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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
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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,
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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 --
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construct 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
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have to answer those 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
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just look to see how many
vectors I needed in that basis,
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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,
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identifying the pivot
columns, the ones that
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have pivots, the ones
that end up with pivots.
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00:13:53,940 --> 00:13:57,500
But now I -- the pivot columns
I'm interested in are columns
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of 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
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and the 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 --
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to prove that this
is a right answer,
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00:14:39,750 --> 00:14:42,460
but you know it's
the right 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
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the dimension 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 matrix --
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let me make them the three
columns of a matrix by just
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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
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is two dimensional, because
I see a basis for the row
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space, this row and that 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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00:16:16,290 --> 00:16:19,270
But the rank is two, and
therefore the column space
253
00:16:19,270 --> 00:16:21,940
is two dimensional by
this wonderful fact
254
00:16:21,940 --> 00:16:25,100
that the row space and column
space have the same dimension.
255
00:16:25,100 --> 00:16:29,130
And therefore there are only
two pivot columns, not three,
256
00:16:29,130 --> 00:16:34,310
and, those, the three
columns are dependent.
257
00:16:34,310 --> 00:16:35,430
OK.
258
00:16:35,430 --> 00:16:45,470
Now let me bury that error
and talk about the row space.
259
00:16:45,470 --> 00:16:48,690
Well, I'm going to give you the
dimensions of all the spaces.
260
00:16:48,690 --> 00:16:52,480
Because that's
such a nice answer.
261
00:16:52,480 --> 00:16:53,100
OK.
262
00:16:53,100 --> 00:16:56,460
So let me come back here.
263
00:16:56,460 --> 00:16:59,640
So we have this great
fact to establish,
264
00:16:59,640 --> 00:17:06,790
that the row space, its
dimension is also the rank.
265
00:17:06,790 --> 00:17:07,910
What about the null space?
266
00:17:07,910 --> 00:17:10,020
OK.
267
00:17:10,020 --> 00:17:13,630
What's a basis for
the null space?
268
00:17:13,630 --> 00:17:15,970
What's the dimension
of the null space?
269
00:17:15,970 --> 00:17:20,274
Let me, I'll put that answer
up here for the null space.
270
00:17:24,200 --> 00:17:27,849
Well, how have we
constructed the null space?
271
00:17:27,849 --> 00:17:32,260
We took the matrix A, we
did those row operations
272
00:17:32,260 --> 00:17:36,170
to get it into a form
U or, or even further.
273
00:17:36,170 --> 00:17:39,730
We got it into the
reduced form R.
274
00:17:39,730 --> 00:17:43,490
And then we read off
special solutions.
275
00:17:43,490 --> 00:17:44,580
Special solutions.
276
00:17:44,580 --> 00:17:48,110
And every special solution
came from a free variable.
277
00:17:48,110 --> 00:17:50,720
And those special solutions
are in the null space,
278
00:17:50,720 --> 00:17:54,360
and the great thing is
they're a basis for it.
279
00:17:54,360 --> 00:17:59,445
So for the null space, a basis
will be the special solutions.
280
00:18:03,280 --> 00:18:07,250
And there's one for every
free variable, right?
281
00:18:07,250 --> 00:18:11,190
For each free variable, we give
that variable the value one,
282
00:18:11,190 --> 00:18:13,350
the other free variables zero.
283
00:18:13,350 --> 00:18:17,840
We get the pivot variables,
we get a vector in the --
284
00:18:17,840 --> 00:18:20,680
we get a special solution.
285
00:18:20,680 --> 00:18:24,140
So we get altogether
n-r of them,
286
00:18:24,140 --> 00:18:30,340
because that's the
number of free variables.
287
00:18:30,340 --> 00:18:32,890
If we have r --
288
00:18:32,890 --> 00:18:38,490
this is the dimension is r, is
the number of pivot variables.
289
00:18:38,490 --> 00:18:40,810
This is the number
of free variables.
290
00:18:40,810 --> 00:18:44,180
So the beauty is that those
special solutions do form
291
00:18:44,180 --> 00:18:51,380
a basis and tell us immediately
that the dimension of the null
292
00:18:51,380 --> 00:18:55,570
space is n --
293
00:18:55,570 --> 00:19:00,590
I better write this well,
because it's so nice -- n-r.
294
00:19:00,590 --> 00:19:04,910
And do you see the nice thing?
295
00:19:04,910 --> 00:19:08,320
That the two dimensions in
this n dimensional space,
296
00:19:08,320 --> 00:19:12,300
one subspace is r dimensional --
297
00:19:12,300 --> 00:19:15,400
to be proved, that's
the row space.
298
00:19:15,400 --> 00:19:18,680
The other subspace
is n-r dimensional,
299
00:19:18,680 --> 00:19:21,150
that's the null space.
300
00:19:21,150 --> 00:19:25,550
And the two dimensions
like together give n.
301
00:19:25,550 --> 00:19:28,870
The sum of r and n-R is n.
302
00:19:28,870 --> 00:19:31,270
And that's just great.
303
00:19:31,270 --> 00:19:35,720
It's really copying the fact
that we have n variables,
304
00:19:35,720 --> 00:19:39,750
r of them are pivot variables
and n-r are free variables,
305
00:19:39,750 --> 00:19:41,380
and n altogether.
306
00:19:41,380 --> 00:19:41,880
OK.
307
00:19:41,880 --> 00:19:47,090
And now what's the dimension
of this poor misbegotten fourth
308
00:19:47,090 --> 00:19:48,500
subspace?
309
00:19:48,500 --> 00:19:53,700
It's got to be m-r.
310
00:19:53,700 --> 00:20:00,110
The dimension of this left null
space, left out practically,
311
00:20:00,110 --> 00:20:01,540
is m-r.
312
00:20:04,120 --> 00:20:08,690
Well, that's really just
saying that this -- again,
313
00:20:08,690 --> 00:20:15,140
the sum of that plus that
is m, and m is correct,
314
00:20:15,140 --> 00:20:21,180
it's the number of
columns in A transpose.
315
00:20:21,180 --> 00:20:25,810
A transpose is just
as good a matrix as A.
316
00:20:25,810 --> 00:20:29,590
It just happens to be n by m.
317
00:20:29,590 --> 00:20:38,360
It happens to have m columns,
so it will have m variables
318
00:20:38,360 --> 00:20:41,990
when I go to A x
equals 0 and m of them,
319
00:20:41,990 --> 00:20:46,950
and r of them will be pivot
variables and m-r will
320
00:20:46,950 --> 00:20:49,920
be free variables.
321
00:20:49,920 --> 00:20:52,650
A transpose is as
good a matrix as A.
322
00:20:52,650 --> 00:20:57,410
It follows the same rule that
the this plus the dimension --
323
00:20:57,410 --> 00:21:00,830
this dimension plus this
dimension adds up to the number
324
00:21:00,830 --> 00:21:03,130
of columns.
325
00:21:03,130 --> 00:21:06,770
And over here, A
transpose has m columns.
326
00:21:06,770 --> 00:21:09,290
OK.
327
00:21:09,290 --> 00:21:09,790
OK.
328
00:21:09,790 --> 00:21:13,580
So I gave you the easy
answer, the dimensions.
329
00:21:13,580 --> 00:21:21,160
Now can I go back
to check on a basis?
330
00:21:21,160 --> 00:21:25,540
We would like to think
that -- say the row space,
331
00:21:25,540 --> 00:21:29,280
because we've got a basis
for the column space.
332
00:21:29,280 --> 00:21:33,570
The pivot columns give a
basis for the column space.
333
00:21:33,570 --> 00:21:36,840
Now I'm asking you to
look at the row space.
334
00:21:36,840 --> 00:21:40,790
And I -- you could say, OK, I
can produce a basis for the row
335
00:21:40,790 --> 00:21:45,540
space by transposing my
matrix, making those columns,
336
00:21:45,540 --> 00:21:48,670
then doing elimination,
row reduction,
337
00:21:48,670 --> 00:21:54,690
and checking out the pivot
columns in this transposed
338
00:21:54,690 --> 00:21:55,350
matrix.
339
00:21:55,350 --> 00:21:57,760
But that means you
had to do all that row
340
00:21:57,760 --> 00:22:00,750
reduction on A transpose.
341
00:22:00,750 --> 00:22:05,590
It ought to be possible,
if we take a matrix A --
342
00:22:05,590 --> 00:22:08,620
let me take the matrix -- maybe
we had this matrix in the last
343
00:22:08,620 --> 00:22:09,120
lecture.
344
00:22:09,120 --> 00:22:15,530
1 1 1, 2 1 2, 3 2 3, 1 1 1.
345
00:22:21,950 --> 00:22:22,670
OK.
346
00:22:22,670 --> 00:22:24,770
That, that matrix was so easy.
347
00:22:24,770 --> 00:22:29,100
We spotted its pivot columns,
one and two, without actually
348
00:22:29,100 --> 00:22:30,930
doing row reduction.
349
00:22:30,930 --> 00:22:35,070
But now let's do
the job properly.
350
00:22:35,070 --> 00:22:38,980
So I subtract this away
from this to produce a zero.
351
00:22:38,980 --> 00:22:42,810
So one 2 3 1 is fine.
352
00:22:42,810 --> 00:22:47,419
Subtracting that away leaves
me minus 1 -1 0, right?
353
00:22:47,419 --> 00:22:49,960
And subtracting that from the
last row, oh, well that's easy.
354
00:22:53,140 --> 00:22:53,690
OK?
355
00:22:53,690 --> 00:22:56,680
I'm doing row reduction.
356
00:22:56,680 --> 00:23:00,700
Now I've -- the first
column is all set.
357
00:23:00,700 --> 00:23:04,230
The second column I
now see the pivot.
358
00:23:04,230 --> 00:23:07,800
And I can clean up, if I --
359
00:23:07,800 --> 00:23:08,480
actually,
360
00:23:08,480 --> 00:23:09,860
OK.
361
00:23:09,860 --> 00:23:13,070
Why don't I make
the pivot into a 1.
362
00:23:13,070 --> 00:23:18,805
I'll multiply that row through
by by -1, and then I have 1 1.
363
00:23:22,200 --> 00:23:24,370
That was an elementary
operation I'm allowed,
364
00:23:24,370 --> 00:23:26,790
multiply a row by a number.
365
00:23:26,790 --> 00:23:28,150
And now I'll do elimination.
366
00:23:28,150 --> 00:23:31,300
Two of those away from that
will knock this guy out
367
00:23:31,300 --> 00:23:33,120
and make this into a 1.
368
00:23:33,120 --> 00:23:36,560
So that's now a 0 and that's a
369
00:23:36,560 --> 00:23:37,620
OK.
370
00:23:37,620 --> 00:23:39,230
Done.
371
00:23:39,230 --> 00:23:42,470
That's R.
372
00:23:42,470 --> 00:23:45,930
I'm seeing the
identity matrix here.
373
00:23:45,930 --> 00:23:48,530
I'm seeing zeros below.
374
00:23:48,530 --> 00:23:49,990
And I'm seeing F there.
375
00:23:53,151 --> 00:23:53,650
OK.
376
00:23:56,340 --> 00:24:00,110
What about its row space?
377
00:24:00,110 --> 00:24:02,650
What happened to its row space
-- well, what happened --
378
00:24:02,650 --> 00:24:04,610
let me first ask, just
because this is, is --
379
00:24:04,610 --> 00:24:06,780
sometimes something does happen.
380
00:24:06,780 --> 00:24:09,010
Its column space changed.
381
00:24:09,010 --> 00:24:18,820
The column space of R is not
the column space of A, right?
382
00:24:18,820 --> 00:24:22,000
Because 1 1 1 is certainly
in the column space of A
383
00:24:22,000 --> 00:24:26,460
and certainly not in
the column space of R.
384
00:24:26,460 --> 00:24:29,770
I did row operations.
385
00:24:29,770 --> 00:24:33,750
Those row operations
preserve the row space.
386
00:24:33,750 --> 00:24:36,640
So the row, so the column
spaces are different.
387
00:24:36,640 --> 00:24:39,550
Different column spaces,
different column spaces.
388
00:24:45,950 --> 00:24:50,480
But I believe that they
have the same row space.
389
00:24:55,080 --> 00:24:55,880
Same row space.
390
00:25:00,030 --> 00:25:04,040
I believe that the row space of
that matrix and the row space
391
00:25:04,040 --> 00:25:06,210
of this matrix are identical.
392
00:25:06,210 --> 00:25:09,640
They have exactly the
same vectors in them.
393
00:25:09,640 --> 00:25:14,400
Those vectors are vectors
with four components, right?
394
00:25:14,400 --> 00:25:17,940
They're all combinations
of those rows.
395
00:25:17,940 --> 00:25:19,800
Or I believe you
get the same thing
396
00:25:19,800 --> 00:25:22,030
by taking all combinations
of these rows.
397
00:25:24,690 --> 00:25:29,960
And if true, what's a basis?
398
00:25:29,960 --> 00:25:32,430
What's a basis for
the row space of R,
399
00:25:32,430 --> 00:25:42,450
and it'll be a basis for the
row space of the original A,
400
00:25:42,450 --> 00:25:45,120
but it's obviously a basis
for the row space of R.
401
00:25:45,120 --> 00:25:47,840
What's a basis for the
row space of that matrix?
402
00:25:47,840 --> 00:25:48,820
The first two rows.
403
00:25:51,930 --> 00:25:57,550
So a basis for the
row -- so a basis is,
404
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.
405
00:26:15,500 --> 00:26:18,050
Not of A.
406
00:26:18,050 --> 00:26:21,960
Sometimes it's true for
A, but not necessarily.
407
00:26:21,960 --> 00:26:29,280
But R, we definitely have a
matrix here whose row space we
408
00:26:29,280 --> 00:26:32,060
can, we can identify.
409
00:26:32,060 --> 00:26:36,790
The row space is spanned
by the three rows,
410
00:26:36,790 --> 00:26:40,580
but if we want a basis
we want independence.
411
00:26:40,580 --> 00:26:43,380
So out goes row three.
412
00:26:43,380 --> 00:26:46,820
The row space is also spanned
by the first two rows.
413
00:26:46,820 --> 00:26:48,920
This guy didn't
contribute anything.
414
00:26:48,920 --> 00:26:52,706
And of course over here
this 1 2 3 1 in the bottom
415
00:26:52,706 --> 00:26:53,830
didn't contribute anything.
416
00:26:53,830 --> 00:26:56,510
We had it already.
417
00:26:56,510 --> 00:26:58,260
So this, here is a basis.
418
00:26:58,260 --> 00:27:01,260
1 0 1 1 and 0 1 1 0.
419
00:27:04,710 --> 00:27:06,860
I believe those are
in the row space.
420
00:27:06,860 --> 00:27:08,090
I know they're independent.
421
00:27:08,090 --> 00:27:10,680
Why are they in the row space?
422
00:27:10,680 --> 00:27:13,720
Why are those two
vectors in the row space?
423
00:27:13,720 --> 00:27:17,440
Because all those
operations we did,
424
00:27:17,440 --> 00:27:22,680
which started with these rows
and took combinations of them
425
00:27:22,680 --> 00:27:23,900
--
426
00:27:23,900 --> 00:27:28,910
I took this row minus this
row, that gave me something
427
00:27:28,910 --> 00:27:30,780
that's still in the row space.
428
00:27:30,780 --> 00:27:32,400
That's the point.
429
00:27:32,400 --> 00:27:36,760
When I took a row minus a
multiple of another row,
430
00:27:36,760 --> 00:27:38,380
I'm staying in the row space.
431
00:27:38,380 --> 00:27:41,330
The row space is not changing.
432
00:27:41,330 --> 00:27:43,240
My little basis
for it is changing,
433
00:27:43,240 --> 00:27:46,680
and I've ended up with,
sort of the best basis.
434
00:27:49,380 --> 00:27:53,240
If the columns of the identity
matrix are the best basis
435
00:27:53,240 --> 00:28:00,490
for R^3 or R^n, the rows of
this matrix are the best basis
436
00:28:00,490 --> 00:28:02,510
for the row space.
437
00:28:02,510 --> 00:28:06,250
Best in the sense of being
as clean as I can make it.
438
00:28:06,250 --> 00:28:09,310
Starting off with the
identity and then finishing up
439
00:28:09,310 --> 00:28:11,420
with whatever has
to be in there.
440
00:28:11,420 --> 00:28:12,730
OK.
441
00:28:12,730 --> 00:28:16,720
Do you see then that
the dimension is r?
442
00:28:16,720 --> 00:28:23,330
For sure, because we've got
r pivots, r non-zero rows.
443
00:28:23,330 --> 00:28:26,420
We've got the right
number of vectors, r.
444
00:28:26,420 --> 00:28:30,080
They're in the row space,
they're independent.
445
00:28:30,080 --> 00:28:31,600
That's it.
446
00:28:31,600 --> 00:28:34,160
They are a basis
for the row space.
447
00:28:34,160 --> 00:28:36,220
And we can even pin
that down further.
448
00:28:36,220 --> 00:28:41,340
How do I know that every
row of A is a combination?
449
00:28:41,340 --> 00:28:45,110
How do I know they
span the row space?
450
00:28:45,110 --> 00:28:48,110
Well, somebody says, I've
got the right number of them,
451
00:28:48,110 --> 00:28:48,740
so they must.
452
00:28:48,740 --> 00:28:49,870
But -- and that's true.
453
00:28:49,870 --> 00:28:54,400
But let me just say, how
do I know that this row is
454
00:28:54,400 --> 00:28:57,300
a combination of these?
455
00:28:57,300 --> 00:29:01,870
By just reversing the
steps of row reduction.
456
00:29:01,870 --> 00:29:07,760
If I just reverse the steps and
go from A -- from R back to A,
457
00:29:07,760 --> 00:29:10,010
then what do I, what I doing?
458
00:29:10,010 --> 00:29:12,030
I'm starting with
these rows, I'm
459
00:29:12,030 --> 00:29:15,730
taking combinations of them.
460
00:29:15,730 --> 00:29:19,670
After a couple of steps,
undoing the subtractions
461
00:29:19,670 --> 00:29:22,710
that I did before, I'm
back to these rows.
462
00:29:22,710 --> 00:29:25,500
So these rows are
combinations of those rows.
463
00:29:25,500 --> 00:29:28,020
Those rows are
combinations of those rows.
464
00:29:28,020 --> 00:29:31,740
The two row spaces are the same.
465
00:29:31,740 --> 00:29:34,830
The bases are the same.
466
00:29:34,830 --> 00:29:38,540
And the natural
basis is this guy.
467
00:29:38,540 --> 00:29:41,530
Is that all right
for the row space?
468
00:29:41,530 --> 00:29:45,030
The row space is
sitting there in R
469
00:29:45,030 --> 00:29:47,750
in its cleanest possible form.
470
00:29:47,750 --> 00:29:48,800
OK.
471
00:29:48,800 --> 00:29:56,875
Now what about the fourth guy,
the null space of A transpose?
472
00:29:59,630 --> 00:30:03,440
First of all, why do I call
that the left null space?
473
00:30:03,440 --> 00:30:11,610
So let me save that
and bring that down.
474
00:30:11,610 --> 00:30:14,050
OK.
475
00:30:14,050 --> 00:30:20,940
So the fourth space is the
null space of A transpose.
476
00:30:23,700 --> 00:30:27,200
So it has in it vectors,
let me call them y,
477
00:30:27,200 --> 00:30:30,470
so that A transpose y equals 0.
478
00:30:30,470 --> 00:30:35,360
If A transpose y
equals 0, then y
479
00:30:35,360 --> 00:30:39,890
is in the null space of
A transpose, of course.
480
00:30:39,890 --> 00:30:47,770
So this is a matrix times
a column equaling zero.
481
00:30:50,480 --> 00:30:56,550
And now, because I want
y to sit on the left
482
00:30:56,550 --> 00:31:00,200
and I want A instead
of A transpose,
483
00:31:00,200 --> 00:31:03,380
I'll just transpose
that equation.
484
00:31:03,380 --> 00:31:06,180
Can I just transpose that?
485
00:31:06,180 --> 00:31:10,705
On the right, it makes
the zero vector lie down.
486
00:31:14,590 --> 00:31:21,510
And on the left, it's a
product, A, A transpose times y.
487
00:31:21,510 --> 00:31:24,510
If I take the transpose, then
they come in opposite order,
488
00:31:24,510 --> 00:31:25,540
right?
489
00:31:25,540 --> 00:31:30,156
So that's y transpose times
A transpose transpose.
490
00:31:33,210 --> 00:31:35,540
But nobody's going to
leave it like that.
491
00:31:35,540 --> 00:31:39,620
That's -- A transpose
transpose is just A, of course.
492
00:31:39,620 --> 00:31:43,360
When I transposed A
transpose I got back to A.
493
00:31:43,360 --> 00:31:45,870
Now do you see what I have now?
494
00:31:45,870 --> 00:31:51,820
I have a row
vector, y transpose,
495
00:31:51,820 --> 00:31:58,540
multiplying A, and
multiplying from the left.
496
00:31:58,540 --> 00:32:02,180
That's why I call it
the left null space.
497
00:32:02,180 --> 00:32:05,590
But by making it --
putting it on the left,
498
00:32:05,590 --> 00:32:09,720
I had to make it into a row
instead of a column vector,
499
00:32:09,720 --> 00:32:15,250
and so my convention is
I usually don't do that.
500
00:32:15,250 --> 00:32:18,960
I usually stay with A
transpose y equals 0.
501
00:32:18,960 --> 00:32:20,290
OK.
502
00:32:20,290 --> 00:32:27,990
And you might ask, how do we
get a basis -- or I might ask,
503
00:32:27,990 --> 00:32:31,470
how do we get a basis
for this fourth space,
504
00:32:31,470 --> 00:32:32,610
this left null space?
505
00:32:36,150 --> 00:32:36,720
OK.
506
00:32:36,720 --> 00:32:40,050
I'll do it in the example.
507
00:32:40,050 --> 00:32:43,350
As always -- not that one.
508
00:32:49,880 --> 00:32:53,310
The left null space is not
jumping out at me here.
509
00:32:57,060 --> 00:33:00,220
I know which are the
free variables --
510
00:33:00,220 --> 00:33:03,890
the special solutions, but those
are special solutions to A x
511
00:33:03,890 --> 00:33:06,230
equals zero, and now I'm
looking at A transpose,
512
00:33:06,230 --> 00:33:08,420
and I'm not seeing it here.
513
00:33:08,420 --> 00:33:12,760
So -- but somehow you feel that
the work that you did which
514
00:33:12,760 --> 00:33:19,310
simplified A to R should have
revealed the left null space
515
00:33:19,310 --> 00:33:20,760
too.
516
00:33:20,760 --> 00:33:25,970
And it's slightly less
immediate, but it's there.
517
00:33:25,970 --> 00:33:31,010
So from A to R, I
took some steps,
518
00:33:31,010 --> 00:33:34,880
and I guess I'm interested
in what were those steps,
519
00:33:34,880 --> 00:33:36,600
or what were all
of them together.
520
00:33:36,600 --> 00:33:43,160
I don't -- I'm not interested in
what particular ones they were.
521
00:33:43,160 --> 00:33:45,910
I'm interested in what
was the whole matrix that
522
00:33:45,910 --> 00:33:51,620
took me from A to R.
523
00:33:51,620 --> 00:33:52,880
How would you find that?
524
00:33:55,640 --> 00:33:58,970
Do you remember
Gauss-Jordan, where you
525
00:33:58,970 --> 00:34:02,260
tack on the identity matrix?
526
00:34:02,260 --> 00:34:03,810
Let's do that again.
527
00:34:03,810 --> 00:34:06,640
So I, I'll do it above, here.
528
00:34:06,640 --> 00:34:13,620
So this is now, this
is now the idea of --
529
00:34:13,620 --> 00:34:17,570
I take the matrix
A, which is m by n.
530
00:34:20,389 --> 00:34:22,639
In Gauss-Jordan, when
we saw him before --
531
00:34:22,639 --> 00:34:25,150
A was a square
invertible matrix and we
532
00:34:25,150 --> 00:34:27,900
were finding its inverse.
533
00:34:27,900 --> 00:34:29,409
Now the matrix isn't square.
534
00:34:29,409 --> 00:34:32,360
It's probably rectangular.
535
00:34:32,360 --> 00:34:36,989
But I'll still tack on the
identity matrix, and of course
536
00:34:36,989 --> 00:34:42,179
since these have length
m it better be m by m.
537
00:34:42,179 --> 00:34:49,374
And now I'll do the reduced row
echelon form of this matrix.
538
00:34:52,480 --> 00:34:56,120
And what do I get?
539
00:35:01,640 --> 00:35:04,950
The reduced row echelon form
starts with these columns,
540
00:35:04,950 --> 00:35:11,680
starts with the first columns,
works like mad, and produces R.
541
00:35:11,680 --> 00:35:13,920
Of course, still that
same size, m by n.
542
00:35:13,920 --> 00:35:15,700
And we did it before.
543
00:35:15,700 --> 00:35:19,140
And then whatever
it did to get R,
544
00:35:19,140 --> 00:35:22,520
something else is
going to show up here.
545
00:35:22,520 --> 00:35:26,690
Let me call it E, m by m.
546
00:35:26,690 --> 00:35:30,170
It's whatever -- do you see
that E is just going to contain
547
00:35:30,170 --> 00:35:32,640
a record of what we did?
548
00:35:32,640 --> 00:35:38,780
We did whatever it took
to get A to become R.
549
00:35:38,780 --> 00:35:40,930
And at the same time,
we were doing it
550
00:35:40,930 --> 00:35:44,410
to the identity matrix.
551
00:35:44,410 --> 00:35:48,590
So we started with the identity
matrix, we buzzed along.
552
00:35:48,590 --> 00:35:51,190
So we took some --
553
00:35:51,190 --> 00:35:55,860
all this row reduction amounted
to multiplying on the left
554
00:35:55,860 --> 00:36:00,040
by some matrix, some series
of elementary matrices
555
00:36:00,040 --> 00:36:05,780
that altogether gave us one
matrix, and that matrix is E.
556
00:36:05,780 --> 00:36:11,450
So all this row reduction stuff
amounted to multiplying by E.
557
00:36:11,450 --> 00:36:13,120
How do I know that?
558
00:36:13,120 --> 00:36:16,680
It certainly amounted to
multiply it by something.
559
00:36:16,680 --> 00:36:21,660
And that something took I to
E, so that something was E.
560
00:36:21,660 --> 00:36:29,790
So now look at the
first part, E A is R.
561
00:36:29,790 --> 00:36:31,520
No big deal.
562
00:36:31,520 --> 00:36:38,710
All I've said is that the row
reduction steps that we all
563
00:36:38,710 --> 00:36:45,680
know -- well, taking A to
R, are in a, in some matrix,
564
00:36:45,680 --> 00:36:49,340
and I can find out what that
matrix is by just tacking I
565
00:36:49,340 --> 00:36:51,810
on and seeing what comes out.
566
00:36:51,810 --> 00:36:54,570
What comes out is E.
567
00:36:54,570 --> 00:36:58,400
Let's just review the
invertible square case.
568
00:36:58,400 --> 00:37:00,860
What happened then?
569
00:37:00,860 --> 00:37:04,350
Because I was interested
in it in chapter two also.
570
00:37:04,350 --> 00:37:08,260
When A was square and
invertible, I took A I,
571
00:37:08,260 --> 00:37:10,480
I did row, row elimination.
572
00:37:10,480 --> 00:37:12,210
And what was the
R that came out?
573
00:37:12,210 --> 00:37:14,770
It was I.
574
00:37:14,770 --> 00:37:24,530
So in chapter two, in
chapter two, R was I.
575
00:37:24,530 --> 00:37:27,310
The row the, the
reduced row echelon
576
00:37:27,310 --> 00:37:31,730
form of a nice invertible
square matrix is the identity.
577
00:37:31,730 --> 00:37:41,510
So if R was I in that case, then
E was -- then E was A inverse,
578
00:37:41,510 --> 00:37:44,430
because E A is I.
579
00:37:44,430 --> 00:37:45,200
Good.
580
00:37:45,200 --> 00:37:48,540
That's, that was good and easy.
581
00:37:48,540 --> 00:37:52,990
Now what I'm saying is
that there still is an E.
582
00:37:52,990 --> 00:37:55,740
It's not A inverse any more,
because A is rectangular.
583
00:37:55,740 --> 00:37:57,730
It hasn't got an inverse.
584
00:37:57,730 --> 00:38:05,010
But there is still some matrix
E that connected this to this --
585
00:38:05,010 --> 00:38:09,260
oh, I should have figured
out in advanced what it was.
586
00:38:09,260 --> 00:38:11,810
Shoot.
587
00:38:11,810 --> 00:38:12,800
I didn't --
588
00:38:12,800 --> 00:38:16,620
I did those steps and sort
of erased as I went --
589
00:38:16,620 --> 00:38:20,370
and, I should have done
them to the identity too.
590
00:38:20,370 --> 00:38:22,500
Can I do that?
591
00:38:22,500 --> 00:38:23,460
Can I do that?
592
00:38:23,460 --> 00:38:26,140
I'll keep the identity matrix,
like I'm supposed to do,
593
00:38:26,140 --> 00:38:29,750
and I'll do the same operations
on it, and see what I end up
594
00:38:29,750 --> 00:38:30,340
with.
595
00:38:30,340 --> 00:38:31,300
OK.
596
00:38:31,300 --> 00:38:32,810
So I'm starting
with the identity --
597
00:38:32,810 --> 00:38:40,511
which I'll write in light,
light enough, but --
598
00:38:40,511 --> 00:38:41,010
OK.
599
00:38:41,010 --> 00:38:42,520
What did I do?
600
00:38:42,520 --> 00:38:45,950
I subtracted that row from that
one and that row from that one.
601
00:38:45,950 --> 00:38:47,950
OK, I'll do that
to the identity.
602
00:38:47,950 --> 00:38:52,990
So I subtract that first row
from row two and row three.
603
00:38:52,990 --> 00:38:55,310
Good.
604
00:38:55,310 --> 00:38:56,950
Then I think I multiplied --
605
00:38:56,950 --> 00:38:57,620
Do you remember?
606
00:38:57,620 --> 00:39:01,890
I multiplied row
two by minus one.
607
00:39:01,890 --> 00:39:05,270
Let me just do that.
608
00:39:05,270 --> 00:39:06,610
Then what did I do?
609
00:39:06,610 --> 00:39:14,880
I subtracted two of row
two away from row one.
610
00:39:14,880 --> 00:39:15,770
I better do that.
611
00:39:15,770 --> 00:39:17,720
Subtract two of
this away from this.
612
00:39:17,720 --> 00:39:24,180
That's minus one, two of these
away leaves a plus 2 and 0.
613
00:39:24,180 --> 00:39:28,440
I believe that's E.
614
00:39:28,440 --> 00:39:35,870
The way to check is to see,
multiply that E by this A,
615
00:39:35,870 --> 00:39:37,420
just to see did I do it right.
616
00:39:40,570 --> 00:39:49,285
So I believe E was -1 2
0, 1 -1 0, and -1 0 1.
617
00:39:53,020 --> 00:39:53,520
OK.
618
00:39:53,520 --> 00:39:58,030
That's my E, that's
my A, and that's R.
619
00:39:58,030 --> 00:40:00,110
All right.
620
00:40:00,110 --> 00:40:02,950
All I'm struggling
to do is right,
621
00:40:02,950 --> 00:40:09,660
the reason I wanted this blasted
E was so that I could figure
622
00:40:09,660 --> 00:40:14,570
out the left null space,
not only its dimension,
623
00:40:14,570 --> 00:40:17,020
which I know --
624
00:40:17,020 --> 00:40:19,427
actually, what is the dimension
of the left null space?
625
00:40:19,427 --> 00:40:20,260
So here's my matrix.
626
00:40:23,180 --> 00:40:24,465
What's the rank of the matrix?
627
00:40:27,560 --> 00:40:30,640
And the dimension of the null
-- of the left null space is
628
00:40:30,640 --> 00:40:33,470
supposed to be m-r.
629
00:40:33,470 --> 00:40:34,930
It's 3 -2, 1.
630
00:40:34,930 --> 00:40:39,090
I believe that the left null
space is one dimensional.
631
00:40:39,090 --> 00:40:42,990
There is one combination
of those three rows
632
00:40:42,990 --> 00:40:46,840
that produces the zero row.
633
00:40:46,840 --> 00:40:52,270
There's a basis -- a basis for
the left null space has only
634
00:40:52,270 --> 00:40:54,200
got one vector in it.
635
00:40:54,200 --> 00:40:55,790
And what is that vector?
636
00:40:55,790 --> 00:40:58,710
It's here in the last row of E.
637
00:40:58,710 --> 00:41:01,400
But we could have
seen it earlier.
638
00:41:01,400 --> 00:41:05,220
What combination of those
rows gives the zero row?
639
00:41:05,220 --> 00:41:09,110
-1 of that plus one of that.
640
00:41:09,110 --> 00:41:14,460
So a basis for the left
null space of this matrix --
641
00:41:14,460 --> 00:41:18,350
I'm looking for combinations
of rows that give the zero row
642
00:41:18,350 --> 00:41:22,080
if I'm looking at
the left null space.
643
00:41:22,080 --> 00:41:24,980
For the null space, I'm looking
at combinations of columns
644
00:41:24,980 --> 00:41:26,780
to get the zero column.
645
00:41:26,780 --> 00:41:29,760
Now I'm looking at combinations
of these three rows
646
00:41:29,760 --> 00:41:34,370
to get the zero row, and of
course there is my zero row,
647
00:41:34,370 --> 00:41:37,160
and here is my vector
that produced it.
648
00:41:37,160 --> 00:41:40,010
-1 of that row and one of that
649
00:41:40,010 --> 00:41:40,510
row.
650
00:41:40,510 --> 00:41:41,650
Obvious.
651
00:41:41,650 --> 00:41:42,210
OK.
652
00:41:42,210 --> 00:41:45,940
So in that example -- and
actually in all examples,
653
00:41:45,940 --> 00:41:51,310
we have seen how to produce a
basis for the left null space.
654
00:41:51,310 --> 00:41:54,800
I won't ask you that
all the time, because --
655
00:41:54,800 --> 00:41:58,790
it didn't come out
immediately from R.
656
00:41:58,790 --> 00:42:03,850
We had to keep track of E
for that left null space.
657
00:42:03,850 --> 00:42:07,520
But at least it didn't require
us to transpose the matrix
658
00:42:07,520 --> 00:42:10,220
and start all over again.
659
00:42:10,220 --> 00:42:12,390
OK, those are the
four subspaces.
660
00:42:12,390 --> 00:42:15,520
Can I review them?
661
00:42:15,520 --> 00:42:18,950
The row space and the
null space are in R^n.
662
00:42:18,950 --> 00:42:22,220
Their dimensions add to n.
663
00:42:22,220 --> 00:42:27,470
The column space and the
left null space are in R^m,
664
00:42:27,470 --> 00:42:30,900
and their dimensions add to m.
665
00:42:30,900 --> 00:42:33,700
OK.
666
00:42:33,700 --> 00:42:39,080
So let me close
these last minutes
667
00:42:39,080 --> 00:42:49,620
by pushing you a little bit more
to a new type of vector space.
668
00:42:49,620 --> 00:42:53,230
All our vector spaces, all the
ones that we took seriously,
669
00:42:53,230 --> 00:43:01,570
have been subspaces of some real
three or n dimensional space.
670
00:43:01,570 --> 00:43:04,880
Now I'm going to write
down another vector
671
00:43:04,880 --> 00:43:06,720
space, a new vector space.
672
00:43:14,120 --> 00:43:18,610
Say all three by three matrices.
673
00:43:26,420 --> 00:43:27,990
My matrices are the vectors.
674
00:43:31,830 --> 00:43:33,240
Is that all right?
675
00:43:33,240 --> 00:43:34,130
I'm just naming them.
676
00:43:34,130 --> 00:43:36,570
You can put quotes
around vectors.
677
00:43:36,570 --> 00:43:40,050
Every three by three matrix
is one of my vectors.
678
00:43:40,050 --> 00:43:43,060
Now how I entitled to
call those things vectors?
679
00:43:43,060 --> 00:43:46,380
I mean, they look very
much like matrices.
680
00:43:46,380 --> 00:43:49,980
But they are vectors in my
vector space because they obey
681
00:43:49,980 --> 00:43:50,640
the rules.
682
00:43:50,640 --> 00:43:55,850
All I'm supposed to be able to
do with vectors is add them --
683
00:43:55,850 --> 00:43:58,190
I can add matrices --
684
00:43:58,190 --> 00:44:01,580
I'm supposed to be able to
multiply them by scalar numbers
685
00:44:01,580 --> 00:44:09,290
like seven -- well, I can
multiply a matrix by And that
686
00:44:09,290 --> 00:44:11,960
-- and I can take
combinations of matrices,
687
00:44:11,960 --> 00:44:15,060
I can take three of one
matrix minus five of another
688
00:44:15,060 --> 00:44:15,990
matrix.
689
00:44:15,990 --> 00:44:21,260
And those combinations, there's
a zero matrix, the matrix
690
00:44:21,260 --> 00:44:23,570
that has all zeros in it.
691
00:44:23,570 --> 00:44:26,050
If I add that to another
matrix, it doesn't change it.
692
00:44:26,050 --> 00:44:26,960
All the good stuff.
693
00:44:26,960 --> 00:44:30,270
If I multiply a matrix by
one it doesn't change it.
694
00:44:30,270 --> 00:44:32,800
All those eight rules
for a vector space
695
00:44:32,800 --> 00:44:37,460
that we never wrote down,
all easily satisfied.
696
00:44:37,460 --> 00:44:41,150
So now we have a different --
697
00:44:41,150 --> 00:44:46,450
now of course you can say you
can multiply those matrices.
698
00:44:46,450 --> 00:44:47,240
I don't care.
699
00:44:47,240 --> 00:44:50,070
For the moment, I'm only
thinking of these matrices
700
00:44:50,070 --> 00:44:57,960
as forming a vector space --
so I only doing A plus B and c
701
00:44:57,960 --> 00:44:59,350
times A.
702
00:44:59,350 --> 00:45:03,050
I'm not interested
in A B for now.
703
00:45:06,580 --> 00:45:09,340
The fact that I
can multiply is not
704
00:45:09,340 --> 00:45:13,721
relevant to th-
to a vector space.
705
00:45:13,721 --> 00:45:14,220
OK.
706
00:45:14,220 --> 00:45:15,636
So I have three
by three matrices.
707
00:45:18,180 --> 00:45:21,600
And how about subspaces?
708
00:45:21,600 --> 00:45:26,420
What's -- tell me a subspace
of this matrix space.
709
00:45:26,420 --> 00:45:30,270
Let me call this matrix space M.
710
00:45:30,270 --> 00:45:34,270
That's my matrix space, my space
of all three by three matrices.
711
00:45:34,270 --> 00:45:37,730
Tell me a subspace of it.
712
00:45:37,730 --> 00:45:40,670
What about the upper
triangular matrices?
713
00:45:40,670 --> 00:45:41,260
OK.
714
00:45:41,260 --> 00:45:43,330
So subspaces.
715
00:45:43,330 --> 00:45:50,660
Subspaces of M.
716
00:45:50,660 --> 00:45:53,675
All, all upper
triangular matrices.
717
00:46:00,370 --> 00:46:01,750
Another subspace.
718
00:46:01,750 --> 00:46:03,085
All symmetric matrices.
719
00:46:11,610 --> 00:46:13,730
The intersection
of two subspaces
720
00:46:13,730 --> 00:46:15,090
is supposed to be a subspace.
721
00:46:15,090 --> 00:46:20,310
We gave a little effort
to the proof of that fact.
722
00:46:20,310 --> 00:46:23,150
If I look at the matrices
that are in this subspace --
723
00:46:23,150 --> 00:46:26,420
they're symmetric, and
they're also in this subspace,
724
00:46:26,420 --> 00:46:31,280
they're upper triangular,
what do they look like?
725
00:46:31,280 --> 00:46:33,550
Well, if they're
symmetric but they
726
00:46:33,550 --> 00:46:35,660
have zeros below the
diagonal, they better
727
00:46:35,660 --> 00:46:38,650
have zeros above the
diagonal, so the intersection
728
00:46:38,650 --> 00:46:40,285
would be diagonal matrices.
729
00:46:44,820 --> 00:46:48,430
That's another subspace,
smaller than those.
730
00:46:50,970 --> 00:46:53,740
How can I use the word smaller?
731
00:46:53,740 --> 00:46:56,390
Well, I'm now entitled
to use the word smaller.
732
00:46:56,390 --> 00:47:00,410
I mean, well, one way
to say is, OK, these
733
00:47:00,410 --> 00:47:02,570
are contained in those.
734
00:47:02,570 --> 00:47:05,210
These are contained in those.
735
00:47:05,210 --> 00:47:09,205
But more precisely, I could give
the dimension of these spaces.
736
00:47:11,710 --> 00:47:14,740
So I could -- we can compute
-- let's compute it next time,
737
00:47:14,740 --> 00:47:17,870
the dimension of all upper
-- of the subspace of upper
738
00:47:17,870 --> 00:47:20,490
triangular three
by three matrices.
739
00:47:20,490 --> 00:47:23,790
The dimension of symmetric
three by three matrices.
740
00:47:23,790 --> 00:47:27,630
The dimension of diagonal
three by three matrices.
741
00:47:27,630 --> 00:47:29,690
Well, to produce
dimension, that means
742
00:47:29,690 --> 00:47:33,370
I'm supposed to produce
a basis, and then
743
00:47:33,370 --> 00:47:37,230
I just count how many vecto-
how many I needed in the basis.
744
00:47:37,230 --> 00:47:39,880
Let me give you the
answer for this one.
745
00:47:39,880 --> 00:47:41,430
What's the dimension?
746
00:47:41,430 --> 00:47:44,610
The dimension of this
-- say, this subspace,
747
00:47:44,610 --> 00:47:47,520
let me call it D, all
diagonal matrices.
748
00:47:47,520 --> 00:47:54,610
The dimension of
this subspace is --
749
00:47:54,610 --> 00:47:57,610
as I write you're
working it out --
750
00:47:57,610 --> 00:47:58,970
three.
751
00:47:58,970 --> 00:48:09,150
Because here's a matrix in
this -- it's a diagonal matrix.
752
00:48:09,150 --> 00:48:10,040
Here's another one.
753
00:48:15,560 --> 00:48:16,610
Here's another one.
754
00:48:20,350 --> 00:48:22,755
Better make it diagonal,
let me put a seven there.
755
00:48:25,970 --> 00:48:28,130
That was not a
very great choice,
756
00:48:28,130 --> 00:48:31,020
but it's three
diagonal matrices,
757
00:48:31,020 --> 00:48:33,950
and I believe that
they're a basis.
758
00:48:33,950 --> 00:48:37,190
I believe that those three
matrices are independent
759
00:48:37,190 --> 00:48:40,680
and I believe that
any diagonal matrix is
760
00:48:40,680 --> 00:48:42,220
a combination of those three.
761
00:48:42,220 --> 00:48:47,310
So they span the subspace
of diagonal matrices.
762
00:48:47,310 --> 00:48:49,040
Do you see that idea?
763
00:48:49,040 --> 00:48:55,170
It's like stretching the
idea from R^n to R^(n by n),
764
00:48:55,170 --> 00:48:57,360
three by three.
765
00:48:57,360 --> 00:49:02,310
But the -- we can still add, we
can still multiply by numbers,
766
00:49:02,310 --> 00:49:06,370
and we just ignore the fact that
we can multiply two matrices
767
00:49:06,370 --> 00:49:07,550
together.
768
00:49:07,550 --> 00:49:09,380
OK, thank you.
769
00:49:09,380 --> 00:49:11,930
That's lecture ten.