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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 is to tell
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you about the four subspaces
that come with a matrix.
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We've seen two subspaces,
the column space and the null
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space.
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There's two to go.
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OK.
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First of all,
and this is a great way to
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recap and correct the previous
lecture -- so you remember I was
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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 that there was nothing
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special, nothing about that
basis that another basis
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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
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five, and those were
independent.
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And then I said three three
seven wouldn't do,
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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
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two, then probably three three
eight sticks a little bit out of
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the plane and it's independent
and it gives a basis.
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But after class,
to my sorrow,
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a student tells me,
"Wait a minute,
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that ba- that third vector,
three three eight,
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is not independent." 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 gives
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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,
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if I look at that matrix,
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, it's got two identical
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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 because that matrix
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is not invertible 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 space
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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
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one, I'm looking at the rows of
the matrix -- oh,
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anyway, I'll have two equal
rows and the row space will be
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only two 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,
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in other words,
something that I should have
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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
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combinations,
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
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independent, but if they're
dependent, as in this example,
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my error from last time,
they're not -- those three rows
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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 of the columns of A
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transpose.
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And that allows me to use the
convenient notation,
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the column 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 space of A
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transpose, that's telling me in
a nice matrix notation what the
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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 the null space of A
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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
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null space, and I'll show you
why in a moment.
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So often I call this the --
just to write that word -- the
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left null space of A.
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So just the way we have the row
space of A and we switch it to
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the column space of A transpose,
so we have this space of guys
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l- that I call the left null
space of A, but the good
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notation 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, which are just a
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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, I see that this
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fourth space is already getting
second, you know,
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second class citizen treatment
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
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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, to remind you that
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they're in this --
which you know,
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how -- 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
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components, is the column space
and what I'm calling the null
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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, we know everything about
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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 -- construct
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a basis?
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What systematic way would
produce a basis?
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And what's their dimension?
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OK.
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So for each of the four spaces,
I have to answer those
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questions.
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How do I produce a basis?
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And then -- which has a
somewhat long answer.
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And what's the dimension,
which is just a number,
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so it has a real short answer.
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Can I give you the short answer
first?
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I shouldn't do it,
but here it is.
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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?
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I have an m by n matrix.
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The dimension of the column
space is the rank,
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r.
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We actually got to that at the
end of the last lecture,
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but only for an example.
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So I really have to say,
OK, what's going on there.
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I should produce a basis and
then I just look to see how many
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vectors I needed in that basis,
and the answer will be r.
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Actually, I'll do that,
before I get on to the others.
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What's a basis for the columns
space?
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We've done all the work of row
reduction, identifying the pivot
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columns, the ones that have
pivots, the ones that end up
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with pivots.
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But now I -- the pivot columns
I'm interested in are columns of
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A, the original A.
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And those pivot columns,
there are r of them.
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The rank r counts those.
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Those are a basis.
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So if I answer this question
for the column space,
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the answer will be a basis is
the pivot columns and the
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dimension is the rank r,
and there are r pivot columns
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and everything great.
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OK.
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So that space we pretty well
understand.
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I probably have a little going
back to see that -- to prove
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that this is a right answer,
but you know it's the right
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answer.
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Now let me look at the row
space.
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OK.
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Shall I tell you the dimension
of the row space?
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Yes.
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Before we do even an example,
let me tell you the dimension
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of the row space.
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Its dimension is also r.
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The row space and the column
space have the same dimension.
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That's a wonderful fact.
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The dimension of the column
space of A transpose --
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that's the row space -- is r.
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That, that space is r
dimensional.
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Snd so is this one.
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OK.
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That's the sort of insight that
got used in this example.
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If those --
are the three columns of a
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matrix -- let me make them the
three columns of a matrix by
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just erasing some brackets.
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OK, those are the three columns
of a matrix.
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The rank of that matrix,
if I look at the columns,
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it wasn't obvious to me anyway.
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But if I look at the rows,
now it's obvious.
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The row space of that matrix
obviously is two dimensional,
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because I see a basis for the
row space, this row and that
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row.
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And of course,
strictly speaking,
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I'm supposed to transpose those
guys, make them stand up.
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But the rank is two,
and therefore the column space
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is two dimensional by this
wonderful fact that the row
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space and column space have the
same dimension.
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And therefore there are only
two pivot columns,
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not three, and,
those, the three columns are
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dependent.
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OK.
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Now let me bury that error and
talk about the row space.
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Well, I'm going to give you the
dimensions of all the spaces.
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Because that's such a nice
answer.
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OK.
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So let me come back here.
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So we have this great fact to
establish, that the row space,
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its dimension is also the rank.
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OK.
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What about the null space?
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What's a basis for the null
space?
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What's the dimension of the
null space?
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Let me, I'll put that answer up
here for the null space.
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Well, how have we constructed
the null space?
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We took the matrix A,
we did those row operations to
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get it into a form U or,
or even further.
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We got it into the reduced form
R.
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And then we read off special
solutions.
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Special solutions.
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And every special solution came
from a free variable.
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And those special solutions are
in the null space,
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and the great thing is they're
a basis for it.
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So for the null space,
a basis will be the special
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solutions.
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And there's one for every free
variable, right?
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For each free variable,
we give that variable the value
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one, the other free variables
zero.
281
00:18:09.66 --> 00:18:13
We get the pivot variables,
we get a vector in the -- we
282
00:18:13 --> 00:18:15
get a special solution.
283
00:18:15 --> 00:18:20
So we get altogether n-r of
them, because that's the number
284
00:18:20 --> 00:18:22
of free variables.
285
00:18:22 --> 00:18:26
If we have r -- this is the
dimension is r,
286
00:18:26 --> 00:18:29
is the number of pivot
variables.
287
00:18:29 --> 00:18:33
This is the number of free
variables.
288
00:18:33 --> 00:18:38
So the beauty is that those
special solutions do form a
289
00:18:38 --> 00:18:44
basis and tell us immediately
that the dimension of the null
290
00:18:44 --> 00:18:49
space is n --
I better write this well,
291
00:18:49 --> 00:18:52
because it's so nice -- n-r.
292
00:18:52 --> 00:18:56
And do you see the nice thing?
293
00:18:56 --> 00:19:01
That the two dimensions in this
n dimensional space,
294
00:19:01 --> 00:19:07
one subspace is r dimensional
-- to be proved,
295
00:19:07 --> 00:19:09.4
that's the row space.
296
00:19:09.4 --> 00:19:15
The other subspace is n-r
dimensional, that's the null
297
00:19:15 --> 00:19:17
space.
298
00:19:17 --> 00:19:21
And the two dimensions like
together give n.
299
00:19:21 --> 00:19:24.44
The sum of r and n-R is n.
300
00:19:24.44 --> 00:19:26
And that's just great.
301
00:19:26 --> 00:19:32
It's really copying the fact
that we have n variables,
302
00:19:32 --> 00:19:37
r of them are pivot variables
and n-r are free variables,
303
00:19:37 --> 00:19:40
and n altogether.
304
00:19:40 --> 00:19:40
OK.
305
00:19:40 --> 00:19:46
And now what's the dimension of
this poor misbegotten fourth
306
00:19:46 --> 00:19:47
subspace?
307
00:19:47 --> 00:19:49
It's got to be m-r.
308
00:19:49 --> 00:19:55
The dimension of this left null
space, left out practically,
309
00:19:55 --> 00:19:56
is m-r.
310
00:19:56 --> 00:20:01
Well, that's really just saying
that this -- again,
311
00:20:01 --> 00:20:06.3
the sum of that plus that is m,
and m is correct,
312
00:20:06.3 --> 00:20:11
it's the number of columns in A
transpose.
313
00:20:11 --> 00:20:16
A transpose is just as good a
matrix as A.
314
00:20:16 --> 00:20:19
It just happens to be n by m.
315
00:20:19 --> 00:20:26
It happens to have m columns,
so it will have m variables
316
00:20:26 --> 00:20:33
when I go to A x equals 0 and m
of them, and r of them will be
317
00:20:33 --> 00:20:39
pivot variables and m-r will be
free variables.
318
00:20:39 --> 00:20:44
A transpose is as good a matrix
as A.
319
00:20:44 --> 00:20:50
It follows the same rule that
the this plus the dimension --
320
00:20:50 --> 00:20:57
this dimension plus this
dimension adds up to the number
321
00:20:57 --> 00:20:58
of columns.
322
00:20:58 --> 00:21:04
And over here,
A transpose has m columns.
323
00:21:04 --> 00:21:04
OK.
324
00:21:04 --> 00:21:04
OK.
325
00:21:04 --> 00:21:09
So I gave you the easy answer,
the dimensions.
326
00:21:09 --> 00:21:13
Now can I go back to check on a
basis?
327
00:21:13 --> 00:21:18
We would like to think that --
say the row space,
328
00:21:18 --> 00:21:23
because we've got a basis for
the column space.
329
00:21:23 --> 00:21:30
The pivot columns give a basis
for the column space.
330
00:21:30 --> 00:21:34
Now I'm asking you to look at
the row space.
331
00:21:34 --> 00:21:38
And I -- you could say,
OK, I can produce a basis for
332
00:21:38 --> 00:21:43.51
the row space by transposing my
matrix, making those columns,
333
00:21:43.51 --> 00:21:46
then doing elimination,
row reduction,
334
00:21:46 --> 00:21:50
and checking out the pivot
columns in this transposed
335
00:21:50 --> 00:21:52
matrix.
336
00:21:52 --> 00:21:58
But that means you had to do
all that row reduction on A
337
00:21:58 --> 00:21:59
transpose.
338
00:21:59 --> 00:22:05.69
It ought to be possible,
if we take a matrix A -- let me
339
00:22:05.69 --> 00:22:12
take the matrix -- maybe we had
this matrix in the last lecture.
340
00:22:12 --> 00:22:16
1 1 1, 2 1 2,
3 2 3, 1 1 1.
341
00:22:16 --> 00:22:16
OK.
342
00:22:16 --> 00:22:20
That, that matrix was so easy.
343
00:22:20 --> 00:22:24
We spotted its pivot columns,
one and two,
344
00:22:24 --> 00:22:28
without actually doing row
reduction.
345
00:22:28 --> 00:22:32
But now let's do the job
properly.
346
00:22:32 --> 00:22:37
So I subtract this away from
this to produce a zero.
347
00:22:37 --> 00:22:41
So one 2 3 1 is fine.
348
00:22:41 --> 00:22:45.72
Subtracting that away leaves me
minus 1 -1 0,
349
00:22:45.72 --> 00:22:46
right?
350
00:22:46 --> 00:22:50
And subtracting that from the
last row, oh,
351
00:22:50 --> 00:22:51
well that's easy.
352
00:22:51 --> 00:22:52.26
OK?
353
00:22:52.26 --> 00:22:54
I'm doing row reduction.
354
00:22:54 --> 00:22:58.32
Now I've -- the first column is
all set.
355
00:22:58.32 --> 00:23:03
The second column I now see the
pivot.
356
00:23:03 --> 00:23:06
And I can clean up,
if I -- actually,
357
00:23:06 --> 00:23:06
OK.
358
00:23:06 --> 00:23:10
Why don't I make the pivot into
a 1.
359
00:23:10 --> 00:23:16
I'll multiply that row through
by by -1, and then I have 1 1.
360
00:23:16 --> 00:23:20
That was an elementary
operation I'm allowed,
361
00:23:20 --> 00:23:23
multiply a row by a number.
362
00:23:23 --> 00:23:27
And now I'll do elimination.
363
00:23:27 --> 00:23:33
Two of those away from that
will knock this guy out and make
364
00:23:33 --> 00:23:35
this into a 1.
365
00:23:35 --> 00:23:36
So that's now a 0 and that's a
366
00:23:37 --> 00:23:38
367
00:23:38 --> 00:23:38
OK.
368
00:23:38 --> 00:23:39
Done.
369
00:23:39 --> 00:23:40
That's R.
370
00:23:40 --> 00:23:44
I'm seeing the identity matrix
here.
371
00:23:44 --> 00:23:47
I'm seeing zeros below.
372
00:23:47 --> 00:23:49
And I'm seeing F there.
373
00:23:49 --> 00:23:50
OK.
374
00:23:50 --> 00:23:52
What about its row space?
375
00:23:52 --> 00:23:58
What happened to its row space
-- well, what happened -- let me
376
00:23:58 --> 00:24:03
first ask, just because this is,
is -- sometimes something does
377
00:24:03 --> 00:24:04
happen.
378
00:24:04 --> 00:24:06
Its column space changed.
379
00:24:06 --> 00:24:11
The column space of R is not
the column space of A,
380
00:24:11 --> 00:24:13
right?
381
00:24:13 --> 00:24:18
Because 1 1 1 is certainly in
the column space of A and
382
00:24:18 --> 00:24:22
certainly not in the column
space of R.
383
00:24:22 --> 00:24:24
I did row operations.
384
00:24:24 --> 00:24:29
Those row operations preserve
the row space.
385
00:24:29 --> 00:24:34
So the row, so the column
spaces are different.
386
00:24:34 --> 00:24:40
Different column spaces,
different column spaces.
387
00:24:40 --> 00:24:45
But I believe that they have
the same row space.
388
00:24:45 --> 00:24:47
Same row space.
389
00:24:47 --> 00:24:55
I believe that the row space of
that matrix and the row space of
390
00:24:55 --> 00:24:58
this matrix are identical.
391
00:24:58 --> 00:25:03
They have exactly the same
vectors in them.
392
00:25:03 --> 00:25:08
Those vectors are vectors with
four components,
393
00:25:08 --> 00:25:10
right?
394
00:25:10 --> 00:25:14
They're all combinations of
those rows.
395
00:25:14 --> 00:25:19.99
Or I believe you get the same
thing by taking all combinations
396
00:25:19.99 --> 00:25:21
of these rows.
397
00:25:21 --> 00:25:23
And if true,
what's a basis?
398
00:25:23 --> 00:25:29
What's a basis for the row
space of R, and it'll be a basis
399
00:25:29 --> 00:25:34
for the row space of the
original A, but it's obviously a
400
00:25:34 --> 00:25:38
basis for the row space of R.
401
00:25:38 --> 00:25:48
What's a basis for the row
space of that matrix?
402
00:25:48 --> 00:25:52.13
The first two rows.
403
00:25:52.13 --> 00:26:05
So a basis for the row -- so a
basis is, for the row space of A
404
00:26:05 --> 00:26:12
or of R, is, is the first R rows
of R.
405
00:26:12 --> 00:26:16
Not of A.
406
00:26:16 --> 00:26:21
Sometimes it's true for A,
but not necessarily.
407
00:26:21 --> 00:26:27
But R, we definitely have a
matrix here whose row space we
408
00:26:27 --> 00:26:29
can, we can identify.
409
00:26:29 --> 00:26:35
The row space is spanned by the
three rows, but if we want a
410
00:26:35 --> 00:26:38
basis we want independence.
411
00:26:38 --> 00:26:42
So out goes row three.
412
00:26:42 --> 00:26:46
The row space is also spanned
by the first two rows.
413
00:26:46 --> 00:26:49
This guy didn't contribute
anything.
414
00:26:49 --> 00:26:54
And of course over here this 1
2 3 1 in the bottom didn't
415
00:26:54 --> 00:26:56
contribute anything.
416
00:26:56 --> 00:26:58
We had it already.
417
00:26:58 --> 00:27:03
So this, here is a basis.
1 0 1 1 and 0 1 1 0.
418
00:27:03 --> 00:27:06
I believe those are in the row
space.
419
00:27:06 --> 00:27:08
I know they're independent.
420
00:27:08 --> 00:27:11
Why are they in the row space?
421
00:27:11 --> 00:27:15
Why are those two vectors in
the row space?
422
00:27:15 --> 00:27:20
Because all those operations we
did, which started with these
423
00:27:20 --> 00:27:24
rows and took combinations of
them --
424
00:27:24 --> 00:27:29.23
I took this row minus this row,
that gave me something that's
425
00:27:29.23 --> 00:27:31
still in the row space.
426
00:27:31 --> 00:27:32
That's the point.
427
00:27:32 --> 00:27:36
When I took a row minus a
multiple of another row,
428
00:27:36 --> 00:27:38
I'm staying in the row space.
429
00:27:38 --> 00:27:40
The row space is not changing.
430
00:27:40 --> 00:27:44
My little basis for it is
changing, and I've ended up
431
00:27:44 --> 00:27:47
with, sort of the best basis.
432
00:27:47 --> 00:27:52.93
If the columns of the identity
matrix are the best basis for
433
00:27:52.93 --> 00:27:57
R^3 or R^n, the rows of this
matrix are the best basis for
434
00:27:57 --> 00:27:58.96
the row space.
435
00:27:58.96 --> 00:28:03
Best in the sense of being as
clean as I can make it.
436
00:28:03 --> 00:28:08
Starting off with the identity
and then finishing up with
437
00:28:08 --> 00:28:10
whatever has to be in there.
438
00:28:10 --> 00:28:11
OK.
439
00:28:11 --> 00:28:15
Do you see then that the
dimension is r?
440
00:28:15 --> 00:28:20
For sure, because we've got r
pivots, r non-zero rows.
441
00:28:20 --> 00:28:24
We've got the right number of
vectors, r.
442
00:28:24 --> 00:28:28.27
They're in the row space,
they're independent.
443
00:28:28.27 --> 00:28:29
That's it.
444
00:28:29 --> 00:28:33
They are a basis for the row
space.
445
00:28:33 --> 00:28:36
And we can even pin that down
further.
446
00:28:36 --> 00:28:39
How do I know that every row of
A is a combination?
447
00:28:39 --> 00:28:42.25
How do I know they span the row
space?
448
00:28:42.25 --> 00:28:45
Well, somebody says,
I've got the right number of
449
00:28:45 --> 00:28:47
them, so they must.
450
00:28:47 --> 00:28:49
But -- and that's true.
451
00:28:49 --> 00:28:53
But let me just say,
how do I know that this row is
452
00:28:53 --> 00:28:55
a combination of these?
453
00:28:55 --> 00:28:59
By just reversing the steps of
row reduction.
454
00:28:59 --> 00:29:05
If I just reverse the steps and
go from A -- from R back to A,
455
00:29:05 --> 00:29:07
then what do I,
what I doing?
456
00:29:07 --> 00:29:12
I'm starting with these rows,
I'm taking combinations of
457
00:29:12 --> 00:29:14
them.
458
00:29:14 --> 00:29:19
After a couple of steps,
undoing the subtractions that I
459
00:29:19 --> 00:29:21.89
did before, I'm back to these
rows.
460
00:29:21.89 --> 00:29:25
So these rows are combinations
of those rows.
461
00:29:25 --> 00:29:29
Those rows are combinations of
those rows.
462
00:29:29 --> 00:29:31
The two row spaces are the
same.
463
00:29:31 --> 00:29:33
The bases are the same.
464
00:29:33 --> 00:29:37
And the natural basis is this
guy.
465
00:29:37 --> 00:29:41
Is that all right for the row
space?
466
00:29:41 --> 00:29:47
The row space is sitting there
in R in its cleanest possible
467
00:29:47 --> 00:29:48
form.
468
00:29:48 --> 00:29:48
OK.
469
00:29:48 --> 00:29:55
Now what about the fourth guy,
the null space of A transpose?
470
00:29:55 --> 00:29:59
First of all,
why do I call that the left
471
00:29:59 --> 00:30:01
null space?
472
00:30:01 --> 00:30:06.75
So let me save that and bring
that down.
473
00:30:06.75 --> 00:30:07
OK.
474
00:30:07 --> 00:30:14
So the fourth space is the null
space of A transpose.
475
00:30:14 --> 00:30:21
So it has in it vectors,
let me call them y,
476
00:30:21 --> 00:30:25
so that A transpose y equals 0.
477
00:30:25 --> 00:30:33
If A transpose y equals 0,
then y is in the null space of
478
00:30:33 --> 00:30:38
A transpose, of course.
479
00:30:38 --> 00:30:44
So this is a matrix times a
column equaling zero.
480
00:30:44 --> 00:30:51
And now, because I want y to
sit on the left and I want A
481
00:30:51 --> 00:30:57
instead of A transpose,
I'll just transpose that
482
00:30:57 --> 00:30:58
equation.
483
00:30:58 --> 00:31:01
Can I just transpose that?
484
00:31:01 --> 00:31:06
On the right,
it makes the zero vector lie
485
00:31:06 --> 00:31:07
down.
486
00:31:07 --> 00:31:11
And on the left,
it's a product,
487
00:31:11 --> 00:31:15
A, A transpose times y.
488
00:31:15 --> 00:31:20.62
If I take the transpose,
then they come in opposite
489
00:31:20.62 --> 00:31:21
order, right?
490
00:31:21 --> 00:31:26
So that's y transpose times A
transpose transpose.
491
00:31:26 --> 00:31:30
But nobody's going to leave it
like that.
492
00:31:30 --> 00:31:36
That's -- A transpose transpose
is just A, of course.
493
00:31:36 --> 00:31:42
When I transposed A transpose I
got back to A.
494
00:31:42 --> 00:31:45
Now do you see what I have now?
495
00:31:45 --> 00:31:49
I have a row vector,
y transpose,
496
00:31:49 --> 00:31:54
multiplying A,
and multiplying from the left.
497
00:31:54 --> 00:31:59
That's why I call it the left
null space.
498
00:31:59 --> 00:32:05
But by making it --
putting it on the left,
499
00:32:05 --> 00:32:10
I had to make it into a row
instead of a column vector,
500
00:32:10 --> 00:32:15
and so my convention is I
usually don't do that.
501
00:32:15 --> 00:32:19
I usually stay with A transpose
y equals 0.
502
00:32:19 --> 00:32:20
OK.
503
00:32:20 --> 00:32:25
And you might ask,
how do we get a basis --
504
00:32:25 --> 00:32:31
or I might ask,
how do we get a basis for this
505
00:32:31 --> 00:32:36
fourth space,
this left null space?
506
00:32:36 --> 00:32:36
OK.
507
00:32:36 --> 00:32:40
I'll do it in the example.
508
00:32:40 --> 00:32:44
As always -- not that one.
509
00:32:44 --> 00:32:51
The left null space is not
jumping out at me here.
510
00:32:51 --> 00:32:57
I know which are the free
variables --
511
00:32:57 --> 00:33:02
the special solutions,
but those are special solutions
512
00:33:02 --> 00:33:06
to A x equals zero,
and now I'm looking at A
513
00:33:06 --> 00:33:09
transpose, and I'm not seeing it
here.
514
00:33:09 --> 00:33:14.54
So -- but somehow you feel that
the work that you did which
515
00:33:14.54 --> 00:33:19
simplified A to R should have
revealed the left null space
516
00:33:19 --> 00:33:19.85
too.
517
00:33:19.85 --> 00:33:25.16
And it's slightly less
immediate, but it's there.
518
00:33:25.16 --> 00:33:28
So from A to R,
I took some steps,
519
00:33:28 --> 00:33:33
and I guess I'm interested in
what were those steps,
520
00:33:33 --> 00:33:37
or what were all of them
together.
521
00:33:37 --> 00:33:42
I don't -- I'm not interested
in what particular ones they
522
00:33:42 --> 00:33:43
were.
523
00:33:43 --> 00:33:49
I'm interested in what was the
whole matrix that took me from A
524
00:33:49 --> 00:33:50
to R.
525
00:33:50 --> 00:33:53
How would you find that?
526
00:33:53 --> 00:34:00
Do you remember Gauss-Jordan,
where you tack on the identity
527
00:34:00 --> 00:34:01
matrix?
528
00:34:01 --> 00:34:04
Let's do that again.
529
00:34:04 --> 00:34:07
So I, I'll do it above,
here.
530
00:34:07 --> 00:34:12
So this is now,
this is now the idea of -- I
531
00:34:12 --> 00:34:16
take the matrix A,
which is m by n.
532
00:34:16 --> 00:34:22
In Gauss-Jordan,
when we saw him before --
533
00:34:22 --> 00:34:29.43
A was a square invertible
matrix and we were finding its
534
00:34:29.43 --> 00:34:30
inverse.
535
00:34:30 --> 00:34:33
Now the matrix isn't square.
536
00:34:33 --> 00:34:36
It's probably rectangular.
537
00:34:36 --> 00:34:42
But I'll still tack on the
identity matrix,
538
00:34:42 --> 00:34:49
and of course since these have
length m it better be m by m.
539
00:34:49 --> 00:34:56
And now I'll do the reduced row
echelon form of this matrix.
540
00:34:56 --> 00:35:00
And what do I get?
541
00:35:00 --> 00:35:04.58
The reduced row echelon form
starts with these columns,
542
00:35:04.58 --> 00:35:08
starts with the first columns,
works like mad,
543
00:35:08 --> 00:35:09
and produces R.
544
00:35:09 --> 00:35:12
Of course, still that same
size, m by n.
545
00:35:12 --> 00:35:14
And we did it before.
546
00:35:14 --> 00:35:19
And then whatever it did to get
R, something else is going to
547
00:35:19 --> 00:35:21
show up here.
548
00:35:21 --> 00:35:23
Let me call it E,
m by m.
549
00:35:23 --> 00:35:28
It's whatever -- do you see
that E is just going to contain
550
00:35:28 --> 00:35:30
a record of what we did?
551
00:35:30 --> 00:35:34
We did whatever it took to get
A to become R.
552
00:35:34 --> 00:35:38
And at the same time,
we were doing it to the
553
00:35:38 --> 00:35:39
identity matrix.
554
00:35:39 --> 00:35:45
So we started with the identity
matrix, we buzzed along.
555
00:35:45 --> 00:35:50
So we took some -- all this row
reduction amounted to
556
00:35:50 --> 00:35:56
multiplying on the left by some
matrix, some series of
557
00:35:56 --> 00:36:01
elementary matrices that
altogether gave us one matrix,
558
00:36:01 --> 00:36:03
and that matrix is E.
559
00:36:03 --> 00:36:10
So all this row reduction stuff
amounted to multiplying by E.
560
00:36:10 --> 00:36:12
How do I know that?
561
00:36:12 --> 00:36:17
It certainly amounted to
multiply it by something.
562
00:36:17 --> 00:36:23
And that something took I to E,
so that something was E.
563
00:36:23 --> 00:36:27
So now look at the first part,
E A is R.
564
00:36:27 --> 00:36:28
No big deal.
565
00:36:28 --> 00:36:35
All I've said is that the row
reduction steps that we all know
566
00:36:35 --> 00:36:38
--
well, taking A to R,
567
00:36:38 --> 00:36:43
are in a, in some matrix,
and I can find out what that
568
00:36:43 --> 00:36:48
matrix is by just tacking I on
and seeing what comes out.
569
00:36:48 --> 00:36:50
What comes out is E.
570
00:36:50 --> 00:36:54.1
Let's just review the
invertible square case.
571
00:36:54.1 --> 00:36:55
What happened then?
572
00:36:55 --> 00:37:01
Because I was interested in it
in chapter two also.
573
00:37:01 --> 00:37:07
When A was square and
invertible, I took A I,
574
00:37:07 --> 00:37:11
I did row, row elimination.
575
00:37:11 --> 00:37:16
And what was the R that came
out?
576
00:37:16 --> 00:37:17
It was I.
577
00:37:17 --> 00:37:22
So in chapter two,
in chapter two,
578
00:37:22 --> 00:37:25
R was I.
579
00:37:25 --> 00:37:29
The row the,
the reduced row echelon form of
580
00:37:29 --> 00:37:34
a nice invertible square matrix
is the identity.
581
00:37:34 --> 00:37:39
So if R was I in that case,
then E was -- then E was A
582
00:37:39 --> 00:37:42
inverse, because E A is I.
583
00:37:42 --> 00:37:42
Good.
584
00:37:42 --> 00:37:46
That's, that was good and easy.
585
00:37:46 --> 00:37:52
Now what I'm saying is that
there still is an E.
586
00:37:52 --> 00:37:56
It's not A inverse any more,
because A is rectangular.
587
00:37:56 --> 00:37:58.49
It hasn't got an inverse.
588
00:37:58.49 --> 00:38:03
But there is still some matrix
E that connected this to this --
589
00:38:03 --> 00:38:07
oh, I should have figured out in
advanced what it was.
590
00:38:07 --> 00:38:08
Shoot.
591
00:38:08 --> 00:38:14
I didn't -- I did those steps
and sort of erased as I went --
592
00:38:14 --> 00:38:18
and, I should have done them to
the identity too.
593
00:38:18 --> 00:38:19
Can I do that?
594
00:38:19 --> 00:38:20
Can I do that?
595
00:38:20 --> 00:38:24
I'll keep the identity matrix,
like I'm supposed to do,
596
00:38:24 --> 00:38:29
and I'll do the same operations
on it, and see what I end up
597
00:38:29 --> 00:38:29
with.
598
00:38:29 --> 00:38:30
OK.
599
00:38:30 --> 00:38:33.9
So I'm starting with the
identity --
600
00:38:33.9 --> 00:38:37
which I'll write in light,
light enough,
601
00:38:37 --> 00:38:38
but -- OK.
602
00:38:38 --> 00:38:39
What did I do?
603
00:38:39 --> 00:38:44
I subtracted that row from that
one and that row from that one.
604
00:38:44 --> 00:38:47
OK, I'll do that to the
identity.
605
00:38:47 --> 00:38:51.7
So I subtract that first row
from row two and row three.
606
00:38:51.7 --> 00:38:52.12
Good.
607
00:38:52.12 --> 00:38:57
Then I think I multiplied --
Do you remember?
608
00:38:57 --> 00:39:00
I multiplied row two by minus
one.
609
00:39:00 --> 00:39:02
Let me just do that.
610
00:39:02 --> 00:39:04
Then what did I do?
611
00:39:04 --> 00:39:09
I subtracted two of row two
away from row one.
612
00:39:09 --> 00:39:10
I better do that.
613
00:39:10 --> 00:39:14
Subtract two of this away from
this.
614
00:39:14 --> 00:39:19
That's minus one,
two of these away leaves a plus
615
00:39:19 --> 00:39:21
2 and 0.
616
00:39:21 --> 00:39:24
I believe that's E.
617
00:39:24 --> 00:39:33
The way to check is to see,
multiply that E by this A,
618
00:39:33 --> 00:39:38
just to see did I do it right.
619
00:39:38 --> 00:39:46
So I believe E was -1 2 0,
1 -1 0, and -1 0 1.
620
00:39:46 --> 00:39:47
OK.
621
00:39:47 --> 00:39:51
That's my E,
that's my A,
622
00:39:51 --> 00:39:55
and that's R.
623
00:39:55 --> 00:39:56
All right.
624
00:39:56 --> 00:40:02.25
All I'm struggling to do is
right, the reason I wanted this
625
00:40:02.25 --> 00:40:08
blasted E was so that I could
figure out the left null space,
626
00:40:08 --> 00:40:13
not only its dimension,
which I know -- actually,
627
00:40:13 --> 00:40:19
what is the dimension of the
left null space?
628
00:40:19 --> 00:40:20
So here's my matrix.
629
00:40:20 --> 00:40:22
What's the rank of the matrix?
630
00:40:22 --> 00:40:23
631
00:40:23 --> 00:40:27
And the dimension of the null
-- of the left null space is
632
00:40:27 --> 00:40:29
supposed to be m-r.
633
00:40:29 --> 00:40:30
It's 3 -2, 1.
634
00:40:30 --> 00:40:34
I believe that the left null
space is one dimensional.
635
00:40:34 --> 00:40:38
There is one combination of
those three rows that produces
636
00:40:38 --> 00:40:40
the zero row.
637
00:40:40 --> 00:40:46
There's a basis -- a basis for
the left null space has only got
638
00:40:46 --> 00:40:48
one vector in it.
639
00:40:48 --> 00:40:50
And what is that vector?
640
00:40:50 --> 00:40:53
It's here in the last row of E.
641
00:40:53 --> 00:40:56.74
But we could have seen it
earlier.
642
00:40:56.74 --> 00:41:02
What combination of those rows
gives the zero row?
643
00:41:02 --> 00:41:04
-1 of that plus one of that.
644
00:41:04 --> 00:41:09.78
So a basis for the left null
space of this matrix -- I'm
645
00:41:09.78 --> 00:41:15
looking for combinations of rows
that give the zero row if I'm
646
00:41:15 --> 00:41:18
looking at the left null space.
647
00:41:18 --> 00:41:22
For the null space,
I'm looking at combinations of
648
00:41:22 --> 00:41:25
columns to get the zero column.
649
00:41:25 --> 00:41:30
Now I'm looking at combinations
of these three rows to get the
650
00:41:30 --> 00:41:35
zero row, and of course there is
my zero row, and here is my
651
00:41:35 --> 00:41:40
vector that produced it.
-1 of that row and one of that
652
00:41:40 --> 00:41:40
row.
653
00:41:40 --> 00:41:41
Obvious.
654
00:41:41 --> 00:41:41
OK.
655
00:41:41 --> 00:41:46
So in that example --
and actually in all examples,
656
00:41:46 --> 00:41:50
we have seen how to produce a
basis for the left null space.
657
00:41:50 --> 00:41:54
I won't ask you that all the
time, because -- it didn't come
658
00:41:54 --> 00:41:56
out immediately from R.
659
00:41:56 --> 00:42:00
We had to keep track of E for
that left null space.
660
00:42:00 --> 00:42:04
But at least it didn't require
us to transpose the matrix and
661
00:42:04 --> 00:42:07
start all over again.
662
00:42:07 --> 00:42:10
OK, those are the four
subspaces.
663
00:42:10 --> 00:42:12
Can I review them?
664
00:42:12 --> 00:42:18
The row space and the null
space are in R^n.
665
00:42:18 --> 00:42:21
Their dimensions add to n.
666
00:42:21 --> 00:42:27
The column space and the left
null space are in R^m,
667
00:42:27 --> 00:42:30
and their dimensions add to m.
668
00:42:30 --> 00:42:32
OK.
669
00:42:32 --> 00:42:41
So let me close these last
minutes by pushing you a little
670
00:42:41 --> 00:42:47.65
bit more to a new type of vector
space.
671
00:42:47.65 --> 00:42:55
All our vector spaces,
all the ones that we took
672
00:42:55 --> 00:43:03
seriously, have been subspaces
of some real three or n
673
00:43:03 --> 00:43:06
dimensional space.
674
00:43:06 --> 00:43:14
Now I'm going to write down
another vector space,
675
00:43:14 --> 00:43:17
a new vector space.
676
00:43:17 --> 00:43:24.96
Say all three by three
matrices.
677
00:43:24.96 --> 00:43:27
My matrices are the vectors.
678
00:43:27 --> 00:43:28
Is that all right?
679
00:43:28 --> 00:43:30
I'm just naming them.
680
00:43:30 --> 00:43:33
You can put quotes around
vectors.
681
00:43:33 --> 00:43:37
Every three by three matrix is
one of my vectors.
682
00:43:37 --> 00:43:41
Now how I entitled to call
those things vectors?
683
00:43:41 --> 00:43:45
I mean, they look very much
like matrices.
684
00:43:45 --> 00:43:50
But they are vectors in my
vector space because they obey
685
00:43:50 --> 00:43:50
the rules.
686
00:43:50 --> 00:43:56
All I'm supposed to be able to
do with vectors is add them -- I
687
00:43:56 --> 00:44:01
can add matrices -- I'm supposed
to be able to multiply them by
688
00:44:01 --> 00:44:06
scalar numbers like seven --
well, I can multiply a matrix by
689
00:44:06 --> 00:44:06
690
00:44:06 --> 00:44:06
691
00:44:06 --> 00:44:10
And that --
and I can take combinations of
692
00:44:10 --> 00:44:14
matrices, I can take three of
one matrix minus five of another
693
00:44:14 --> 00:44:14
matrix.
694
00:44:14 --> 00:44:17
And those combinations,
there's a zero matrix,
695
00:44:17 --> 00:44:19
the matrix that has all zeros
in it.
696
00:44:19 --> 00:44:22
If I add that to another
matrix, it doesn't change it.
697
00:44:22 --> 00:44:24
All the good stuff.
698
00:44:24 --> 00:44:28
If I multiply a matrix by one
it doesn't change it.
699
00:44:28 --> 00:44:33
All those eight rules for a
vector space that we never wrote
700
00:44:33 --> 00:44:35
down, all easily satisfied.
701
00:44:35 --> 00:44:40
So now we have a different --
now of course you can say you
702
00:44:40 --> 00:44:42
can multiply those matrices.
703
00:44:42 --> 00:44:43
I don't care.
704
00:44:43 --> 00:44:47
For the moment,
I'm only thinking of these
705
00:44:47 --> 00:44:51
matrices as forming a vector
space --
706
00:44:51 --> 00:44:55
so I only doing A plus B and c
times A.
707
00:44:55 --> 00:44:59
I'm not interested in A B for
now.
708
00:44:59 --> 00:45:07.53
The fact that I can multiply is
not relevant to th- to a vector
709
00:45:07.53 --> 00:45:08
space.
710
00:45:08 --> 00:45:08
OK.
711
00:45:08 --> 00:45:12
So I have three by three
matrices.
712
00:45:12 --> 00:45:15
And how about subspaces?
713
00:45:15 --> 00:45:23
What's -- tell me a subspace of
this matrix space.
714
00:45:23 --> 00:45:26
Let me call this matrix space
M.
715
00:45:26 --> 00:45:31
That's my matrix space,
my space of all three by three
716
00:45:31 --> 00:45:32
matrices.
717
00:45:32 --> 00:45:35.43
Tell me a subspace of it.
718
00:45:35.43 --> 00:45:35
OK.
719
00:45:35 --> 00:45:39
What about the upper triangular
matrices?
720
00:45:39 --> 00:45:42
So subspaces.
721
00:45:42 --> 00:45:44
Subspaces of M.
722
00:45:44 --> 00:45:49
All, all upper triangular
matrices.
723
00:45:49 --> 00:45:52
Another subspace.
724
00:45:52 --> 00:45:55
All symmetric matrices.
725
00:45:55 --> 00:46:03
The intersection of two
subspaces is supposed to be a
726
00:46:03 --> 00:46:05
subspace.
727
00:46:05 --> 00:46:12
We gave a little effort to the
proof of that fact.
728
00:46:12 --> 00:46:22
If I look at the matrices that
are in this subspace --
729
00:46:22 --> 00:46:26
they're symmetric,
and they're also in this
730
00:46:26 --> 00:46:30
subspace, they're upper
triangular, what do they look
731
00:46:30 --> 00:46:30.45
like?
732
00:46:30.45 --> 00:46:34
Well, if they're symmetric but
they have zeros below the
733
00:46:34 --> 00:46:38
diagonal, they better have zeros
above the diagonal,
734
00:46:38 --> 00:46:42
so the intersection would be
diagonal matrices.
735
00:46:42 --> 00:46:46.38
That's another subspace,
smaller than those.
736
00:46:46.38 --> 00:46:49
How can I use the word smaller?
737
00:46:49 --> 00:46:53
Well, I'm now entitled to use
the word smaller.
738
00:46:53 --> 00:46:56
I mean, well,
one way to say is,
739
00:46:56 --> 00:46:59
OK, these are contained in
those.
740
00:46:59 --> 00:47:01
These are contained in those.
741
00:47:01 --> 00:47:06
But more precisely,
I could give the dimension of
742
00:47:06 --> 00:47:08
these spaces.
743
00:47:08 --> 00:47:12
So I could -- we can compute --
let's compute it next time,
744
00:47:12 --> 00:47:15
the dimension of all upper --
of the subspace of upper
745
00:47:15 --> 00:47:17
triangular three by three
matrices.
746
00:47:17 --> 00:47:21
The dimension of symmetric
three by three matrices.
747
00:47:21 --> 00:47:24.92
The dimension of diagonal three
by three matrices.
748
00:47:24.92 --> 00:47:29
Well, to produce dimension,
that means I'm supposed to
749
00:47:29 --> 00:47:32
produce a basis,
and then I just count how many
750
00:47:32 --> 00:47:35
vecto- how many I needed in the
basis.
751
00:47:35 --> 00:47:38
Let me give you the answer for
this one.
752
00:47:38 --> 00:47:40
What's the dimension?
753
00:47:40 --> 00:47:45
The dimension of this --
say, this subspace,
754
00:47:45 --> 00:47:50
let me call it D,
all diagonal matrices.
755
00:47:50 --> 00:47:58
The dimension of this subspace
is -- as I write you're working
756
00:47:58 --> 00:48:00
it out -- three.
757
00:48:00 --> 00:48:08
Because here's a matrix in this
-- it's a diagonal matrix.
758
00:48:08 --> 00:48:10
Here's another one.
759
00:48:10 --> 00:48:13
Here's another one.
760
00:48:13 --> 00:48:21
Better make it diagonal,
let me put a seven there.
761
00:48:21 --> 00:48:26
That was not a very great
choice, but it's three diagonal
762
00:48:26 --> 00:48:30
matrices, and I believe that
they're a basis.
763
00:48:30 --> 00:48:35
I believe that those three
matrices are independent and I
764
00:48:35 --> 00:48:41
believe that any diagonal matrix
is a combination of those three.
765
00:48:41 --> 00:48:46
So they span the subspace of
diagonal matrices.
766
00:48:46 --> 00:48:47
Do you see that idea?
767
00:48:47 --> 00:48:52
It's like stretching the idea
from R^n to R^(n by n),
768
00:48:52 --> 00:48:53.96
three by three.
769
00:48:53.96 --> 00:48:58
But the -- we can still add,
we can still multiply by
770
00:48:58 --> 00:49:03
numbers, and we just ignore the
fact that we can multiply two
771
00:49:03 --> 00:49:05
matrices together.
772
00:49:05 --> 00:49:06
OK, thank you.
773
00:49:06 --> 00:49:09
That's lecture ten.