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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

202
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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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00:12:53 --> 00:12:56
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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00:13:01 --> 00:13:04
is invertible,
we just got a true for this

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00:13:04 --> 00:13:07
one, and we got a false,
we got a z- we got a

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00:13:07 --> 00:13:10.98
non-invertible one for this one.

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00:13:10.98 --> 00:13:14
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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00:13:19 --> 00:13:22.02
I mean, why was it tricky?

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Because what is true about
determinants?

223
00:13:26 --> 00:13:31
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,

225
00:13:35 --> 00:13:37
could be A transpose,
could be somebody else's

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00:13:37 --> 00:13:38
matrix.

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00:13:38 --> 00:13:41
Then it would be true that the
determinant of B A would equal

228
00:13:41 --> 00:13:42
the determinant of A B.

229
00:13:42 --> 00:13:46
But if the matrices are not
square and it would actually be

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00:13:46 --> 00:13:49
true that it would be equal --
that this would equal the

231
00:13:49 --> 00:13:53
determinant of A times the
determinant of A transpose.

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00:13:53 --> 00:13:57
We could even split up those
two separate determinants.

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00:13:57 --> 00:14:00.77
And, of course,
those would be equal.

234
00:14:00.77 --> 00:14:02.92
But only when A is square.

235
00:14:02.92 --> 00:14:06
So that's just,
that's a question that rests on

236
00:14:06 --> 00:14:11
the, the falseness rests on the
fact that the matrix isn't

237
00:14:11 --> 00:14:14
square in the first place.

238
00:14:14 --> 00:14:15
OK, good.

239
00:14:15 --> 00:14:16
Let's see.

240
00:14:16 --> 00:14:20
Oh, now, even asks more.

241
00:14:20 --> 00:14:26
Prove that A transpose y equals
c -- hah-God,

242
00:14:26 --> 00:14:30
it's -- this question goes on
and on.

243
00:14:30 --> 00:14:35
now I ask you about A transpose
y=c.

244
00:14:35 --> 00:14:43
So I'm asking you about the
equation -- about the matrix A

245
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

247
00:14:54 --> 00:15:00
solution for every c,
every right-hand side c,

248
00:15:00 --> 00:15:07
and, in fact -- in fact,
infinitely many solutions for

249
00:15:07 --> 00:15:09
every c.

250
00:15:09 --> 00:15:09.46
OK.

251
00:15:09.46 --> 00:15:15
Well, none -- none of this is
difficult, but,

252
00:15:15 --> 00:15:20
it's been a little while.

253
00:15:20 --> 00:15:23
So we just have to think again.

254
00:15:23 --> 00:15:28
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
00:15:35 --> 00:15:36
Of course.

258
00:15:36 --> 00:15:41
To show that a system has at
least one solution,

259
00:15:41 --> 00:15:45
when does this,
when does this system -- when

260
00:15:45 --> 00:15:49
is the system always solvable?

261
00:15:49 --> 00:15:58
When it has full row rank,
when the rows are independent.

262
00:15:58 --> 00:16:04
Here, we have n rows,
and that's the rank.

263
00:16:04 --> 00:16:12
So at least one solution,
because the number of rows,

264
00:16:12 --> 00:16:19
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,

269
00:16:34 --> 00:16:38
it has independent rows,
so there's at least one

270
00:16:38 --> 00:16:39
solution.

271
00:16:39 --> 00:16:44
But now, how do I even know
that there are infinitely many

272
00:16:44 --> 00:16:46
solutions?

273
00:16:46 --> 00:16:52
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

276
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
00:17:14 --> 00:17:17
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.