
facts.
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Some possible facts.
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So this is really true or
false.
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The determinant -- this is part
one, the determinant of A
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transpose A is the same as the
determinant of A A transpose.
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Is that true or not?
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Second one, A transpose A,
is invertible.
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Is invertible.
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Third possible fact,
A A transpose is positive
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definite.
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So you see how,
on an exam question,
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I try to connect the different
parts of the course.
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So, well, I mean,
the simplest way would be to
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try it with that matrix as a
good example,
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but maybe we can answer,
even directly.
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Let me take number two first.
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Because I'm -- you know,
I'm very, very fond of that
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matrix, A transpose A.
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And when is it invertible?
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When is the matrix A transpose
A, invertible?
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The great thing is that I can
tell from the rank of A that I
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don't have to multiply out A
transpose A.
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A transpose A,
is invertible -- well,
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if A has a null space other
than the zero vector,
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then it -- it's -- no way it's
going to be invertible.
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But the beauty is,
if the null space of A is just
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the zero vector,
so the fact -- the key fact is,
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this is invertible if r=n,
by which I mean,
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independent columns of A.
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In A.
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In the matrix A.
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If r=n -- if the matrix A has
independent columns,
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then this combination,
A transpose A,
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is square and still that same
null space, only the zero
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vector, independent columns all
good, and so,
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what's the true/false?
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Is it -- is this middle one T
or F for this,
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in this setup?
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Well, we discovered that -- we
discovered that -- that r was n,
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from that second fact.
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So this is a true.
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That's a true.
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And, of course,
A transpose A,
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in this example,
would probably be -- what would
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A transpose A,
be, for that matrix?
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Can you multiply A transpose A,
and see what it looks like for
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that matrix?
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What shape would it be?
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It will be two by two.
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And what matrix will it be?
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The identity.
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So, it checks out.
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OK, what about A A transpose?
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Well, depending on the shape of
A, it could be good or not so
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good.
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It's always symmetric,
it's always square,
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but what's the size,
now?
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This is three by n,
and this is n by three,
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so the result is three by
three.
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Is it positive definite?
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I don't think so.
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False.
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If I multiply that by A
transpose, A A transpose,
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what would the rank be?
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It would be the same as the
rank of A, that's -- it would be
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just rank two.
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And if it's three-by-three,
and it's only rank two,
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it's certainly not positive
definite.
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So what could I say about A A
transpose, if I wanted to,
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like, say something true about
it?
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It's true that it is positive
semi-definite.
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If I made this semi-definite,
it would always be true,
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always.
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But if I'm looking for positive
definite, then I'm looking at
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the null space of whatever's
here, and, in this case,
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it's got a null space.
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So A, A -- eh,
shall we just figure it out,
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here?
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A A transpose,
for that matrix,
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will be three-by-three.
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If I multiplied A by A
transpose, what would the first
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row be?
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All zeroes, right?
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First row of A A transpose,
could only be all zeroes,
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so it's probably a one there
and a one there,
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or something like that.
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But, I don't even know if
that's right.
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But it's all zeroes there,
so it's certainly not positive
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definite.
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Let me not put anything up I'm
not sh- don't check.
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What about this determinant?
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Oh, well, I guess -- that's a
sort of tricky question.
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Is it true or false in this
case?
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It's false, apparently,
because A transpose A,
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is invertible,
we just got a true for this
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one, and we got a false,
we got a z- we got a
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non-invertible one for this one.
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So actually,
this one is false,
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number one.
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That surprises us,
actually, because it's,
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I mean, why was it tricky?
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Because what is true about
determinants?
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This would be true if those
matrices were square.
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If I have two square matrices,
A and any other matrix B,
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could be A transpose,
could be somebody else's
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matrix.
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Then it would be true that the
determinant of B A would equal
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the determinant of A B.
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But if the matrices are not
square and it would actually be
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true that it would be equal --
that this would equal the
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determinant of A times the
determinant of A transpose.
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We could even split up those
two separate determinants.
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And, of course,
those would be equal.
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But only when A is square.
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So that's just,
that's a question that rests on
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the, the falseness rests on the
fact that the matrix isn't
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square in the first place.
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OK, good.
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Let's see.
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Oh, now, even asks more.
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Prove that A transpose y equals
c -- hah-God,
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00:14:26 --> 00:14:30
it's -- this question goes on
and on.
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now I ask you about A transpose
y=c.
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So I'm asking you about the
equation -- about the matrix A
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00:14:43 --> 00:14:46.02
transpose.
246
00:14:46.02 --> 00:14:54
And I want you to prove that it
has at least one solution -- one
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00:14:54 --> 00:15:00
solution for every c,
every right-hand side c,
248
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and, in fact -- in fact,
infinitely many solutions for
249
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every c.
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OK.
251
00:15:09.46 --> 00:15:15
Well, none -- none of this is
difficult, but,
252
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it's been a little while.
253
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So we just have to think again.
254
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When I have a system of
equations -- this is -- this
255
00:15:28 --> 00:15:33
matrix A transpose is now,
instead of being three by n,
256
00:15:33 --> 00:15:35
it's n by three,
it's n by m.
257
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Of course.
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To show that a system has at
least one solution,
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when does this,
when does this system -- when
260
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is the system always solvable?
261
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When it has full row rank,
when the rows are independent.
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Here, we have n rows,
and that's the rank.
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So at least one solution,
because the number of rows,
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which is n, for the transpose,
is equal to r,
265
00:16:19 --> 00:16:22
the rank.
266
00:16:22 --> 00:16:27
This A transpose had
independent rows because A had
267
00:16:27 --> 00:16:29
independent columns,
right?
268
00:16:29 --> 00:16:34
The original A had independent
columns, when we transpose it,
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00:16:34 --> 00:16:38
it has independent rows,
so there's at least one
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solution.
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But now, how do I even know
that there are infinitely many
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solutions?
273
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Oh, what do I -- I want to know
something about the null space.
274
00:16:52 --> 00:16:57.34
What's the dimension of the
null space of A transpose?
275
00:16:57.34 --> 00:17:03
So the answer has got to be the
dimension of the null space of A
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00:17:03 --> 00:17:07.28
transpose, what's the general
fact?
277
00:17:07.28 --> 00:17:13
If A is an m by n matrix of
rank r, what's the dimension of
278
00:17:13 --> 00:17:14
A transpose?
279
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The null space of A transpose?
280
00:17:17 --> 00:17:23
Do you remember that little
fourth subspace that's tagging
281
00:17:23 --> 00:17:26
along down in our big picture?
282
00:17:26 --> 00:17:28
It's dimension was m-r.
283
00:17:28 --> 00:17:34
And, that's bigger than zero.
m is bigger than r.
284
00:17:34 --> 00:17:38
So there's a lot in that null
space.
285
00:17:38 --> 00:17:44
So there's always one solution
because n i- this is speaking
286
00:17:44 --> 00:17:46
about A transpose.
287
00:17:46 --> 00:17:51
So for A transpose,
the roles of m and n are
288
00:17:51 --> 00:17:56
reversed, of course,
so I'm -- keep in mind that
289
00:17:56 --> 00:18:02
this board was about A
transpose, so the roles -- so
290
00:18:02 --> 00:18:07
it's the null space of a
transpose, and there are m-r
291
00:18:07 --> 00:18:09
free variables.
292
00:18:09 --> 00:18:12
OK, that's, like,
just some, review.
293
00:18:12 --> 00:18:18
Can I take another problem
that's also sort of
294
00:18:18 --> 00:18:26
-- suppose the matrix A has
three columns,
295
00:18:26 --> 00:18:28.43
v1, v2, v3.
296
00:18:28.43 --> 00:18:35
Those are the columns of the
matrix.
297
00:18:35 --> 00:18:37
All right.
298
00:18:37 --> 00:18:39
Question A.
299
00:18:39 --> 00:18:42
Solve Ax=v1-v2+v3.
300
00:18:42 --> 00:18:46
Tell me what x is.
301
00:18:46 --> 00:18:55
Well, there,
you're seeing the most
302
00:18:55 --> 00:18:59
-- the one absolutely essential
fact about matrix
303
00:18:59 --> 00:19:01
multiplication,
how does it work,
304
00:19:01 --> 00:19:05
when we do it a column at a
time, the very,
305
00:19:05 --> 00:19:08.69
very first day,
way back in September,
306
00:19:08.69 --> 00:19:12
we did multiplication a column
at a time.
307
00:19:12 --> 00:19:13
So what's x?
308
00:19:13 --> 00:19:14
Just tell me?
309
00:19:14 --> 00:19:15
One minus one,
one.
310
00:19:15 --> 00:19:17
Thanks.
311
00:19:17 --> 00:19:17
OK.
312
00:19:17 --> 00:19:20
Everybody's got that.
313
00:19:20 --> 00:19:20
OK?
314
00:19:20 --> 00:19:26
Then the next question is,
suppose that combination is
315
00:19:26 --> 00:19:31
zero -- oh, yes,
OK, so question (b) says --
316
00:19:31 --> 00:19:36
part (b) says,
suppose this thing is zero.
317
00:19:36 --> 00:19:39
Suppose that's zero.
318
00:19:39 --> 00:19:42
Then the solution is not
unique.
319
00:19:42 --> 00:19:49
Suppose I want true or false.
-- and a reason.
320
00:19:49 --> 00:19:52
Suppose this combination is
zero.
321
00:19:52 --> 00:19:53
v1-v2+v3.
322
00:19:53 --> 00:19:57
Show that -- what does that
tell me?
323
00:19:57 --> 00:20:02
So it's a separate question,
maybe I sort of saved time by
324
00:20:02 --> 00:20:07
writing it that way,
but it's a totally separate
325
00:20:07 --> 00:20:09
question.
326
00:20:09 --> 00:20:15
If I have a matrix,
and I know that column one
327
00:20:15 --> 00:20:21
minus column two plus column
three is zero,
328
00:20:21 --> 00:20:28
what does that tell me about
whether the solution is unique
329
00:20:28 --> 00:20:30
or not?
330
00:20:30 --> 00:20:33.94
Is there more than one
solution?
331
00:20:33.94 --> 00:20:36
What's uniqueness about?
332
00:20:36 --> 00:20:42
Uniqueness is about,
is there anything in the null
333
00:20:42 --> 00:20:43.5
space, right?
334
00:20:43.5 --> 00:20:50
The solution is unique when
there's nobody in the null space
335
00:20:50 --> 00:20:53
except the zero vector.
336
00:20:53 --> 00:21:00
And, if that's zero,
then this guy would be in the
337
00:21:00 --> 00:21:01
null space.
338
00:21:01 --> 00:21:07
So if this were zero,
then this x is in the null
339
00:21:07 --> 00:21:09
space of A.
340
00:21:09 --> 00:21:16
So solutions are never unique,
because I could always add that
341
00:21:16 --> 00:21:22
to any solution,
and Ax wouldn't change.
342
00:21:22 --> 00:21:27
So it's always that question.
343
00:21:27 --> 00:21:31
Is there somebody in the null
space?
344
00:21:31 --> 00:21:32
OK.
345
00:21:32 --> 00:21:37
Oh, now, here's a totally
different question.
346
00:21:37 --> 00:21:44
Suppose those three vectors,
v1, v2, v3, are orthonormal.
347
00:21:44 --> 00:21:52
So this isn't going to happen
for orthonormal vectors.
348
00:21:52 --> 00:21:55
OK, so part (c),
forget part (b).
349
00:21:55 --> 00:21:55
c.
350
00:21:55 --> 00:22:00.18
If v1, v2, v3,
are orthonormal -- so that I
351
00:22:00.18 --> 00:22:04
would usually have called them
q1, q2, q3.
352
00:22:04 --> 00:22:09
Now, what combination -- oh,
here's a nice question,
353
00:22:09 --> 00:22:14
if I say so myself -- what
combination of v1 and v2 is
354
00:22:14 --> 00:22:17
closest to v3?
355
00:22:17 --> 00:22:23
What point on the plane of v1
and v2 is the closest point to
356
00:22:23 --> 00:22:27
v3 if these vectors are
orthonormal?
357
00:22:27 --> 00:22:33
So let me -- I'll start the
sentence -- then the combination
358
00:22:33 --> 00:22:39
something times v1 plus
something times v2 is the
359
00:22:39 --> 00:22:42
closest combination to v3?
360
00:22:42 --> 00:22:45
And what's the answer?
361
00:22:45 --> 00:22:49.75
What's the closest vector on
that plane to v3?
362
00:22:49.75 --> 00:22:50
Zeroes.
363
00:22:50 --> 00:22:51
Right.
364
00:22:51 --> 00:22:55
We just imagine the x,
y, z axes, the v1,
365
00:22:55 --> 00:22:59.17
v2, th- v3 could be the
standard basis,
366
00:22:59.17 --> 00:23:02
the x, y, z vectors,
and, of course,
367
00:23:02 --> 00:23:08
the point on the xy plane
that's closest to v3 on the z
368
00:23:08 --> 00:23:11
axis is zero.
369
00:23:11 --> 00:23:18
So if we're orthonormal,
then the projection of v3 onto
370
00:23:18 --> 00:23:25
that plane is perpendicular,
it hits right at zero.
371
00:23:25 --> 00:23:33
OK, so that's like a quick --
you know, an easy question,
372
00:23:33 --> 00:23:37
but still brings it out.
373
00:23:37 --> 00:23:38
OK.
374
00:23:38 --> 00:23:46
Let me see what,
shall I write down a Markov
375
00:23:46 --> 00:23:54
matrix, and I'll ask you for its
eigenvalues.
376
00:23:54 --> 00:23:55
OK.
377
00:23:55 --> 00:24:08
Here's a Markov matrix -- this
-- and, tell me its eigenvalues.
378
00:24:08 --> 00:24:16
So here -- I'll call the matrix
A, and I'll call this as point
379
00:24:16 --> 00:24:21
two, point four,
point four, point four,
380
00:24:21 --> 00:24:26
point four, point two,
point four, point three,
381
00:24:26 --> 00:24:29
point three,
point four.
382
00:24:29 --> 00:24:30
OK.
383
00:24:30 --> 00:24:36.99
Let's see -- it helps out to
notice that column one plus
384
00:24:36.99 --> 00:24:41
column two
-- what's interesting about
385
00:24:41 --> 00:24:44
column one plus column two?
386
00:24:44 --> 00:24:46
It's twice as much as column
three.
387
00:24:46 --> 00:24:51
So column one plus column two
equals two times column three.
388
00:24:51 --> 00:24:55
I put that in there,
column one plus column two
389
00:24:55 --> 00:24:57
equals twice column three.
390
00:24:57 --> 00:25:00
That's observation.
391
00:25:00 --> 00:25:00
OK.
392
00:25:00 --> 00:25:04
Tell me the eigenvalues of the
matrix.
393
00:25:04 --> 0.
OK, tell me one eigenvalue?
394
0. --> 00:25:06.93
395
00:25:06.93 --> 00:25:09
Because the matrix is singular.
396
00:25:09 --> 00:25:12
Tell me another eigenvalue?
397
00:25:12 --> 00:25:18
One, because it's a Markov
matrix, the columns add to the
398
00:25:18 --> 00:25:22
all ones vector,
and that will be an eigenvector
399
00:25:22 --> 00:25:25
of A transpose.
400
00:25:25 --> 00:25:28
And tell me the third
eigenvalue?
401
00:25:28 --> 00:25:33
Let's see, to make the trace
come out right,
402
00:25:33 --> 00:25:37
which is point eight,
we need minus point two.
403
00:25:37 --> 00:25:38
OK.
404
00:25:38 --> 00:25:42
And now, suppose I start the
Markov process.
405
00:25:42 --> 00:25:49
Suppose I start with u(0) -- so
I'm going to look at the powers
406
00:25:49 --> 00:25:52
of A applied to u(0).
407
00:25:52 --> 00:25:54
This is uk.
408
00:25:54 --> 00:26:04
And there's my matrix,
and I'm going to let u(0) be --
409
00:26:04 --> 00:26:10
this is going to be zero,
ten, zero.
410
00:26:10 --> 00:26:19
And my question is,
what does that approach?
411
00:26:19 --> 00:26:24
If u(0) is equal to this --
there is u(0).
412
00:26:24 --> 00:26:27
Shall I write it in?
413
00:26:27 --> 00:26:30
Maybe I'll just write in u(0).
414
00:26:30 --> 00:26:37
A to the k, starting with ten
people in state two,
415
00:26:37 --> 00:26:43
and every step follows the
Markov rule, what does the
416
00:26:43 --> 00:26:49
solution look like after k
steps?
417
00:26:49 --> 00:26:51
Let me just ask you that.
418
00:26:51 --> 00:26:54.85
And then, what happens as k
goes to infinity?
419
00:26:54.85 --> 00:26:57
This is a steady-state
question, right?
420
00:26:57 --> 00:27:00.4
I'm looking for the steady
state.
421
00:27:00.4 --> 00:27:04
Actually, the question doesn't
ask for the k step answer,
422
00:27:04 --> 00:27:09
it just jumps right away to
infinity -- but how would I
423
00:27:09 --> 00:27:12
express the solution after k
steps?
424
00:27:12 --> 00:27:19
It would be some multiple of
the first eigenvalue to the k-th
425
00:27:19 --> 00:27:24
power -- times the first
eigenvector, plus some other
426
00:27:24 --> 00:27:28
multiple of the second
eigenvalue, times its
427
00:27:28 --> 00:27:34
eigenvector, and some multiple
of the third eigenvalue,
428
00:27:34 --> 00:27:37
times its eigenvector.
429
00:27:37 --> 00:27:38
OK.
430
00:27:38 --> 00:27:38
Good.
431
00:27:38 --> 00:27:45
And these eigenvalues are zero,
one, and minus point two.
432
00:27:45 --> 00:27:49
So what happens as k goes to
infinity?
433
00:27:49 --> 00:27:55
The only thing that survives
the steady state -- so at u
434
00:27:55 --> 00:28:00
infinity, this is gone,
this is gone,
435
00:28:00 --> 00:28:02
all that's left is c2x2.
436
00:28:02 --> 00:28:05
So I'd better find x2.
437
00:28:05 --> 00:28:11
I've got to find that
eigenvector to complete the
438
00:28:11 --> 00:28:13
answer.
439
00:28:13 --> 00:28:18
What's the eigenvector that
corresponds to lambda equal one?
440
00:28:18 --> 00:28:22
That's the key eigenvector in
any Markov process,
441
00:28:22 --> 00:28:24
is that eigenvector.
442
00:28:24 --> 00:28:28
Lambda equal one is an
eigenvalue, I need its
443
00:28:28 --> 00:28:31
eigenvector x2,
and then I need to know how
444
00:28:31 --> 00:28:36
much of it is in the starting
vector u0.
445
00:28:36 --> 00:28:36.6
OK.
446
00:28:36.6 --> 00:28:40
So, how do I find that
eigenvector?
447
00:28:40 --> 00:28:45.86
I guess I subtract one from the
diagonal, right?
448
00:28:45.86 --> 00:28:51
So I have minus point eight,
minus point eight,
449
00:28:51 --> 00:28:54
minus point six,
and the rest,
450
00:28:54 --> 00:28:59
of course, is just
-- still point four,
451
00:28:59 --> 00:29:04
point four, point four,
point four, point three,
452
00:29:04 --> 00:29:07
point three,
and hopefully,
453
00:29:07 --> 00:29:12
that's a singular matrix,
so I'm looking to solve A minus
454
00:29:12 --> 00:29:14
Ix equal zero.
455
00:29:14 --> 00:29:18
Let's see -- can anybody spot
the solution here?
456
00:29:18 --> 00:29:22
I don't know,
I didn't make it easy for
457
00:29:22 --> 00:29:24
myself.
458
00:29:24 --> 00:29:27
What do you think there?
459
00:29:27 --> 00:29:33
Maybe those first two entries
might be -- oh,
460
00:29:33 --> 00:29:36
no, what do you think?
461
00:29:36 --> 00:29:37
Anybody see it?
462
00:29:37 --> 00:29:43
We could use elimination if we
were desperate.
463
00:29:43 --> 00:29:47
Are we that desperate?
464
00:29:47 --> 00:29:53
Anybody just call out if you
see the vector that's in that
465
00:29:53 --> 00:29:54.41
null space.
466
00:29:54.41 --> 00:29:59
Eh, there better be a vector in
that null space,
467
00:29:59 --> 00:30:00
or I'm quitting.
468
00:30:00 --> 00:30:04
Uh, ha- OK, well,
I guess we could use
469
00:30:04 --> 00:30:05
elimination.
470
00:30:05 --> 00:30:12
I thought maybe somebody might
see it from further away.
471
00:30:12 --> 00:30:18
Is there a chance that these
guys are -- could it be that
472
00:30:18 --> 00:30:24
these two are equal and this is
whatever it takes,
473
00:30:24 --> 00:30:28
like, something like three,
three, two?
474
00:30:28 --> 00:30:31
Would that possibly work?
475
00:30:31 --> 00:30:36
I mean, that's great for this
-- no, it's not that great.
476
00:30:36 --> 00:30:39
Three, three,
four -- this is,
477
00:30:39 --> 00:30:42
deeper mathematics you're
watching now.
478
00:30:42 --> 00:30:46
Three, three,
four, is that -- it works!
479
00:30:46 --> 00:30:48
Don't mess with it!
480
00:30:48 --> 00:30:49
It works!
481
00:30:49 --> 00:30:50
Uh, yes.
482
00:30:50 --> 00:30:53
OK, it works,
all right.
483
00:30:53 --> 00:30:57
And, yes, OK,
and, so that's x2,
484
00:30:57 --> 00:31:01
three, three,
four, and, how much of that
485
00:31:01 --> 00:31:05
vector is in the starting
vector?
486
00:31:05 --> 00:31:10
Well, we could go through a
complicated process.
487
00:31:10 --> 00:31:16
But what's the beauty of Markov
things?
488
00:31:16 --> 00:31:21
That the total number of the
total population,
489
00:31:21 --> 00:31:24
the sum of these doesn't
change.
490
00:31:24 --> 00:31:30
That the total number of
people, they're moving around,
491
00:31:30 --> 00:31:35
but they don't get born or die
or get dead.
492
00:31:35 --> 00:31:39
So there's ten of them at the
start, so there's ten of them
493
00:31:39 --> 00:31:42.35
there, so c2 is actually one,
yes.
494
00:31:42.35 --> 00:31:45
So that would be the correct
solution.
495
00:31:45 --> 00:31:45
OK.
496
00:31:45 --> 00:31:47
That would be the u infinity.
497
00:31:47 --> 00:31:48
OK.
498
00:31:48 --> 00:31:50
So I used there,
in that process,
499
00:31:50 --> 00:31:54
sort of, the main facts about
Markov matrices to,
500
00:31:54 --> 00:31:57
to get a jump on the answer.
501
00:31:57 --> 00:31:58
OK.
let's see.
502
00:31:58 --> 00:32:01
OK, here's some,
kind of quick,
503
00:32:01 --> 00:32:02
short questions.
504
00:32:02 --> 00:32:07
Uh, maybe I'll move over to
this board, and leave that for
505
00:32:07 --> 00:32:07
the moment.
506
00:32:07 --> 00:32:10
I'm looking for two-by-two
matrices.
507
00:32:10 --> 00:32:15
And I'll read out the property
I want, and you give me an
508
00:32:15 --> 00:32:20
example, or tell me there isn't
such a matrix.
509
00:32:20 --> 00:32:21
All right.
510
00:32:21 --> 00:32:23
Here we go.
511
00:32:23 --> 00:32:27
First -- so two-by-twos.
512
00:32:27 --> 00:32:37
First, I want the projection
onto the line through A equals
513
00:32:37 --> 00:32:40
four minus three.
514
00:32:40 --> 00:32:49
So it's a one-dimensional
projection matrix I'm looking
515
00:32:49 --> 00:32:51
for.
516
00:32:51 --> 00:32:53
And what's the formula for it?
517
00:32:53 --> 00:32:57
What's the formula for the
projection matrix P onto a line
518
00:32:57 --> 00:32:57
through A.
519
00:32:57 --> 00:33:00
And then we'd just plug in this
particular A.
520
00:33:00 --> 00:33:02
Do you remember that formula?
521
00:33:02 --> 00:33:06
There's an A and an A
transpose, and normally we would
522
00:33:06 --> 00:33:10
have an A transpose A inverse in
the middle, but here we've just
523
00:33:10 --> 00:33:14
got numbers, so we just divide
by it.
524
00:33:14 --> 00:33:20.09
And then plug in A and we've
got it.
525
00:33:20.09 --> 00:33:20
OK.
526
00:33:20 --> 00:33:22
So, equals.
527
00:33:22 --> 00:33:26
You can put in the numbers.
528
00:33:26 --> 00:33:29
Trivial, right.
529
00:33:29 --> 00:33:29
OK.
530
00:33:29 --> 00:33:31
Number two.
531
00:33:31 --> 00:33:37
So this is a new problem.
532
00:33:37 --> 00:33:43
The matrix with eigenvalue zero
and three and eigenvectors --
533
00:33:43 --> 00:33:47
well, let me write these down.
eigenvalue zero,
534
00:33:47 --> 00:33:51
eigenvector one,
two, eigenvalue three,
535
00:33:51 --> 00:33:53
eigenvector two,
one.
536
00:33:53 --> 00:33:59
I'm giving you the eigenvalues
and eigenvectors instead of
537
00:33:59 --> 00:34:00
asking for them.
538
00:34:00 --> 00:34:05
Now I'm asking for the matrix.
539
00:34:05 --> 00:34:08
What's the matrix,
then?
540
00:34:08 --> 00:34:09
What's A?
541
00:34:09 --> 00:34:15
Here was a formula,
then we just put in some
542
00:34:15 --> 00:34:23
numbers, what's the formula
here, into which we'll just put
543
00:34:23 --> 00:34:25
the given numbers?
544
00:34:25 --> 00:34:31
It's the S lambda S inverse,
right?
545
00:34:31 --> 00:34:35.57
So it's S, which is this
eigenvector matrix,
546
00:34:35.57 --> 00:34:39
it's the lambda,
which is the eigenvalue matrix,
547
00:34:39 --> 00:34:43
it's the S inverse,
whatever that turns out to be,
548
00:34:43 --> 00:34:46
let me just leave it as
inverse.
549
00:34:46 --> 00:34:48
That has to be it,
right?
550
00:34:48 --> 00:34:54
Because if we went in the other
direction, that matrix S would
551
00:34:54 --> 00:34:56
diagonalize A to produce lambda.
552
00:34:56 --> 00:35:00
So it's S lambda S inverse.
553
00:35:00 --> 00:35:00
Good.
554
00:35:00 --> 00:35:03
OK, ready for number three.
555
00:35:03 --> 00:35:09
A real matrix that cannot be
factored into A -- I'm looking
556
00:35:09 --> 00:35:15
for a matrix A that never could
equal B transpose B,
557
00:35:15 --> 00:35:16.06
for any B.
558
00:35:16.06 --> 00:35:22
A two-by-two matrix that could
not be factored in the form B
559
00:35:22 --> 00:35:24
transpose B.
560
00:35:24 --> 00:35:30
So all you have to do is think,
well, what does B transpose B,
561
00:35:30 --> 00:35:34
look like, and then pick
something different.
562
00:35:34 --> 00:35:36
What do you suggest?
563
00:35:36 --> 00:35:37
Let's see.
564
00:35:37 --> 00:35:43
What shall we take for a matrix
that could not have this form,
565
00:35:43 --> 00:35:46
B transpose B.
566
00:35:46 --> 00:35:48
Well, what do we know about B
transpose B?
567
00:35:48 --> 00:35:50
It's always symmetric.
568
00:35:50 --> 00:35:52
So just give me any
non-symmetric matrix,
569
00:35:52 --> 00:35:54
it couldn't possibly have that
form.
570
00:35:54 --> 00:35:55
OK.
571
00:35:55 --> 00:35:58
And let me ask the fourth part
of this question -- a matrix
572
00:35:58 --> 00:36:00
that has orthogonal
eigenvectors,
573
00:36:00 --> 00:36:03
but it's not symmetric.
574
00:36:03 --> 00:36:10
What matrices have orthogonal
eigenvectors,
575
00:36:10 --> 00:36:15
but they're not symmetric
matrices?
576
00:36:15 --> 00:36:27
What other families of matrices
have orthogonal eigenvectors?
577
00:36:27 --> 00:36:31
We know symmetric matrices do,
but others, also.
578
00:36:31 --> 00:36:35
So I'm looking for orthogonal
eigenvectors,
579
00:36:35 --> 00:36:37
and, what do you suggest?
580
00:36:37 --> 00:36:40
The matrix could be
skew-symmetric.
581
00:36:40 --> 00:36:43
It could be an orthogonal
matrix.
582
00:36:43 --> 00:36:47.67
It could be symmetric,
but that was too easy,
583
00:36:47.67 --> 00:36:50
so I ruled that out.
584
00:36:50 --> 00:37:00
It could be skew-symmetric like
one minus one,
585
00:37:00 --> 00:37:02
like that.
586
00:37:02 --> 00:37:14
Or it could be an orthogonal
matrix like cosine sine,
587
00:37:14 --> 00:37:19
minus sine, cosine.
588
00:37:19 --> 00:37:27
All those matrices would have
complex orthogonal eigenvectors.
589
00:37:27 --> 00:37:34
But they would be orthogonal,
and so those examples are fine.
590
00:37:34 --> 00:37:34
OK.
591
00:37:34 --> 00:37:41
We can continue a little longer
if you would like to,
592
00:37:41 --> 00:37:44
with these -- from this exam.
593
00:37:44 --> 00:37:47
From these exams.
594
00:37:47 --> 00:37:49
Least squares?
595
00:37:49 --> 00:37:53
OK, here's a least squares
problem in which,
596
00:37:53 --> 00:37:58
to make life quick,
I've given the answer -- it's
597
00:37:58 --> 00:38:01
like Jeopardy!,
right?
598
00:38:01 --> 00:38:06
I just give the answer,
and you give the question.
599
00:38:06 --> 00:38:06
OK.
600
00:38:06 --> 00:38:08
Whoops, sorry.
601
00:38:08 --> 00:38:14
Let's see, can I stay over here
for the next question?
602
00:38:14 --> 00:38:17.44
OK.
least squares.
603
00:38:17.44 --> 00:38:23
So I'm giving you the problem,
one, one, one,
604
00:38:23 --> 00:38:28
zero, one, two,
c d equals three,
605
00:38:28 --> 00:38:35
four, one, and that's b,
of course, this is Ax=b.
606
00:38:35 --> 00:38:41
And the least squares solution
--
607
00:38:41 --> 00:38:45
Maybe I put c hat d hat to
emphasize it's not the true
608
00:38:45 --> 00:38:46
solution.
609
00:38:46 --> 00:38:51
So the least square solution --
the hats really go here -- is
610
00:38:51 --> 00:38:53
eleven-thirds and minus one.
611
00:38:53 --> 00:38:57
Of course, you could have
figured that out in no time.
612
00:38:57 --> 00:39:00
So this year,
I'll ask you to do it,
613
00:39:00 --> 00:39:02
probably.
614
00:39:02 --> 00:39:08
But, suppose we're given the
answer, then let's just remember
615
00:39:08 --> 00:39:10
what happened.
616
00:39:10 --> 00:39:12
OK, good question.
617
00:39:12 --> 00:39:19
What's the projection P of this
vector onto the column space of
618
00:39:19 --> 00:39:20
that matrix?
619
00:39:20 --> 00:39:26.17
So I'll write that question
down, one.
620
00:39:26.17 --> 00:39:27
What is P?
621
00:39:27 --> 00:39:28
The projection.
622
00:39:28 --> 00:39:35
The projection of b onto the
column space of A is what?
623
00:39:35 --> 00:39:41
Hopefully, that's what the
least squares problem solved.
624
00:39:41 --> 00:39:42
What is it?
625
00:39:42 --> 00:39:48
This was the best solution,
it's eleven-thirds times column
626
00:39:48 --> 00:39:55
one, plus -- or rather,
minus one times column two.
627
00:39:55 --> 00:39:56
Right?
628
00:39:56 --> 00:40:00
That's what least squares did.
629
00:40:00 --> 00:40:08
It found the combination of the
columns that was as close as
630
00:40:08 --> 00:40:10
possible to b.
631
00:40:10 --> 00:40:15
That's what least squares was
doing.
632
00:40:15 --> 00:40:18
It found the projection.
633
00:40:18 --> 00:40:20
OK?
634
00:40:20 --> 00:40:23
Secondly, draw the straight
line problem that corresponds to
635
00:40:23 --> 00:40:23
this system.
636
00:40:23 --> 00:40:26
So I guess that the straight
line fitting a straight line
637
00:40:26 --> 00:40:28
problem, we kind of recognize.
638
00:40:28 --> 00:40:30
So we recognize,
these are the heights,
639
00:40:30 --> 00:40:32
and these are the points,
and so at zero,
640
00:40:32 --> 00:40:34
one, two, the heights are
three, and at t equal to one,
641
00:40:34 --> 00:40:36
the height is four,
one, two, three,
642
00:40:36 --> 00:40:39
four, and at t equal to two,
the height is one.
643
00:40:39 --> 00:40:48
So I'm trying to fit the best
straight line through those
644
00:40:48 --> 00:40:50
points.
645
00:40:50 --> 00:40:50.72
God.
646
00:40:50.72 --> 00:40:59
I could fit a triangle very
well, but, I don't even know
647
00:40:59 --> 00:41:06
which way the best straight line
goes.
648
00:41:06 --> 00:41:14
Oh, I do know how it goes,
because there's the answer,yes.
649
00:41:14 --> 00:41:21
It has a height eleven-thirds,
and it has slope minus one,
650
00:41:21 --> 00:41:26.17
so it's something like that.
651
00:41:26.17 --> 00:41:26
OK.
652
00:41:26 --> 00:41:27.21
Great.
653
00:41:27.21 --> 00:41:33.45
Now, finally -- and this
completes the course -- find a
654
00:41:33.45 --> 00:41:37.38
different vector b,
not all zeroes,
655
00:41:37.38 --> 00:41:43
for which the least square
solution would be zero.
656
00:41:43 --> 00:41:49
So I want you to find a
different B so that the least
657
00:41:49 --> 00:41:53
square solution changes to all
zeroes.
658
00:41:53 --> 00:41:59
So tell me what I'm really
looking for here.
659
00:41:59 --> 00:42:04
I'm looking for a b where the
best combination of these two
660
00:42:04 --> 00:42:07
columns is the zero combination.
661
00:42:07 --> 00:42:10
So what kind of a vector b I
looking for?
662
00:42:10 --> 00:42:14
I'm looking for a vector b
that's orthogonal to those
663
00:42:14 --> 00:42:15
columns.
664
00:42:15 --> 00:42:19
It's orthogonal to those
columns, it's orthogonal to the
665
00:42:19 --> 00:42:24
column space,
the best possible answer is
666
00:42:24 --> 00:42:24
zero.
667
00:42:24 --> 00:42:30
So a vector b that's orthogonal
to those columns -- let's see,
668
00:42:30 --> 00:42:34
maybe one of those minus two of
those, and one of those?
669
00:42:34 --> 00:42:38
That would be orthogonal to
those columns,
670
00:42:38 --> 00:42:41
and the best vector would be
zero, zero.
671
00:42:41 --> 00:42:43
OK.
672
00:42:43 --> 00:42:47
So that's as many questions as
I can do in an hour,
673
00:42:47 --> 00:42:50
but you get three hours,
and, let me just say,
674
00:42:50 --> 00:42:54
as I've said by e-mail,
thanks very much for your
675
00:42:54 --> 00:42:58.38
patience as this series of
lectures was videotaped,
676
00:42:58.38 --> 00:43:02
and, thanks for filling out
these forms, maybe just leave
677
00:43:02 --> 00:43:06.8
them on the table up there as
you go out
678
00:43:06.8 --> 00:43:10
-- and above all,
thanks for taking the course.
679
00:43:10 --> 00:43:11
Thank you.
680
00:43:11 --> 00:43:14
Thanks.