WEBVTT
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OK, here is lecture
ten in linear algebra.
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Two important things
to do in this lecture.
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One is to correct an
error from lecture nine.
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So the blackboard with that
awful error is still with us.
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And the second,
the big thing to do
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is to tell you about
the four subspaces that
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come with a matrix.
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We've seen two subspaces,
the column space
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and the null space.
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There's two to go.
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First of all, and
this is a great way to
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OK. recap and correct
the previous lecture --
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so you remember I
was just doing R^3.
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I couldn't have taken a
simpler example than R^3.
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And I wrote down
the standard basis.
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That's the standard basis.
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The basis -- the obvious basis
for the whole three dimensional
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space.
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And then I wanted
to make the point
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that there was nothing special,
nothing about that basis
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that another basis
couldn't have.
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It could have
linear independence,
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it could span a space.
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There's lots of other bases.
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So I started with these vectors,
one one two and two two five,
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and those were independent.
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And then I said
three three seven
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wouldn't do, because three
three seven is the sum of those.
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So in my innocence, I
put in three three eight.
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I figured probably if three
three seven is on the plane,
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is -- which I know, it's in
the plane with these two,
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then probably three three
eight sticks a little bit out
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of the plane and it's
independent and it gives
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a basis.
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But after class, to my
sorrow, a student tells me,
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"Wait a minute, that ba- that
third vector, three three
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eight, is not independent."
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And why did she say that?
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She didn't actually take
the time, didn't have to,
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to find w- w- what combination
of this one and this one
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gives three three eight.
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She did something else.
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In other words,
she looked ahead,
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because she said, wait a minute,
if I look at that matrix,
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it's not invertible.
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That third column can't be
independent of the first two,
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because when I look
at that matrix,
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it's got two identical rows.
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I have a square matrix.
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Its rows are
obviously dependent.
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And that makes the
columns dependent.
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So there's my error.
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When I look at the matrix A
that has those three columns,
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those three columns
can't be independent
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because that matrix
is not invertible
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because it's got two equal rows.
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And today's lecture will
reach the conclusion,
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the great conclusion,
that connects the column
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space with the row space.
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So those are -- the row space
is now going to be another one
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of my fundamental subspaces.
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The row space of this matrix,
or of this one -- well,
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the row space of this one is OK,
but the row space of this one,
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I'm looking at the rows of
the matrix -- oh, anyway,
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I'll have two equal rows and
the row space will be only two
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dimensional.
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The rank of the matrix with
these columns will only be two.
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So only two of those columns,
columns can be independent too.
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The rows tell me something about
the columns, in other words,
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something that I should
have noticed and I didn't.
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OK.
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So now let me pin down these
four fundamental subspaces.
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So here are the four
fundamental subspaces.
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This is really the heart of
this approach to linear algebra,
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to see these four subspaces,
how they're related.
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So what are they?
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The column space, C of A.
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The null space, N of A.
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And now comes the row
space, something new.
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The row space, what's in that?
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It's all combinations
of the rows.
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That's natural.
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We want a space, so we have
to take all combinations,
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and we start with the rows.
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So the rows span the row space.
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Are the rows a basis
for the row space?
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Maybe so, maybe no.
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The rows are a basis for the row
space when they're independent,
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but if they're dependent,
as in this example,
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my error from last
time, they're not --
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those three rows
are not a basis.
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The row space wouldn't --
would only be two dimensional.
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I only need two
rows for a basis.
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So the row space,
now what's in it?
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It's all combinations
of the rows of A.
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All combinations
of the rows of A.
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But I don't like working
with row vectors.
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All my vectors have
been column vectors.
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I'd like to stay
with column vectors.
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How can I get to column
vectors out of these rows?
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I transpose the matrix.
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So if that's OK with you,
I'm going to transpose the
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matrix.
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I'm, I'm going to
say all combinations
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of the columns of A transpose.
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And that allows me to use the
convenient notation, the column
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space of A transpose.
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Nothing, no mathematics
went on there.
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We just got some vectors that
were lying down to stand up.
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But it means that we
can use this column
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space of A transpose, that's
telling me in a nice matrix
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notation what the row space is.
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OK.
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And finally is
another null space.
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The fourth fundamental
space will be
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the null space of A transpose.
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The fourth guy is the
null space of A transpose.
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And of course my notation
is N of A transpose.
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That's the null
space of A transpose.
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Eh, we don't have a perfect
name for this space as a --
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connecting with A, but our usual
name is the left null space,
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and I'll show you
why in a moment.
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So often I call this the --
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just to write that word --
the left null space of A.
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So just the way we
have the row space of A
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and we switch it to the
column space of A transpose,
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so we have this space
of guys l- that I
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call the left null space
of A, but the good notation
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is it's the null
space of A transpose.
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OK.
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Those are four spaces.
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Where are those spaces?
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What, what big space are they
in for -- when A is m by n?
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In that case, the
null space of A,
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what's in the null space of A?
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Vectors with n components,
solutions to A x equals zero.
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So the null space
of A is in R^n.
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What's in the column space of A?
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Well, columns.
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How many components
dothose columns have?
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m.
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So this column space is in R^m.
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What about the column
space of A transpose,
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which are just a disguised
way of saying the rows of A?
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The rows of A, in this
three by six matrix,
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have six components,
n components.
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The column space is in R^n.
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And the null space
of A transpose,
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I see that this fourth space
is already getting second,
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you know, second class
citizen treatment
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and it doesn't deserve it.
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It's, it should be
there, it is there,
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and shouldn't be squeezed.
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The null space of A transpose --
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well, if the null space of A
had vectors with n components,
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the null space of A
transpose will be in R^m.
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I want to draw a picture
of the four spaces.
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OK.
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OK.
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Here are the four spaces.
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OK, Let me put n dimensional
space over on this side.
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Then which were the
subspaces in R^n?
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The null space was
and the row space was.
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So here we have the -- can I
make that picture of the row
00:10:53.471 --> 00:10:53.970
space?
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And can I make this kind of
picture of the null space?
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That's just meant
to be a sketch,
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to remind you that they're in
this -- which you know, how --
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what type of vectors are in it?
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Vectors with n components.
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Over here, inside, consisting
of vectors with m components,
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is the column space
and what I'm calling
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the null space of A transpose.
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Those are the ones
with m components.
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OK.
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To understand these spaces
is our, is our job now.
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Because by understanding
those spaces,
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we know everything about
this half of linear algebra.
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What do I mean by
understanding those spaces?
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I would like to know a
basis for those spaces.
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For each one of those
spaces, how would I create --
00:12:10.090 --> 00:12:11.440
construct a basis?
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What systematic way
would produce a basis?
00:12:15.260 --> 00:12:17.180
And what's their dimension?
00:12:24.611 --> 00:12:25.110
OK.
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So for each of
the four spaces, I
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have to answer those questions.
00:12:32.500 --> 00:12:34.780
How do I produce a basis?
00:12:34.780 --> 00:12:38.620
And then -- which has
a somewhat long answer.
00:12:38.620 --> 00:12:42.210
And what's the dimension,
which is just a number,
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so it has a real short answer.
00:12:44.030 --> 00:12:46.250
Can I give you the
short answer first?
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I shouldn't do it,
but here it is.
00:12:50.930 --> 00:12:56.206
I can tell you the dimension
of the column space.
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Let me start with this guy.
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What's its dimension?
00:12:58.590 --> 00:13:01.360
I have an m by n matrix.
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The dimension of the
column space is the rank,
00:13:09.041 --> 00:13:09.540
r.
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We actually got to that at
the end of the last lecture,
00:13:17.210 --> 00:13:19.570
but only for an example.
00:13:19.570 --> 00:13:24.520
So I really have to say,
OK, what's going on there.
00:13:24.520 --> 00:13:28.340
I should produce
a basis and then I
00:13:28.340 --> 00:13:31.260
just look to see how many
vectors I needed in that basis,
00:13:31.260 --> 00:13:34.360
and the answer will be r.
00:13:34.360 --> 00:13:38.980
Actually, I'll do that,
before I get on to the others.
00:13:38.980 --> 00:13:41.050
What's a basis for
the columns space?
00:13:43.770 --> 00:13:46.980
We've done all the
work of row reduction,
00:13:46.980 --> 00:13:50.810
identifying the pivot
columns, the ones that
00:13:50.810 --> 00:13:53.940
have pivots, the ones
that end up with pivots.
00:13:53.940 --> 00:13:57.500
But now I -- the pivot columns
I'm interested in are columns
00:13:57.500 --> 00:14:00.530
of A, the original A.
00:14:00.530 --> 00:14:03.610
And those pivot columns,
there are r of them.
00:14:03.610 --> 00:14:05.840
The rank r counts those.
00:14:05.840 --> 00:14:07.550
Those are a basis.
00:14:07.550 --> 00:14:12.520
So if I answer this question
for the column space,
00:14:12.520 --> 00:14:18.100
the answer will be a
basis is the pivot columns
00:14:18.100 --> 00:14:23.790
and the dimension is the rank
r, and there are r pivot columns
00:14:23.790 --> 00:14:26.360
and everything great.
00:14:26.360 --> 00:14:26.950
OK.
00:14:26.950 --> 00:14:29.920
So that space we
pretty well understand.
00:14:32.560 --> 00:14:36.090
I probably have a little
going back to see that --
00:14:36.090 --> 00:14:39.750
to prove that this
is a right answer,
00:14:39.750 --> 00:14:42.460
but you know it's
the right answer.
00:14:42.460 --> 00:14:45.255
Now let me look
at the row space.
00:14:49.340 --> 00:14:51.830
OK.
00:14:51.830 --> 00:14:55.190
Shall I tell you the
dimension of the row space?
00:14:55.190 --> 00:14:55.830
Yes.
00:14:55.830 --> 00:14:58.180
Before we do even an
example, let me tell you
00:14:58.180 --> 00:15:00.220
the dimension of the row space.
00:15:00.220 --> 00:15:03.650
Its dimension is also r.
00:15:03.650 --> 00:15:06.830
The row space and the column
space have the same dimension.
00:15:06.830 --> 00:15:08.440
That's a wonderful fact.
00:15:08.440 --> 00:15:12.950
The dimension of the column
space of A transpose --
00:15:12.950 --> 00:15:17.170
that's the row space -- is r.
00:15:17.170 --> 00:15:19.430
That, that space
is r dimensional.
00:15:22.100 --> 00:15:24.790
Snd so is this one.
00:15:24.790 --> 00:15:27.570
OK.
00:15:27.570 --> 00:15:35.430
That's the sort of insight
that got used in this example.
00:15:35.430 --> 00:15:40.990
If those -- are the three
columns of a matrix --
00:15:40.990 --> 00:15:44.720
let me make them the three
columns of a matrix by just
00:15:44.720 --> 00:15:47.000
erasing some brackets.
00:15:47.000 --> 00:15:51.320
OK, those are the three
columns of a matrix.
00:15:51.320 --> 00:15:54.130
The rank of that matrix,
if I look at the columns,
00:15:54.130 --> 00:15:57.560
it wasn't obvious to me anyway.
00:15:57.560 --> 00:16:01.250
But if I look at the
rows, now it's obvious.
00:16:01.250 --> 00:16:03.860
The row space of
that matrix obviously
00:16:03.860 --> 00:16:07.380
is two dimensional, because
I see a basis for the row
00:16:07.380 --> 00:16:10.650
space, this row and that row.
00:16:10.650 --> 00:16:12.370
And of course,
strictly speaking,
00:16:12.370 --> 00:16:16.290
I'm supposed to transpose
those guys, make them stand up.
00:16:16.290 --> 00:16:19.270
But the rank is two, and
therefore the column space
00:16:19.270 --> 00:16:21.940
is two dimensional by
this wonderful fact
00:16:21.940 --> 00:16:25.100
that the row space and column
space have the same dimension.
00:16:25.100 --> 00:16:29.130
And therefore there are only
two pivot columns, not three,
00:16:29.130 --> 00:16:34.310
and, those, the three
columns are dependent.
00:16:34.310 --> 00:16:35.430
OK.
00:16:35.430 --> 00:16:45.470
Now let me bury that error
and talk about the row space.
00:16:45.470 --> 00:16:48.690
Well, I'm going to give you the
dimensions of all the spaces.
00:16:48.690 --> 00:16:52.480
Because that's
such a nice answer.
00:16:52.480 --> 00:16:53.100
OK.
00:16:53.100 --> 00:16:56.460
So let me come back here.
00:16:56.460 --> 00:16:59.640
So we have this great
fact to establish,
00:16:59.640 --> 00:17:06.790
that the row space, its
dimension is also the rank.
00:17:06.790 --> 00:17:07.910
What about the null space?
00:17:07.910 --> 00:17:10.020
OK.
00:17:10.020 --> 00:17:13.630
What's a basis for
the null space?
00:17:13.630 --> 00:17:15.970
What's the dimension
of the null space?
00:17:15.970 --> 00:17:20.274
Let me, I'll put that answer
up here for the null space.
00:17:24.200 --> 00:17:27.849
Well, how have we
constructed the null space?
00:17:27.849 --> 00:17:32.260
We took the matrix A, we
did those row operations
00:17:32.260 --> 00:17:36.170
to get it into a form
U or, or even further.
00:17:36.170 --> 00:17:39.730
We got it into the
reduced form R.
00:17:39.730 --> 00:17:43.490
And then we read off
special solutions.
00:17:43.490 --> 00:17:44.580
Special solutions.
00:17:44.580 --> 00:17:48.110
And every special solution
came from a free variable.
00:17:48.110 --> 00:17:50.720
And those special solutions
are in the null space,
00:17:50.720 --> 00:17:54.360
and the great thing is
they're a basis for it.
00:17:54.360 --> 00:17:59.445
So for the null space, a basis
will be the special solutions.
00:18:03.280 --> 00:18:07.250
And there's one for every
free variable, right?
00:18:07.250 --> 00:18:11.190
For each free variable, we give
that variable the value one,
00:18:11.190 --> 00:18:13.350
the other free variables zero.
00:18:13.350 --> 00:18:17.840
We get the pivot variables,
we get a vector in the --
00:18:17.840 --> 00:18:20.680
we get a special solution.
00:18:20.680 --> 00:18:24.140
So we get altogether
n-r of them,
00:18:24.140 --> 00:18:30.340
because that's the
number of free variables.
00:18:30.340 --> 00:18:32.890
If we have r --
00:18:32.890 --> 00:18:38.490
this is the dimension is r, is
the number of pivot variables.
00:18:38.490 --> 00:18:40.810
This is the number
of free variables.
00:18:40.810 --> 00:18:44.180
So the beauty is that those
special solutions do form
00:18:44.180 --> 00:18:51.380
a basis and tell us immediately
that the dimension of the null
00:18:51.380 --> 00:18:55.570
space is n --
00:18:55.570 --> 00:19:00.590
I better write this well,
because it's so nice -- n-r.
00:19:00.590 --> 00:19:04.910
And do you see the nice thing?
00:19:04.910 --> 00:19:08.320
That the two dimensions in
this n dimensional space,
00:19:08.320 --> 00:19:12.300
one subspace is r dimensional --
00:19:12.300 --> 00:19:15.400
to be proved, that's
the row space.
00:19:15.400 --> 00:19:18.680
The other subspace
is n-r dimensional,
00:19:18.680 --> 00:19:21.150
that's the null space.
00:19:21.150 --> 00:19:25.550
And the two dimensions
like together give n.
00:19:25.550 --> 00:19:28.870
The sum of r and n-R is n.
00:19:28.870 --> 00:19:31.270
And that's just great.
00:19:31.270 --> 00:19:35.720
It's really copying the fact
that we have n variables,
00:19:35.720 --> 00:19:39.750
r of them are pivot variables
and n-r are free variables,
00:19:39.750 --> 00:19:41.380
and n altogether.
00:19:41.380 --> 00:19:41.880
OK.
00:19:41.880 --> 00:19:47.090
And now what's the dimension
of this poor misbegotten fourth
00:19:47.090 --> 00:19:48.500
subspace?
00:19:48.500 --> 00:19:53.700
It's got to be m-r.
00:19:53.700 --> 00:20:00.110
The dimension of this left null
space, left out practically,
00:20:00.110 --> 00:20:01.540
is m-r.
00:20:04.120 --> 00:20:08.690
Well, that's really just
saying that this -- again,
00:20:08.690 --> 00:20:15.140
the sum of that plus that
is m, and m is correct,
00:20:15.140 --> 00:20:21.180
it's the number of
columns in A transpose.
00:20:21.180 --> 00:20:25.810
A transpose is just
as good a matrix as A.
00:20:25.810 --> 00:20:29.590
It just happens to be n by m.
00:20:29.590 --> 00:20:38.360
It happens to have m columns,
so it will have m variables
00:20:38.360 --> 00:20:41.990
when I go to A x
equals 0 and m of them,
00:20:41.990 --> 00:20:46.950
and r of them will be pivot
variables and m-r will
00:20:46.950 --> 00:20:49.920
be free variables.
00:20:49.920 --> 00:20:52.650
A transpose is as
good a matrix as A.
00:20:52.650 --> 00:20:57.410
It follows the same rule that
the this plus the dimension --
00:20:57.410 --> 00:21:00.830
this dimension plus this
dimension adds up to the number
00:21:00.830 --> 00:21:03.130
of columns.
00:21:03.130 --> 00:21:06.770
And over here, A
transpose has m columns.
00:21:06.770 --> 00:21:09.290
OK.
00:21:09.290 --> 00:21:09.790
OK.
00:21:09.790 --> 00:21:13.580
So I gave you the easy
answer, the dimensions.
00:21:13.580 --> 00:21:21.160
Now can I go back
to check on a basis?
00:21:21.160 --> 00:21:25.540
We would like to think
that -- say the row space,
00:21:25.540 --> 00:21:29.280
because we've got a basis
for the column space.
00:21:29.280 --> 00:21:33.570
The pivot columns give a
basis for the column space.
00:21:33.570 --> 00:21:36.840
Now I'm asking you to
look at the row space.
00:21:36.840 --> 00:21:40.790
And I -- you could say, OK, I
can produce a basis for the row
00:21:40.790 --> 00:21:45.540
space by transposing my
matrix, making those columns,
00:21:45.540 --> 00:21:48.670
then doing elimination,
row reduction,
00:21:48.670 --> 00:21:54.690
and checking out the pivot
columns in this transposed
00:21:54.690 --> 00:21:55.350
matrix.
00:21:55.350 --> 00:21:57.760
But that means you
had to do all that row
00:21:57.760 --> 00:22:00.750
reduction on A transpose.
00:22:00.750 --> 00:22:05.590
It ought to be possible,
if we take a matrix A --
00:22:05.590 --> 00:22:08.620
let me take the matrix -- maybe
we had this matrix in the last
00:22:08.620 --> 00:22:09.120
lecture.
00:22:09.120 --> 00:22:15.530
1 1 1, 2 1 2, 3 2 3, 1 1 1.
00:22:21.950 --> 00:22:22.670
OK.
00:22:22.670 --> 00:22:24.770
That, that matrix was so easy.
00:22:24.770 --> 00:22:29.100
We spotted its pivot columns,
one and two, without actually
00:22:29.100 --> 00:22:30.930
doing row reduction.
00:22:30.930 --> 00:22:35.070
But now let's do
the job properly.
00:22:35.070 --> 00:22:38.980
So I subtract this away
from this to produce a zero.
00:22:38.980 --> 00:22:42.810
So one 2 3 1 is fine.
00:22:42.810 --> 00:22:47.419
Subtracting that away leaves
me minus 1 -1 0, right?
00:22:47.419 --> 00:22:49.960
And subtracting that from the
last row, oh, well that's easy.
00:22:53.140 --> 00:22:53.690
OK?
00:22:53.690 --> 00:22:56.680
I'm doing row reduction.
00:22:56.680 --> 00:23:00.700
Now I've -- the first
column is all set.
00:23:00.700 --> 00:23:04.230
The second column I
now see the pivot.
00:23:04.230 --> 00:23:07.800
And I can clean up, if I --
00:23:07.800 --> 00:23:08.480
actually,
00:23:08.480 --> 00:23:09.860
OK.
00:23:09.860 --> 00:23:13.070
Why don't I make
the pivot into a 1.
00:23:13.070 --> 00:23:18.805
I'll multiply that row through
by by -1, and then I have 1 1.
00:23:22.200 --> 00:23:24.370
That was an elementary
operation I'm allowed,
00:23:24.370 --> 00:23:26.790
multiply a row by a number.
00:23:26.790 --> 00:23:28.150
And now I'll do elimination.
00:23:28.150 --> 00:23:31.300
Two of those away from that
will knock this guy out
00:23:31.300 --> 00:23:33.120
and make this into a 1.
00:23:33.120 --> 00:23:36.560
So that's now a 0 and that's a
00:23:36.560 --> 00:23:37.620
OK.
00:23:37.620 --> 00:23:39.230
Done.
00:23:39.230 --> 00:23:42.470
That's R.
00:23:42.470 --> 00:23:45.930
I'm seeing the
identity matrix here.
00:23:45.930 --> 00:23:48.530
I'm seeing zeros below.
00:23:48.530 --> 00:23:49.990
And I'm seeing F there.
00:23:53.151 --> 00:23:53.650
OK.
00:23:56.340 --> 00:24:00.110
What about its row space?
00:24:00.110 --> 00:24:02.650
What happened to its row space
-- well, what happened --
00:24:02.650 --> 00:24:04.610
let me first ask, just
because this is, is --
00:24:04.610 --> 00:24:06.780
sometimes something does happen.
00:24:06.780 --> 00:24:09.010
Its column space changed.
00:24:09.010 --> 00:24:18.820
The column space of R is not
the column space of A, right?
00:24:18.820 --> 00:24:22.000
Because 1 1 1 is certainly
in the column space of A
00:24:22.000 --> 00:24:26.460
and certainly not in
the column space of R.
00:24:26.460 --> 00:24:29.770
I did row operations.
00:24:29.770 --> 00:24:33.750
Those row operations
preserve the row space.
00:24:33.750 --> 00:24:36.640
So the row, so the column
spaces are different.
00:24:36.640 --> 00:24:39.550
Different column spaces,
different column spaces.
00:24:45.950 --> 00:24:50.480
But I believe that they
have the same row space.
00:24:55.080 --> 00:24:55.880
Same row space.
00:25:00.030 --> 00:25:04.040
I believe that the row space of
that matrix and the row space
00:25:04.040 --> 00:25:06.210
of this matrix are identical.
00:25:06.210 --> 00:25:09.640
They have exactly the
same vectors in them.
00:25:09.640 --> 00:25:14.400
Those vectors are vectors
with four components, right?
00:25:14.400 --> 00:25:17.940
They're all combinations
of those rows.
00:25:17.940 --> 00:25:19.800
Or I believe you
get the same thing
00:25:19.800 --> 00:25:22.030
by taking all combinations
of these rows.
00:25:24.690 --> 00:25:29.960
And if true, what's a basis?
00:25:29.960 --> 00:25:32.430
What's a basis for
the row space of R,
00:25:32.430 --> 00:25:42.450
and it'll be a basis for the
row space of the original A,
00:25:42.450 --> 00:25:45.120
but it's obviously a basis
for the row space of R.
00:25:45.120 --> 00:25:47.840
What's a basis for the
row space of that matrix?
00:25:47.840 --> 00:25:48.820
The first two rows.
00:25:51.930 --> 00:25:57.550
So a basis for the
row -- so a basis is,
00:25:57.550 --> 00:26:15.500
for the row space of A or of R,
is, is the first R rows of R.
00:26:15.500 --> 00:26:18.050
Not of A.
00:26:18.050 --> 00:26:21.960
Sometimes it's true for
A, but not necessarily.
00:26:21.960 --> 00:26:29.280
But R, we definitely have a
matrix here whose row space we
00:26:29.280 --> 00:26:32.060
can, we can identify.
00:26:32.060 --> 00:26:36.790
The row space is spanned
by the three rows,
00:26:36.790 --> 00:26:40.580
but if we want a basis
we want independence.
00:26:40.580 --> 00:26:43.380
So out goes row three.
00:26:43.380 --> 00:26:46.820
The row space is also spanned
by the first two rows.
00:26:46.820 --> 00:26:48.920
This guy didn't
contribute anything.
00:26:48.920 --> 00:26:52.706
And of course over here
this 1 2 3 1 in the bottom
00:26:52.706 --> 00:26:53.830
didn't contribute anything.
00:26:53.830 --> 00:26:56.510
We had it already.
00:26:56.510 --> 00:26:58.260
So this, here is a basis.
00:26:58.260 --> 00:27:01.260
1 0 1 1 and 0 1 1 0.
00:27:04.710 --> 00:27:06.860
I believe those are
in the row space.
00:27:06.860 --> 00:27:08.090
I know they're independent.
00:27:08.090 --> 00:27:10.680
Why are they in the row space?
00:27:10.680 --> 00:27:13.720
Why are those two
vectors in the row space?
00:27:13.720 --> 00:27:17.440
Because all those
operations we did,
00:27:17.440 --> 00:27:22.680
which started with these rows
and took combinations of them
00:27:22.680 --> 00:27:23.900
--
00:27:23.900 --> 00:27:28.910
I took this row minus this
row, that gave me something
00:27:28.910 --> 00:27:30.780
that's still in the row space.
00:27:30.780 --> 00:27:32.400
That's the point.
00:27:32.400 --> 00:27:36.760
When I took a row minus a
multiple of another row,
00:27:36.760 --> 00:27:38.380
I'm staying in the row space.
00:27:38.380 --> 00:27:41.330
The row space is not changing.
00:27:41.330 --> 00:27:43.240
My little basis
for it is changing,
00:27:43.240 --> 00:27:46.680
and I've ended up with,
sort of the best basis.
00:27:49.380 --> 00:27:53.240
If the columns of the identity
matrix are the best basis
00:27:53.240 --> 00:28:00.490
for R^3 or R^n, the rows of
this matrix are the best basis
00:28:00.490 --> 00:28:02.510
for the row space.
00:28:02.510 --> 00:28:06.250
Best in the sense of being
as clean as I can make it.
00:28:06.250 --> 00:28:09.310
Starting off with the
identity and then finishing up
00:28:09.310 --> 00:28:11.420
with whatever has
to be in there.
00:28:11.420 --> 00:28:12.730
OK.
00:28:12.730 --> 00:28:16.720
Do you see then that
the dimension is r?
00:28:16.720 --> 00:28:23.330
For sure, because we've got
r pivots, r non-zero rows.
00:28:23.330 --> 00:28:26.420
We've got the right
number of vectors, r.
00:28:26.420 --> 00:28:30.080
They're in the row space,
they're independent.
00:28:30.080 --> 00:28:31.600
That's it.
00:28:31.600 --> 00:28:34.160
They are a basis
for the row space.
00:28:34.160 --> 00:28:36.220
And we can even pin
that down further.
00:28:36.220 --> 00:28:41.340
How do I know that every
row of A is a combination?
00:28:41.340 --> 00:28:45.110
How do I know they
span the row space?
00:28:45.110 --> 00:28:48.110
Well, somebody says, I've
got the right number of them,
00:28:48.110 --> 00:28:48.740
so they must.
00:28:48.740 --> 00:28:49.870
But -- and that's true.
00:28:49.870 --> 00:28:54.400
But let me just say, how
do I know that this row is
00:28:54.400 --> 00:28:57.300
a combination of these?
00:28:57.300 --> 00:29:01.870
By just reversing the
steps of row reduction.
00:29:01.870 --> 00:29:07.760
If I just reverse the steps and
go from A -- from R back to A,
00:29:07.760 --> 00:29:10.010
then what do I, what I doing?
00:29:10.010 --> 00:29:12.030
I'm starting with
these rows, I'm
00:29:12.030 --> 00:29:15.730
taking combinations of them.
00:29:15.730 --> 00:29:19.670
After a couple of steps,
undoing the subtractions
00:29:19.670 --> 00:29:22.710
that I did before, I'm
back to these rows.
00:29:22.710 --> 00:29:25.500
So these rows are
combinations of those rows.
00:29:25.500 --> 00:29:28.020
Those rows are
combinations of those rows.
00:29:28.020 --> 00:29:31.740
The two row spaces are the same.
00:29:31.740 --> 00:29:34.830
The bases are the same.
00:29:34.830 --> 00:29:38.540
And the natural
basis is this guy.
00:29:38.540 --> 00:29:41.530
Is that all right
for the row space?
00:29:41.530 --> 00:29:45.030
The row space is
sitting there in R
00:29:45.030 --> 00:29:47.750
in its cleanest possible form.
00:29:47.750 --> 00:29:48.800
OK.
00:29:48.800 --> 00:29:56.875
Now what about the fourth guy,
the null space of A transpose?
00:29:59.630 --> 00:30:03.440
First of all, why do I call
that the left null space?
00:30:03.440 --> 00:30:11.610
So let me save that
and bring that down.
00:30:11.610 --> 00:30:14.050
OK.
00:30:14.050 --> 00:30:20.940
So the fourth space is the
null space of A transpose.
00:30:23.700 --> 00:30:27.200
So it has in it vectors,
let me call them y,
00:30:27.200 --> 00:30:30.470
so that A transpose y equals 0.
00:30:30.470 --> 00:30:35.360
If A transpose y
equals 0, then y
00:30:35.360 --> 00:30:39.890
is in the null space of
A transpose, of course.
00:30:39.890 --> 00:30:47.770
So this is a matrix times
a column equaling zero.
00:30:50.480 --> 00:30:56.550
And now, because I want
y to sit on the left
00:30:56.550 --> 00:31:00.200
and I want A instead
of A transpose,
00:31:00.200 --> 00:31:03.380
I'll just transpose
that equation.
00:31:03.380 --> 00:31:06.180
Can I just transpose that?
00:31:06.180 --> 00:31:10.705
On the right, it makes
the zero vector lie down.
00:31:14.590 --> 00:31:21.510
And on the left, it's a
product, A, A transpose times y.
00:31:21.510 --> 00:31:24.510
If I take the transpose, then
they come in opposite order,
00:31:24.510 --> 00:31:25.540
right?
00:31:25.540 --> 00:31:30.156
So that's y transpose times
A transpose transpose.
00:31:33.210 --> 00:31:35.540
But nobody's going to
leave it like that.
00:31:35.540 --> 00:31:39.620
That's -- A transpose
transpose is just A, of course.
00:31:39.620 --> 00:31:43.360
When I transposed A
transpose I got back to A.
00:31:43.360 --> 00:31:45.870
Now do you see what I have now?
00:31:45.870 --> 00:31:51.820
I have a row
vector, y transpose,
00:31:51.820 --> 00:31:58.540
multiplying A, and
multiplying from the left.
00:31:58.540 --> 00:32:02.180
That's why I call it
the left null space.
00:32:02.180 --> 00:32:05.590
But by making it --
putting it on the left,
00:32:05.590 --> 00:32:09.720
I had to make it into a row
instead of a column vector,
00:32:09.720 --> 00:32:15.250
and so my convention is
I usually don't do that.
00:32:15.250 --> 00:32:18.960
I usually stay with A
transpose y equals 0.
00:32:18.960 --> 00:32:20.290
OK.
00:32:20.290 --> 00:32:27.990
And you might ask, how do we
get a basis -- or I might ask,
00:32:27.990 --> 00:32:31.470
how do we get a basis
for this fourth space,
00:32:31.470 --> 00:32:32.610
this left null space?
00:32:36.150 --> 00:32:36.720
OK.
00:32:36.720 --> 00:32:40.050
I'll do it in the example.
00:32:40.050 --> 00:32:43.350
As always -- not that one.
00:32:49.880 --> 00:32:53.310
The left null space is not
jumping out at me here.
00:32:57.060 --> 00:33:00.220
I know which are the
free variables --
00:33:00.220 --> 00:33:03.890
the special solutions, but those
are special solutions to A x
00:33:03.890 --> 00:33:06.230
equals zero, and now I'm
looking at A transpose,
00:33:06.230 --> 00:33:08.420
and I'm not seeing it here.
00:33:08.420 --> 00:33:12.760
So -- but somehow you feel that
the work that you did which
00:33:12.760 --> 00:33:19.310
simplified A to R should have
revealed the left null space
00:33:19.310 --> 00:33:20.760
too.
00:33:20.760 --> 00:33:25.970
And it's slightly less
immediate, but it's there.
00:33:25.970 --> 00:33:31.010
So from A to R, I
took some steps,
00:33:31.010 --> 00:33:34.880
and I guess I'm interested
in what were those steps,
00:33:34.880 --> 00:33:36.600
or what were all
of them together.
00:33:36.600 --> 00:33:43.160
I don't -- I'm not interested in
what particular ones they were.
00:33:43.160 --> 00:33:45.910
I'm interested in what
was the whole matrix that
00:33:45.910 --> 00:33:51.620
took me from A to R.
00:33:51.620 --> 00:33:52.880
How would you find that?
00:33:55.640 --> 00:33:58.970
Do you remember
Gauss-Jordan, where you
00:33:58.970 --> 00:34:02.260
tack on the identity matrix?
00:34:02.260 --> 00:34:03.810
Let's do that again.
00:34:03.810 --> 00:34:06.640
So I, I'll do it above, here.
00:34:06.640 --> 00:34:13.620
So this is now, this
is now the idea of --
00:34:13.620 --> 00:34:17.570
I take the matrix
A, which is m by n.
00:34:20.389 --> 00:34:22.639
In Gauss-Jordan, when
we saw him before --
00:34:22.639 --> 00:34:25.150
A was a square
invertible matrix and we
00:34:25.150 --> 00:34:27.900
were finding its inverse.
00:34:27.900 --> 00:34:29.409
Now the matrix isn't square.
00:34:29.409 --> 00:34:32.360
It's probably rectangular.
00:34:32.360 --> 00:34:36.989
But I'll still tack on the
identity matrix, and of course
00:34:36.989 --> 00:34:42.179
since these have length
m it better be m by m.
00:34:42.179 --> 00:34:49.374
And now I'll do the reduced row
echelon form of this matrix.
00:34:52.480 --> 00:34:56.120
And what do I get?
00:35:01.640 --> 00:35:04.950
The reduced row echelon form
starts with these columns,
00:35:04.950 --> 00:35:11.680
starts with the first columns,
works like mad, and produces R.
00:35:11.680 --> 00:35:13.920
Of course, still that
same size, m by n.
00:35:13.920 --> 00:35:15.700
And we did it before.
00:35:15.700 --> 00:35:19.140
And then whatever
it did to get R,
00:35:19.140 --> 00:35:22.520
something else is
going to show up here.
00:35:22.520 --> 00:35:26.690
Let me call it E, m by m.
00:35:26.690 --> 00:35:30.170
It's whatever -- do you see
that E is just going to contain
00:35:30.170 --> 00:35:32.640
a record of what we did?
00:35:32.640 --> 00:35:38.780
We did whatever it took
to get A to become R.
00:35:38.780 --> 00:35:40.930
And at the same time,
we were doing it
00:35:40.930 --> 00:35:44.410
to the identity matrix.
00:35:44.410 --> 00:35:48.590
So we started with the identity
matrix, we buzzed along.
00:35:48.590 --> 00:35:51.190
So we took some --
00:35:51.190 --> 00:35:55.860
all this row reduction amounted
to multiplying on the left
00:35:55.860 --> 00:36:00.040
by some matrix, some series
of elementary matrices
00:36:00.040 --> 00:36:05.780
that altogether gave us one
matrix, and that matrix is E.
00:36:05.780 --> 00:36:11.450
So all this row reduction stuff
amounted to multiplying by E.
00:36:11.450 --> 00:36:13.120
How do I know that?
00:36:13.120 --> 00:36:16.680
It certainly amounted to
multiply it by something.
00:36:16.680 --> 00:36:21.660
And that something took I to
E, so that something was E.
00:36:21.660 --> 00:36:29.790
So now look at the
first part, E A is R.
00:36:29.790 --> 00:36:31.520
No big deal.
00:36:31.520 --> 00:36:38.710
All I've said is that the row
reduction steps that we all
00:36:38.710 --> 00:36:45.680
know -- well, taking A to
R, are in a, in some matrix,
00:36:45.680 --> 00:36:49.340
and I can find out what that
matrix is by just tacking I
00:36:49.340 --> 00:36:51.810
on and seeing what comes out.
00:36:51.810 --> 00:36:54.570
What comes out is E.
00:36:54.570 --> 00:36:58.400
Let's just review the
invertible square case.
00:36:58.400 --> 00:37:00.860
What happened then?
00:37:00.860 --> 00:37:04.350
Because I was interested
in it in chapter two also.
00:37:04.350 --> 00:37:08.260
When A was square and
invertible, I took A I,
00:37:08.260 --> 00:37:10.480
I did row, row elimination.
00:37:10.480 --> 00:37:12.210
And what was the
R that came out?
00:37:12.210 --> 00:37:14.770
It was I.
00:37:14.770 --> 00:37:24.530
So in chapter two, in
chapter two, R was I.
00:37:24.530 --> 00:37:27.310
The row the, the
reduced row echelon
00:37:27.310 --> 00:37:31.730
form of a nice invertible
square matrix is the identity.
00:37:31.730 --> 00:37:41.510
So if R was I in that case, then
E was -- then E was A inverse,
00:37:41.510 --> 00:37:44.430
because E A is I.
00:37:44.430 --> 00:37:45.200
Good.
00:37:45.200 --> 00:37:48.540
That's, that was good and easy.
00:37:48.540 --> 00:37:52.990
Now what I'm saying is
that there still is an E.
00:37:52.990 --> 00:37:55.740
It's not A inverse any more,
because A is rectangular.
00:37:55.740 --> 00:37:57.730
It hasn't got an inverse.
00:37:57.730 --> 00:38:05.010
But there is still some matrix
E that connected this to this --
00:38:05.010 --> 00:38:09.260
oh, I should have figured
out in advanced what it was.
00:38:09.260 --> 00:38:11.810
Shoot.
00:38:11.810 --> 00:38:12.800
I didn't --
00:38:12.800 --> 00:38:16.620
I did those steps and sort
of erased as I went --
00:38:16.620 --> 00:38:20.370
and, I should have done
them to the identity too.
00:38:20.370 --> 00:38:22.500
Can I do that?
00:38:22.500 --> 00:38:23.460
Can I do that?
00:38:23.460 --> 00:38:26.140
I'll keep the identity matrix,
like I'm supposed to do,
00:38:26.140 --> 00:38:29.750
and I'll do the same operations
on it, and see what I end up
00:38:29.750 --> 00:38:30.340
with.
00:38:30.340 --> 00:38:31.300
OK.
00:38:31.300 --> 00:38:32.810
So I'm starting
with the identity --
00:38:32.810 --> 00:38:40.511
which I'll write in light,
light enough, but --
00:38:40.511 --> 00:38:41.010
OK.
00:38:41.010 --> 00:38:42.520
What did I do?
00:38:42.520 --> 00:38:45.950
I subtracted that row from that
one and that row from that one.
00:38:45.950 --> 00:38:47.950
OK, I'll do that
to the identity.
00:38:47.950 --> 00:38:52.990
So I subtract that first row
from row two and row three.
00:38:52.990 --> 00:38:55.310
Good.
00:38:55.310 --> 00:38:56.950
Then I think I multiplied --
00:38:56.950 --> 00:38:57.620
Do you remember?
00:38:57.620 --> 00:39:01.890
I multiplied row
two by minus one.
00:39:01.890 --> 00:39:05.270
Let me just do that.
00:39:05.270 --> 00:39:06.610
Then what did I do?
00:39:06.610 --> 00:39:14.880
I subtracted two of row
two away from row one.
00:39:14.880 --> 00:39:15.770
I better do that.
00:39:15.770 --> 00:39:17.720
Subtract two of
this away from this.
00:39:17.720 --> 00:39:24.180
That's minus one, two of these
away leaves a plus 2 and 0.
00:39:24.180 --> 00:39:28.440
I believe that's E.
00:39:28.440 --> 00:39:35.870
The way to check is to see,
multiply that E by this A,
00:39:35.870 --> 00:39:37.420
just to see did I do it right.
00:39:40.570 --> 00:39:49.285
So I believe E was -1 2
0, 1 -1 0, and -1 0 1.
00:39:53.020 --> 00:39:53.520
OK.
00:39:53.520 --> 00:39:58.030
That's my E, that's
my A, and that's R.
00:39:58.030 --> 00:40:00.110
All right.
00:40:00.110 --> 00:40:02.950
All I'm struggling
to do is right,
00:40:02.950 --> 00:40:09.660
the reason I wanted this blasted
E was so that I could figure
00:40:09.660 --> 00:40:14.570
out the left null space,
not only its dimension,
00:40:14.570 --> 00:40:17.020
which I know --
00:40:17.020 --> 00:40:19.427
actually, what is the dimension
of the left null space?
00:40:19.427 --> 00:40:20.260
So here's my matrix.
00:40:23.180 --> 00:40:24.465
What's the rank of the matrix?
00:40:27.560 --> 00:40:30.640
And the dimension of the null
-- of the left null space is
00:40:30.640 --> 00:40:33.470
supposed to be m-r.
00:40:33.470 --> 00:40:34.930
It's 3 -2, 1.
00:40:34.930 --> 00:40:39.090
I believe that the left null
space is one dimensional.
00:40:39.090 --> 00:40:42.990
There is one combination
of those three rows
00:40:42.990 --> 00:40:46.840
that produces the zero row.
00:40:46.840 --> 00:40:52.270
There's a basis -- a basis for
the left null space has only
00:40:52.270 --> 00:40:54.200
got one vector in it.
00:40:54.200 --> 00:40:55.790
And what is that vector?
00:40:55.790 --> 00:40:58.710
It's here in the last row of E.
00:40:58.710 --> 00:41:01.400
But we could have
seen it earlier.
00:41:01.400 --> 00:41:05.220
What combination of those
rows gives the zero row?
00:41:05.220 --> 00:41:09.110
-1 of that plus one of that.
00:41:09.110 --> 00:41:14.460
So a basis for the left
null space of this matrix --
00:41:14.460 --> 00:41:18.350
I'm looking for combinations
of rows that give the zero row
00:41:18.350 --> 00:41:22.080
if I'm looking at
the left null space.
00:41:22.080 --> 00:41:24.980
For the null space, I'm looking
at combinations of columns
00:41:24.980 --> 00:41:26.780
to get the zero column.
00:41:26.780 --> 00:41:29.760
Now I'm looking at combinations
of these three rows
00:41:29.760 --> 00:41:34.370
to get the zero row, and of
course there is my zero row,
00:41:34.370 --> 00:41:37.160
and here is my vector
that produced it.
00:41:37.160 --> 00:41:40.010
-1 of that row and one of that
00:41:40.010 --> 00:41:40.510
row.
00:41:40.510 --> 00:41:41.650
Obvious.
00:41:41.650 --> 00:41:42.210
OK.
00:41:42.210 --> 00:41:45.940
So in that example -- and
actually in all examples,
00:41:45.940 --> 00:41:51.310
we have seen how to produce a
basis for the left null space.
00:41:51.310 --> 00:41:54.800
I won't ask you that
all the time, because --
00:41:54.800 --> 00:41:58.790
it didn't come out
immediately from R.
00:41:58.790 --> 00:42:03.850
We had to keep track of E
for that left null space.
00:42:03.850 --> 00:42:07.520
But at least it didn't require
us to transpose the matrix
00:42:07.520 --> 00:42:10.220
and start all over again.
00:42:10.220 --> 00:42:12.390
OK, those are the
four subspaces.
00:42:12.390 --> 00:42:15.520
Can I review them?
00:42:15.520 --> 00:42:18.950
The row space and the
null space are in R^n.
00:42:18.950 --> 00:42:22.220
Their dimensions add to n.
00:42:22.220 --> 00:42:27.470
The column space and the
left null space are in R^m,
00:42:27.470 --> 00:42:30.900
and their dimensions add to m.
00:42:30.900 --> 00:42:33.700
OK.
00:42:33.700 --> 00:42:39.080
So let me close
these last minutes
00:42:39.080 --> 00:42:49.620
by pushing you a little bit more
to a new type of vector space.
00:42:49.620 --> 00:42:53.230
All our vector spaces, all the
ones that we took seriously,
00:42:53.230 --> 00:43:01.570
have been subspaces of some real
three or n dimensional space.
00:43:01.570 --> 00:43:04.880
Now I'm going to write
down another vector
00:43:04.880 --> 00:43:06.720
space, a new vector space.
00:43:14.120 --> 00:43:18.610
Say all three by three matrices.
00:43:26.420 --> 00:43:27.990
My matrices are the vectors.
00:43:31.830 --> 00:43:33.240
Is that all right?
00:43:33.240 --> 00:43:34.130
I'm just naming them.
00:43:34.130 --> 00:43:36.570
You can put quotes
around vectors.
00:43:36.570 --> 00:43:40.050
Every three by three matrix
is one of my vectors.
00:43:40.050 --> 00:43:43.060
Now how I entitled to
call those things vectors?
00:43:43.060 --> 00:43:46.380
I mean, they look very
much like matrices.
00:43:46.380 --> 00:43:49.980
But they are vectors in my
vector space because they obey
00:43:49.980 --> 00:43:50.640
the rules.
00:43:50.640 --> 00:43:55.850
All I'm supposed to be able to
do with vectors is add them --
00:43:55.850 --> 00:43:58.190
I can add matrices --
00:43:58.190 --> 00:44:01.580
I'm supposed to be able to
multiply them by scalar numbers
00:44:01.580 --> 00:44:09.290
like seven -- well, I can
multiply a matrix by And that
00:44:09.290 --> 00:44:11.960
-- and I can take
combinations of matrices,
00:44:11.960 --> 00:44:15.060
I can take three of one
matrix minus five of another
00:44:15.060 --> 00:44:15.990
matrix.
00:44:15.990 --> 00:44:21.260
And those combinations, there's
a zero matrix, the matrix
00:44:21.260 --> 00:44:23.570
that has all zeros in it.
00:44:23.570 --> 00:44:26.050
If I add that to another
matrix, it doesn't change it.
00:44:26.050 --> 00:44:26.960
All the good stuff.
00:44:26.960 --> 00:44:30.270
If I multiply a matrix by
one it doesn't change it.
00:44:30.270 --> 00:44:32.800
All those eight rules
for a vector space
00:44:32.800 --> 00:44:37.460
that we never wrote down,
all easily satisfied.
00:44:37.460 --> 00:44:41.150
So now we have a different --
00:44:41.150 --> 00:44:46.450
now of course you can say you
can multiply those matrices.
00:44:46.450 --> 00:44:47.240
I don't care.
00:44:47.240 --> 00:44:50.070
For the moment, I'm only
thinking of these matrices
00:44:50.070 --> 00:44:57.960
as forming a vector space --
so I only doing A plus B and c
00:44:57.960 --> 00:44:59.350
times A.
00:44:59.350 --> 00:45:03.050
I'm not interested
in A B for now.
00:45:06.580 --> 00:45:09.340
The fact that I
can multiply is not
00:45:09.340 --> 00:45:13.721
relevant to th-
to a vector space.
00:45:13.721 --> 00:45:14.220
OK.
00:45:14.220 --> 00:45:15.636
So I have three
by three matrices.
00:45:18.180 --> 00:45:21.600
And how about subspaces?
00:45:21.600 --> 00:45:26.420
What's -- tell me a subspace
of this matrix space.
00:45:26.420 --> 00:45:30.270
Let me call this matrix space M.
00:45:30.270 --> 00:45:34.270
That's my matrix space, my space
of all three by three matrices.
00:45:34.270 --> 00:45:37.730
Tell me a subspace of it.
00:45:37.730 --> 00:45:40.670
What about the upper
triangular matrices?
00:45:40.670 --> 00:45:41.260
OK.
00:45:41.260 --> 00:45:43.330
So subspaces.
00:45:43.330 --> 00:45:50.660
Subspaces of M.
00:45:50.660 --> 00:45:53.675
All, all upper
triangular matrices.
00:46:00.370 --> 00:46:01.750
Another subspace.
00:46:01.750 --> 00:46:03.085
All symmetric matrices.
00:46:11.610 --> 00:46:13.730
The intersection
of two subspaces
00:46:13.730 --> 00:46:15.090
is supposed to be a subspace.
00:46:15.090 --> 00:46:20.310
We gave a little effort
to the proof of that fact.
00:46:20.310 --> 00:46:23.150
If I look at the matrices
that are in this subspace --
00:46:23.150 --> 00:46:26.420
they're symmetric, and
they're also in this subspace,
00:46:26.420 --> 00:46:31.280
they're upper triangular,
what do they look like?
00:46:31.280 --> 00:46:33.550
Well, if they're
symmetric but they
00:46:33.550 --> 00:46:35.660
have zeros below the
diagonal, they better
00:46:35.660 --> 00:46:38.650
have zeros above the
diagonal, so the intersection
00:46:38.650 --> 00:46:40.285
would be diagonal matrices.
00:46:44.820 --> 00:46:48.430
That's another subspace,
smaller than those.
00:46:50.970 --> 00:46:53.740
How can I use the word smaller?
00:46:53.740 --> 00:46:56.390
Well, I'm now entitled
to use the word smaller.
00:46:56.390 --> 00:47:00.410
I mean, well, one way
to say is, OK, these
00:47:00.410 --> 00:47:02.570
are contained in those.
00:47:02.570 --> 00:47:05.210
These are contained in those.
00:47:05.210 --> 00:47:09.205
But more precisely, I could give
the dimension of these spaces.
00:47:11.710 --> 00:47:14.740
So I could -- we can compute
-- let's compute it next time,
00:47:14.740 --> 00:47:17.870
the dimension of all upper
-- of the subspace of upper
00:47:17.870 --> 00:47:20.490
triangular three
by three matrices.
00:47:20.490 --> 00:47:23.790
The dimension of symmetric
three by three matrices.
00:47:23.790 --> 00:47:27.630
The dimension of diagonal
three by three matrices.
00:47:27.630 --> 00:47:29.690
Well, to produce
dimension, that means
00:47:29.690 --> 00:47:33.370
I'm supposed to produce
a basis, and then
00:47:33.370 --> 00:47:37.230
I just count how many vecto-
how many I needed in the basis.
00:47:37.230 --> 00:47:39.880
Let me give you the
answer for this one.
00:47:39.880 --> 00:47:41.430
What's the dimension?
00:47:41.430 --> 00:47:44.610
The dimension of this
-- say, this subspace,
00:47:44.610 --> 00:47:47.520
let me call it D, all
diagonal matrices.
00:47:47.520 --> 00:47:54.610
The dimension of
this subspace is --
00:47:54.610 --> 00:47:57.610
as I write you're
working it out --
00:47:57.610 --> 00:47:58.970
three.
00:47:58.970 --> 00:48:09.150
Because here's a matrix in
this -- it's a diagonal matrix.
00:48:09.150 --> 00:48:10.040
Here's another one.
00:48:15.560 --> 00:48:16.610
Here's another one.
00:48:20.350 --> 00:48:22.755
Better make it diagonal,
let me put a seven there.
00:48:25.970 --> 00:48:28.130
That was not a
very great choice,
00:48:28.130 --> 00:48:31.020
but it's three
diagonal matrices,
00:48:31.020 --> 00:48:33.950
and I believe that
they're a basis.
00:48:33.950 --> 00:48:37.190
I believe that those three
matrices are independent
00:48:37.190 --> 00:48:40.680
and I believe that
any diagonal matrix is
00:48:40.680 --> 00:48:42.220
a combination of those three.
00:48:42.220 --> 00:48:47.310
So they span the subspace
of diagonal matrices.
00:48:47.310 --> 00:48:49.040
Do you see that idea?
00:48:49.040 --> 00:48:55.170
It's like stretching the
idea from R^n to R^(n by n),
00:48:55.170 --> 00:48:57.360
three by three.
00:48:57.360 --> 00:49:02.310
But the -- we can still add, we
can still multiply by numbers,
00:49:02.310 --> 00:49:06.370
and we just ignore the fact that
we can multiply two matrices
00:49:06.370 --> 00:49:07.550
together.
00:49:07.550 --> 00:49:09.380
OK, thank you.
00:49:09.380 --> 00:49:11.930
That's lecture ten.