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OK.
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Uh this is the review lecture
for the first part of the
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course, the Ax=b part of the
course.
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And the exam will emphasize
chapter three.
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Because those are the --0
chapter three was about the
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rectangular matrices where we
had null spaces and null spaces
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of A transpose,
and ranks, and all the things
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that are so clear when the
matrix is square and invertible,
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they became things to think
about for rectangular matrices.
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So, and vector spaces and
subspaces and above all those
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four subspaces.
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OK, I'm thinking to start at
least I'll just look at old
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exams, read out questions,
write on the board what I need
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to and we can see what the
answers are.
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The first one I see is one I
can just read out.
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Well, I'll write a little.
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Suppose u, v and w are nonzero
vectors in R^7.
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What are the possible -- they
span a -- a vector space.
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They span a subspace of R^7,
and what are the possible
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dimensions?
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So that's a straightforward
question, what are the possible
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dimensions of the subspace
spanned by u,
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v and w?
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OK, one, two,
or three, right.
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One, two or three.
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Couldn't be more because we've
only got three vectors,
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and couldn't be zero because --
because I told you the vectors
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were nonzero.
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Otherwise if I allowed the
possibility that those were all
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the zero vector --
then the zero-dimensional
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subspace would have been in
there.
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OK.
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Now can I jump to a more
serious question?
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OK.
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We have a five by three matrix.
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And I'm calling it U.
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I'm saying it's in echelon
form.
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And it has three pivots,
r=3.
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Three pivots.
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OK.
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First question,
what's the null space?
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What's the null space of this
matrix U, so this matrix is five
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by three, and I find it helpful
to just see visually what five
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by three means,
what that shape is.
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Three columns.
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Three columns in U then,
five rows, three pivots,
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and what's the null space?
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The null space of U is -- and
it asks for a spec-of course I'm
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looking for an answer that isn't
just the definition of the null
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space, but is the null space of
this matrix, with this
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information.
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And what is it?
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It's only the zero vector.
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Because we're told that the
rank is three,
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so those three columns must be
independent, no combination --
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of those columns is the zero
vector except -- so the only
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thing in this null space is the
zero vector, and I --
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I could even say what that
vector is, zero,
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zero, zero.
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That's OK.
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So that's what's in the null
space.
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All right?
let me go on with -- this
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question has multiple parts.
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What's the -- oh now it asks
you about a ten by three matrix,
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B, which is the matrix U and
two U.
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It actually -- I would probably
be writing R -- and maybe I
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should be writing R here now.
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This exam goes back a few years
when I emphasized U more than R.
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Now, what's the echelon form
for that matrix?
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So the echelon form,
what's the rank,
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and what's the echelon form?
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Let's suppose this is in
reduced echelon form,
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so that I could be using the
letter R.
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So I'll ask for the reduced row
echelon form so imagine that
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these are --
U is in reduced row echelon
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form but now I've doubled the
height of the matrix,
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what will happen when we do row
reduction?
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What row reduction will take us
to what matrix here?
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So you start doing elimination.
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You're doing elimination on
single rows.
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But of course we're allowed to
think of blocks.
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So what, well,
what's the answer look like?
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U and z- or R -- let's -- I'll
stay with this letter U but I'm
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really thinking it's in reduced
form, and zero.
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OK.
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Fine.
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Then it asks oh,
further, it asks about this
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matrix.
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U, U, U, and zero.
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OK, what's the echelon form of
this?
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So it's just like practice in
thinking through what would row
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elimination, what would row
reduction do.
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Have I thought this through,
so what -- what are we -- if we
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start doing elimination,
basically we're going to
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subtract these rows from these
--
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so it's going to take us to U,
U, zero, and minus U,
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I guess, right?
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Take the thing all the way to R
-- let's suppose U is really R.
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Suppose that we're really going
for the reduced row echelon
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form.
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Then would we stop there?
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No.
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We would clean out,
we would -- we could use this
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to -- is that right,
can I so I took this row --
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these rows away from these to
get there.
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Now I take these rows away from
these, so that gives me zero.
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There.
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And now what more would I do if
I'm really shooting for R,
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the reduced row echelon form?
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I would -- then I want plus
ones in the pivot,
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so I would multiply through by
minus one to get plus there.
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So essentially I'm seeing
reduced row echelon form there
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and there, and there's just one
little twist still to go.
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Do you see what that final
twist might be?
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To have if -- if U is in
reduced row echelon form and now
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I'm looking at U,
U, there's one little step to
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take, this isn't like a big deal
at all, but -- but if I really
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want this to be in reduced form,
what would I still -- might I
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still have to do?
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I might have some zero rows
here, I might have some zero
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rows here that strictly should
move to the bottom.
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Well, I'm not going to make a
project out of that.
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OK.
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What's the rank of that matrix?
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What's the rank of this matrix
C?
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Given that I know that the
original U has rank three,
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what's the rank of this guy?
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Six, right.
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That has rank six,
I can tell.
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What was -- what's the rank of
this B, while -- while we're at
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it?
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The rank of B,
is that six or three?
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Three is right.
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Three is right.
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Because we actually got it to
where we could just see three
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pivots.
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OK, and oh, now finally this
easy one, what's the dimension
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of the null space -- of the null
space of C transpose?
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Oh, boy.
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OK, so what do I -- if I want
the dimension of a null space,
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I want to know the size of the
matrix --
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so what's the size of the
matrix C?
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It looks like it's ten by six,
is it?
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Ten by six, so C is ten by six,
so m is ten,
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so C has ten rows,
C transpose has ten columns,
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so there are ten columns there.
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So how many free variables have
I got, once I --
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if I start with the ten columns
in C transpose,
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that's the m for the original
C.
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And what do I subtract off?
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Six.
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Because we said that was the
rank.
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So I'm left with four.
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Thanks.
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OK.
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So I think that's the right
answer --
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the dimension of the null space
of C transpose would be four.
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Right.
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OK.
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Yeah.
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OK.
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So that's one question,
at least it brought in some --
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some of the dimension counts.
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OK.
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Here's another type of
question.
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I give you an equation,
Ax equals two four two.
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And I give you the complete
solution.
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But I don't give you the
matrix.
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And another -- there's another
vector, zero,
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zero, one.
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OK.
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All right.
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My first question is what's the
dimension of the row space?
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Of the matrix A?
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So the main thing that you want
to get from this question is
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that a question could start this
way.
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Sort of backward way.
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By giving you the answer and
not telling you what the problem
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is.
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But we can get a lot of
information, and sometimes we
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can get complete information
about that matrix A.
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OK.
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So what's the dimension of the
row space of A?
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What's the rank?
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Tell me about what's the size
of the matrix,
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yeah, just --
These are the things we want to
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think about, what's the --
what's the shape of the matrix,
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first of all?
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It certainly has three rows.
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But is it a- is it three by
three?
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So the (x)-s that it multiplies
have three components,
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so that --1
does the matrix have three
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columns also?
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Yes.
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So I'm seeing the same length
in b, three, and also in x.
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So A is a three by three
matrix.
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And what's its rank?
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Its rank -- tell me something
about its null space,
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I heard the right answer for
the rank, the rank is one in
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this case.
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Why?
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Because the dimension of the
null space, so the dimension of
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the null space of A is from
knowing that that's the complete
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solution, it's two.
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I'm seeing two vectors here,
and they're independent in the
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null space of A,
because they have to be in the
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null space of A if I'm allowed
to throw into the solution any
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amount of those vectors,
that tells me that's the null
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space part then.
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So the dimension of the null
space is two,
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and then I -- of course I know
the dimensions of all the --
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four subspaces.
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Now actually it asks what's the
matrix?
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Well, what's the matrix in this
case?
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Do we want to -- shall I try to
figure that out?
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Sure.
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Let's -- you'd like me to do
it, OK.
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So what about the matrix,
or let me at least start it,
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OK.
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If A times this x gives two,
four, two, what does that tell
221
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me about the matrix A?
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If A times that x solves that
equation then it tells me that
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the first column is -- the first
column of A is -- one,
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two, one, right.
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The first column of A has to be
one, two, one,
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because if I multiply by x,
that's going to multiply just
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the first column,
and give me two,
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four, two.
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And then I've got two more
columns to find,
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and what information have I got
to find them with?
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I've got the null space.
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So the fact that this is in the
null space, what does that tell
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me about the matrix?
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A matrix that has zero,
zero, one in its null space?
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That tells me that the last
column of the matrix is zeroes.
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Because this is in the null
space, the last column has to be
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zeroes.
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And because this is in the null
space, what's the second column?
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Well, this in the null space
means that if I multiply A by
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that vector I must be getting
zeroes, so I think that better
241
00:15:38 --> 00:15:41
be minus one,
minus two, and minus one.
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OK.
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That's a type of question that
just brings out the information
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that's in that complete
solution.
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OK.
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00:15:49 --> 00:15:56
And then actually I go on to
ask what vectors -- for what
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00:15:56 --> 00:16:00
vectors B can Ax=b be solved?
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00:16:00 --> 00:16:08
Ax=b can be solved if what --
so I'm looking for a condition
249
00:16:08 --> 00:16:10
on b, if any.
250
00:16:10 --> 00:16:17
Can it be solved for every
right-hand side b?
251
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No.
252
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Definitely not.
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00:15:21 --> 00:14:08
When could it be solved?
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00:14:08 --> 00:11:35
Well, what's the ---- I
actually say in this in the
255
00:11:35 --> 00:10:18
exam, don't just tell me.
256
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If b is in the -- and what
would -- what does the exam say
257
00:07:23 --> 00:06:38
there?1
258
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In the column space,
because I do know that it can
259
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be solved exactly when B is in
the column space,
260
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so I guess I'm asking you what
is the column space for this
261
00:14:38 --> 00:15:00
matrix?
262
00:15:00 --> 00:17:04
So what is if b has the form --
so I guess I'm asking what's
263
00:17:04 --> 00:17:07
the column space of this matrix,
and what is it?
264
00:17:07 --> 00:17:12
It's so the column space of
that matrix is all multiples b
265
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-- b is a multiple of one,
two, one.
266
00:17:15 --> 00:17:16
Right?
267
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I can solve the thing if it's a
multiple of one,
268
00:17:19 --> 00:17:24
two, one, and of course sure
enough --
269
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yeah, that was a multiple of
one, two, one,
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00:17:30 --> 00:17:34
and so I had a solution.
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00:17:34 --> 00:17:43
So this is a case where we've
got lots of null space.
272
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Let me just recall rank is big,
don't forget those cases,
273
00:17:50 --> 00:17:58
don't forget the other cases
when r is as big as it can be,
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00:17:58 --> 00:18:01
r equal m or r equal n.
275
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Those are -- we had a full
lecture on that,
276
00:18:06 --> 00:18:10
the full rank,
full lecture,
277
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and important -- important
case.
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OK.
279
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I'll just move on.
280
00:18:18 --> 00:18:22
I think this is the best type
of review.
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00:18:22 --> 00:18:26
It's just brings these ideas
out.
282
00:18:26 --> 00:18:33
Apologies to the camera while I
recover glasses and exam.
283
00:18:33 --> 00:18:33
OK.
284
00:18:33 --> 00:18:38.5
How about a few true-false
ones?
285
00:18:38.5 --> 00:18:42
Actually there won't be a
true-false on the quiz.
286
00:18:42 --> 00:18:46
But it gives us a moment of
quick review.
287
00:18:46 --> 00:18:47
Here's one.
288
00:18:47 --> 00:18:50
If the null space -- I have a
square matrix.
289
00:18:50 --> 00:18:55
If its null space is just the
zero vector, what about the null
290
00:18:55 --> 00:18:58
space of A transpose?
291
00:18:58 --> 00:19:03
If the null space of A is just
the zero vector,
292
00:19:03 --> 00:19:08
and the matrix is square,
what do I know about the null
293
00:19:08 --> 00:19:10
space of A transpose?
294
00:19:10 --> 00:19:12
Also the zero vector.
295
00:19:12 --> 00:19:13
Good.
296
00:19:13 --> 00:19:16
And that's a very very
important fact.
297
00:19:16 --> 00:19:17
OK.
298
00:19:17 --> 00:19:19
How about this?
299
00:19:19 --> 00:19:24
These -- look at the space of
five by five matrices as a
300
00:19:24 --> 00:19:25
vector space.
301
00:19:25 --> 00:19:30
So it's actually a
twenty-five-dimensional vector
302
00:19:30 --> 00:19:31
space.
303
00:19:31 --> 00:19:33
All five by five matrices.
304
00:19:33 --> 00:19:36
Look at the invertible
matrices.
305
00:19:36 --> 00:19:38
Do they form a subspace?
306
00:19:38 --> 00:19:44
So I have this five by --
a space of all five by five
307
00:19:44 --> 00:19:45
matrices.
308
00:19:45 --> 00:19:49.64
I can add them,
I can multiply by numbers.
309
00:19:49.64 --> 00:19:53.81
But now I narrow down to the
invertible ones.
310
00:19:53.81 --> 00:19:56
And I ask are they a subspace?
311
00:19:56 --> 00:20:01
And you -- your answer is --
quiet, but nevertheless
312
00:20:01 --> 00:20:03
definite, no.
313
00:20:03 --> 00:20:04.11
Right?
314
00:20:04.11 --> 00:20:04
No.
315
00:20:04 --> 00:20:08
Because if I add two invertible
matrices I have no idea if the
316
00:20:08 --> 00:20:10
answer is invertible.
317
00:20:10 --> 00:20:14
If I multiply that invertible
-- well, it doesn't even have
318
00:20:14 --> 00:20:18
the zero matrix in it,
it couldn't be a subspace.
319
00:20:18 --> 00:20:21
I have to be able to multiply
by zero --
320
00:20:21 --> 00:20:24
and stay in my subspace,
and the invertible ones
321
00:20:24 --> 00:20:25
wouldn't work.
322
00:20:25 --> 00:20:28
Well, the singular ones
wouldn't work either.
323
00:20:28 --> 00:20:31
They have zero -- the zero
matrix is in the singular
324
00:20:31 --> 00:20:35
matrices, but if I add two
singular matrices I don't know
325
00:20:35 --> 00:20:38
if the answer is singular or
not.
326
00:20:38 --> 00:20:38
OK.
327
00:20:38 --> 00:20:41
So another true-false.
328
00:20:41 --> 00:20:46
If b squared equals zero then b
equals zero.
329
00:20:46 --> 00:20:48
True or false?
330
00:20:48 --> 00:20:52
If b squared equals zero,
true, false?
331
00:20:52 --> 00:20:59
b squared equals zero,
b has to be a square --
332
00:20:59 --> 00:21:04
square matrix,
so that I can multiply it by
333
00:21:04 --> 00:21:08
itself, does that imply that B
is zero?
334
00:21:08 --> 00:21:15
Are there matrices whose square
could be the zero matrix?
335
00:21:15 --> 00:21:16
Yes or no?
336
00:21:16 --> 00:21:18
Yes there are.
337
00:21:18 --> 00:21:25
There are matrices whose square
is the zero matrix.
338
00:21:25 --> 00:21:27
So this statement is false.
339
00:21:27 --> 00:21:32
If b squared is zero,
we don't know that b is zero.
340
00:21:32 --> 00:21:36
For example -- the best example
is that matrix.
341
00:21:36 --> 00:21:39
That matrix is a dangerous
matrix.
342
00:21:39 --> 00:21:44
It will come up in later parts
of this course as an example of
343
00:21:44 --> 00:21:47
what can go wrong.
344
00:21:47 --> 00:22:00
And here is a real simple -- so
this -- so if I square that
345
00:22:00 --> 00:22:12
matrix, I do get the zero
matrix,and it shows --
346
00:22:12 --> 00:22:13
OK.
347
00:22:13 --> 00:22:20.86
A system of m equations in m
unknowns is solvable for every
348
00:22:20.86 --> 00:22:27.04
right-hand side if the columns
are independent.
349
00:22:27.04 --> 00:22:27
OK.
350
00:22:27 --> 00:22:34
So can I say that again,
I'm -- I'll write it down then
351
00:22:34 --> 00:20:45
for short.
m by m matrix independent
352
00:20:45 --> 00:16:58
columns then the question is
does Ax=b, is it always
353
00:16:58 --> 00:16:18
solvable?
354
00:16:18 --> 00:13:25
And the answer is -- yes or no
-- right.
355
00:13:25 --> 00:08:53
OK, which -- did you watch that
quiz, there was a quiz program
356
00:08:53 --> 00:05:59
on TV for a few weeks,
did you see that,
357
00:05:59 --> 00:03:09.8
what was the name of that --2
winning a million dollars or
358
00:03:09.8 --> 00:03:11
something?
359
00:03:11 --> 00:03:14
How to be a millionaire?
360
00:03:14 --> 00:03:19
It was some crazy guy,
what was his name?
361
00:03:19 --> 00:03:22
It's -- Regis,
right.
362
00:03:22 --> 00:03:23
Regis.
363
00:03:23 --> 00:03:25
OK.2
364
00:03:25 --> 00:07:00
If you saw this,
like when you should have been
365
00:07:00 --> 00:11:42.18
doing linear algebra of course
-- but I didn't have to do the
366
00:11:42.18 --> 00:16:09
homework, so I was watching it,
so there were three -- the
367
00:16:09 --> 00:20:50
interesting -- the novel point
was there were three ways that
368
00:20:50 --> 00:23:44.5
you could get help,
right --
369
00:23:44.5 --> 00:23:47
but you could only use each way
once, so you couldn't like use
370
00:23:47 --> 00:23:48
them all the time.
371
00:23:48 --> 00:23:49
So remember that?
372
00:23:49 --> 00:23:52
You could -- so you could poll
the audience,
373
00:23:52 --> 00:23:55.52
and that was a very -- that was
a hundred percent successful
374
00:23:55.52 --> 00:23:57
way, so I'll poll the audience
on this.
375
00:23:57 --> 00:24:01
If the other possibility --
another possibility you could
376
00:24:01 --> 00:24:04
call your friend,
right, or he's your friend
377
00:24:04 --> 00:24:08
until he gives you the wrong
answer, which -- that turned out
378
00:24:08 --> 00:24:11
to be very unreliable,
you know, you'd call up your
379
00:24:11 --> 00:24:15.71
brother or something and ask him
for the capital of whatever,
380
00:24:15.71 --> 00:24:16
Bosnia.
381
00:24:16 --> 00:24:24
What does he know,
he makes some guess,
382
00:24:24 --> 00:24:34
matrix, independent columns,
is Ax=b always solvable?
383
00:24:34 --> 00:24:39
Maybe just hands up for that?
384
00:24:39 --> 00:24:40
A few.
385
00:24:40 --> 00:24:42
And who says no?
386
00:24:42 --> 00:24:48
Oh, gosh, this audience is not
reliable.
387
00:24:48 --> 00:24:50
Fifty fifty.
388
00:24:50 --> 00:24:56
I guess I'd say,
I'd vote yes.
389
00:24:56 --> 00:25:01
Because independent columns,
that means that the rank is the
390
00:25:01 --> 00:25:05
full size m, I have a matrix of
rank m.
391
00:25:05 --> 00:25:10
That means it's -- I mean it's
square, so it's an invertible
392
00:25:10 --> 00:25:13
matrix, and nothing could go
wrong.
393
00:25:13 --> 00:25:14
Yes.
394
00:25:14 --> 00:25:22
So that's the good case and we
always expect it in chapter two,
395
00:25:22 --> 00:25:29
but of course chapter three is
-- only one of the
396
00:25:29 --> 00:25:31
possibilities.
397
00:25:31 --> 00:25:31
OK.
398
00:25:31 --> 00:25:38
Let me move on to another
question from an old quiz.
399
00:25:38 --> 00:25:40
OK.
400
00:25:40 --> 00:25:40
OK.
401
00:25:40 --> 00:25:42.42
Let's see.
402
00:25:42.42 --> 00:25:42
OK.
403
00:25:42 --> 00:25:52.28
I'm going to give you a matrix,
but I'm going to give it to you
404
00:25:52.28 --> 00:25:58
as a product of a couple of
matrices, one,
405
00:25:58 --> 00:26:03
one, zero, zero,
one, zero, one,
406
00:26:03 --> 00:26:09
zero, one, times another
matrix, one, zero,
407
00:26:09 --> 00:26:14
minus one, two,
zero, one, one,
408
00:26:14 --> 00:26:20
minus one, and all zeroes.
409
00:26:20 --> 00:26:05
OK.
410
00:26:05 --> 00:21:55
I would like to ask you
questions about that matrix
411
00:21:55 --> 00:17:10
without doing the multiplication
and finding the matrix b.
412
00:17:10 --> 00:12:10
Can you tell me something -- so
I'll ask questions about this
413
00:12:10 --> 00:06:45
matrix b and I'll answer them
without multiplying it out.2
414
00:06:45 --> 00:11:38
For example,
I'm going to ask you for a
415
00:11:38 --> 00:14:52
basis for the null space.
416
00:14:52 --> 00:18:21.17
A basis for the null space.
417
00:18:21.17 --> 00:22:59
So I'm going to solve Bx equals
zero.
418
00:22:59 --> 00:27:10
So give me a basis for --
for the null space of B.
419
00:27:10 --> 00:27:17
Let's see, what dimension I in
-- the null space of B is a
420
00:27:17 --> 00:27:19
subspace of R.
421
00:27:19 --> 00:27:24
What size vectors I looking for
here?
422
00:27:24 --> 00:27:32
Because if we don't know the
size, we aren't going to find
423
00:27:32 --> 00:27:37.32
it, right?
the null -- this matrix is
424
00:27:37.32 --> 00:27:39
three by four obviously.
425
00:27:39 --> 00:27:45
So if we're looking for the
null space we're looking for
426
00:27:45 --> 00:27:47
those vectors x in R^4.
427
00:27:47 --> 00:27:48
OK.
428
00:27:48 --> 00:27:53
So the null space of B is
certainly a subspace of R^4.
429
00:27:53 --> 00:27:57.94
What do you think its dimension
is?
430
00:27:57.94 --> 00:28:02
Of course once we find the
basis we would know the
431
00:28:02 --> 00:28:05
dimension immediately,
but let's stop first,
432
00:28:05 --> 00:28:08
what's the rank of this matrix
B?
433
00:28:08 --> 00:28:12
Let's see, what -- is that
matrix invertible,
434
00:28:12 --> 00:28:13
that square one there?
435
00:28:13 --> 00:28:16
Let's say sure,
I think it is,
436
00:28:16 --> 00:28:20
yes, that matrix B looks
invertible.
437
00:28:20 --> 00:28:22
Is that pretty clear?
438
00:28:22 --> 00:28:23
Yeah.
439
00:28:23 --> 00:28:23
Yeah.
440
00:28:23 --> 00:28:28
So I've gone wrong in this
course already,
441
00:28:28 --> 00:28:33
but I'll still hope that that
matrix is invertible.
442
00:28:33 --> 00:28:40
Yeah, yeah, because if I look
for a combination of those three
443
00:28:40 --> 00:28:44
columns --
well, I couldn't use this
444
00:28:44 --> 00:28:49
middle column because it would
have a one and in a position
445
00:28:49 --> 00:28:53
that I -- column is otherwise
all zero, so a combination that
446
00:28:53 --> 00:28:58
gives zero can't give us that
problem, and then the other two
447
00:28:58 --> 00:29:03
are clearly independent sets --
so that matrix is invertible.
448
00:29:03 --> 00:29:08
Later we could take a
determinant or other things.
449
00:29:08 --> 00:29:08.5
OK.
450
00:29:08.5 --> 00:29:09.99
What's the setup?
451
00:29:09.99 --> 00:29:14
If I have an invertible matrix,
a nice invertible square
452
00:29:14 --> 00:29:19
matrix, times this guy,
times this second factor,
453
00:29:19 --> 00:29:23
and I'm looking for the null
space, does this have any
454
00:29:23 --> 00:29:25
effect?
455
00:29:25 --> 00:29:32
Is the null -- so what I'm
asking is is the null space of B
456
00:29:32 --> 00:29:37
the same as the null space of
just this part?
457
00:29:37 --> 00:29:38
I think so.
458
00:29:38 --> 00:29:39
I think so.
459
00:29:39 --> 00:29:44
If Bx is zero,
then multiplying by that guy
460
00:29:44 --> 00:29:48.5
I'll still have zero.
461
00:29:48.5 --> 00:29:53
But also if this times some x
give zero, I could always
462
00:29:53 --> 00:29:57
multiply on the left by the
inverse of that,
463
00:29:57 --> 00:30:03
because it is invertible,
and I would discover that this
464
00:30:03 --> 00:30:05
kind of Bx is zero.
465
00:30:05 --> 00:30:09
You want me to write some of
that down --
466
00:30:09 --> 00:30:15
if I have a product here,
C times -- times D,
467
00:30:15 --> 00:30:21
say, and if C is invertible,
the null space of CD,
468
00:30:21 --> 00:30:26.6
well, it will the same as the
null space of D.
469
00:30:26.6 --> 00:30:28
If C is invertible.
470
00:30:28 --> 00:30:36.04
Multiplying by an invertible
matrix on the left can't change
471
00:30:36.04 --> 00:30:37
the null space.
472
00:30:37 --> 00:30:39
OK.
473
00:30:39 --> 00:30:44
So basically I'm asking you for
the null space of this.
474
00:30:44 --> 00:30:48
I don't have to do the
multiplication because I have C
475
00:30:48 --> 00:30:49
is invertible.
476
00:30:49 --> 00:30:52.52
That first factor C is
invertible.
477
00:30:52.52 --> 00:30:55
It's not going to change the
null space.
478
00:30:55 --> 00:30:56.9
OK.
479
00:30:56.9 --> 00:31:02
So can we just write down a
basis now for the null space?
480
00:31:02 --> 00:31:07
So what's the basis for the
null space of -- of that?
481
00:31:07 --> 00:31:13
So basis for the null space I'm
looking for the two -- there are
482
00:31:13 --> 00:31:16
two pivot columns obviously.
483
00:31:16 --> 00:31:19.5
It clearly has rank two.
484
00:31:19.5 --> 00:31:23
I'm looking for the two special
solutions.
485
00:31:23 --> 00:31:27
They'll come from the third and
the fourth.
486
00:31:27 --> 00:31:29
The free variables.
487
00:31:29 --> 00:31:32
OK.
so if the third free variable
488
00:31:32 --> 00:31:38
is a one, then I think probably
I need a minus one there and a
489
00:31:38 --> 00:31:41
one there, it looks like.
490
00:31:41 --> 00:31:45
Do you agree that if I then do
that multiplication I'll get
491
00:31:45 --> 00:31:45
zero?
492
00:31:45 --> 00:31:49.2
And if I have one in the fourth
variable, then maybe I need a
493
00:31:49.2 --> 00:31:52
one in the second variable and
maybe a minus two in the third.
494
00:31:52 --> 00:31:56
So I just reasoned that through
and then if I look back I see
495
00:31:56 --> 00:31:59
sure enough that the free
variable part that I sometimes
496
00:31:59 --> 00:32:04.09
call F, that up --
that two by two corner,
497
00:32:04.09 --> 00:32:08
is sitting here with all its
signs reversed.
498
00:32:08 --> 00:32:15
So that's -- here I'm seeing
minus F, and here I'm seeing the
499
00:32:15 --> 00:32:19.41
identity in the null space
matrix.
500
00:32:19.41 --> 00:32:23
OK, so that's the null space.
501
00:32:23 --> 00:32:30
Another question is solve Bx
equal one, zero,
502
00:32:30 --> 00:32:31
one.
503
00:32:31 --> 00:32:31
OK.
504
00:32:31 --> 00:32:39
So that's one question,
now solve complete solutions.
505
00:32:39 --> 00:32:44
To Bx equal one,
zero, one.
506
00:32:44 --> 00:32:44
OK.
507
00:32:44 --> 00:32:52
Yeah, so I guess I'm seeing if
I wanted to get one,
508
00:32:52 --> 00:32:58
zero, one -
What's our particular solution?
509
00:32:58 --> 00:33:03
So I'm looking for a particular
solution and then the null space
510
00:33:03 --> 00:33:03
part.
511
00:33:03 --> 00:33:03.77
OK.
512
00:33:03.77 --> 00:33:08
I-- actually the first column
of B, so what's the first column
513
00:33:08 --> 00:33:09
of our matrix B?
514
00:33:09 --> 00:33:11.88
It's the vector one,
zero, one.
515
00:33:11.88 --> 00:33:17
The first column of our matrix
agrees with the right-hand side.
516
00:33:17 --> 00:33:22
So I guess I'm thinking x
particular plus x null space
517
00:33:22 --> 00:33:28
will be the particular solution,
since the first column of B is
518
00:33:28 --> 00:33:30.58
exactly right,
that's great.
519
00:33:30.58 --> 00:33:35
And then I have C times that
first null space vector and D
520
00:33:35 --> 00:33:39
times the other null space
vector.
521
00:33:39 --> 00:33:40
Right?
522
00:33:40 --> 00:33:44
The two -- the null space part
of the solution,
523
00:33:44 --> 00:33:49
as always has the arbitrary
constants, the particular
524
00:33:49 --> 00:33:53
solution doesn't have any
arbitrary constants,
525
00:33:53 --> 00:33:59
it's one particular solution,
and in this case it'll --
526
00:33:59 --> 00:34:02
that one would do.
527
00:34:02 --> 00:34:02
OK.
528
00:34:02 --> 00:34:07.03
Fine.
so those are questions taken
529
00:34:07.03 --> 00:34:13
from old quizzes,
any questions coming to mind?
530
00:34:13 --> 00:34:14
Yeah.
531
00:34:14 --> 00:34:16
Q: value.
532
00:34:16 --> 00:34:16
OK.
533
00:34:16 --> 00:34:24
Well, so that particular x
particular, it says that let's
534
00:34:24 --> 00:34:33
see, when I multiply by this
guy, I'm going to get the first
535
00:34:33 --> 00:34:34
column of B.
536
00:34:34 --> 00:34:38
That --
if that's a solution,
537
00:34:38 --> 00:34:41
I multiply B,
B times this x will be the
538
00:34:41 --> 00:34:45
first column of B,
and so I'm saying that the
539
00:34:45 --> 00:34:49
first column of this B agrees
with the right-hand side.
540
00:34:49 --> 00:34:55
So I'm saying that look at the
first column of that matrix B.
541
00:34:55 --> 00:34:58
If you do the multiplication,
it's -- so what's the first
542
00:34:58 --> 00:35:00
column of that matrix?
543
00:35:00 --> 00:35:02
Is that how you do that
multiplication?
544
00:35:02 --> 00:35:05
I multiply that matrix by that
first column.
545
00:35:05 --> 00:35:07
And it picks out one,
zero, one.
546
00:35:07 --> 00:35:11
So the first column of B is
exactly that.
547
00:35:11 --> 00:35:22
And therefore a particular
solution will be this guy.
548
00:35:22 --> 00:35:23
Yeah.
549
00:35:23 --> 00:35:23.95
OK.
550
00:35:23.95 --> 00:35:24
Yes.
551
00:35:24 --> 00:35:30
Q: particular solution.
552
00:35:30 --> 00:35:40
Any of the solutions can be the
particular one that we pick out.
553
00:35:40 --> 00:35:48
So like this plus -- plus this
would be another particular
554
00:35:48 --> 00:35:49
solution.
555
00:35:49 --> 00:35:54
It would be another solution.
556
00:35:54 --> 00:35:58.93
The particular is just telling
us only take one.
557
00:35:58.93 --> 00:36:03
But it's not telling us which
one we have to take.
558
00:36:03 --> 00:36:06
We take the most convenient
one.
559
00:36:06 --> 00:36:10
I guess in this -- in this
problem that was that one.
560
00:36:10 --> 00:36:11.36
Good.
561
00:36:11.36 --> 00:36:12
Other questions?
562
00:36:12 --> 00:36:14
Yes.
563
00:36:14 --> 00:36:18
And this pattern of particular
plus null space,
564
00:36:18 --> 00:36:22
of course, that's going
throughout mathematics of linear
565
00:36:22 --> 00:36:23
systems.
566
00:36:23 --> 00:36:28
We're really doing mathematics
of linear systems here.
567
00:36:28 --> 00:36:33
Our systems are discrete and
they're finite-dimensional --
568
00:36:33 --> 00:36:38
and so it's linear algebra,
but this particular plus null
569
00:36:38 --> 00:36:43
space goes -- that doesn't
depend on having finite matrices
570
00:36:43 --> 00:36:47
-- that spreads much -- that
spreads everywhere.
571
00:36:47 --> 00:36:51.92
OK, I'm going to just like to
encourage you to take problems
572
00:36:51.92 --> 00:36:56
out of the book,
let me do the same myself.
573
00:36:56 --> 00:37:00
OK well here's some easy true
or falses.
574
00:37:00 --> 00:37:05
I don't know why the author put
these in here.
575
00:37:05 --> 00:37:05
OK.
576
00:37:05 --> 00:37:11
If m=n, then the row space
equals the column space.
577
00:37:11 --> 00:37:15
So these are true or falses.
578
00:37:15 --> 00:37:18
If m equals n,
so that means the matrix is
579
00:37:18 --> 00:37:23
square, then the row space
equals the column space?
580
00:37:23 --> 00:37:24
False, good.
581
00:37:24 --> 00:37:27.49
Good, what is equal there?
582
00:37:27.49 --> 00:37:31
What can I say is equal,
if M -- well,
583
00:37:31 --> 00:37:31
yeah.
584
00:37:31 --> 00:37:36
Yeah it -- so that's definitely
false --
585
00:37:36 --> 00:37:41
the row space and the column
space, and this matrix is like
586
00:37:41 --> 00:37:43
always a good example to
consider.
587
00:37:43 --> 00:37:48
So there's a square matrix but
it's row space is the multiples
588
00:37:48 --> 00:37:51
of zero, one,
and its column space is the
589
00:37:51 --> 00:37:53
multiples of one,
zero.
590
00:37:53 --> 00:37:55
Very different.
591
00:37:55 --> 00:37:59
The row space and the column
space are totally different for
592
00:37:59 --> 00:38:00
that matrix.
593
00:38:00 --> 00:38:04
Now of course if the matrix was
symmetric, well,
594
00:38:04 --> 00:38:08
then clearly the row space
equals the column space.
595
00:38:08 --> 00:38:08
OK.
596
00:38:08 --> 00:38:10
How about this question?
597
00:38:10 --> 00:38:15
The matrices A and minus A
share the same four subspaces?
598
00:38:15 --> 00:38:20
Do the matrices A and minus A
have the same column space,
599
00:38:20 --> 00:38:25
do they have the same null
space, do they have the same row
600
00:38:25 --> 00:38:25
space?
601
00:38:25 --> 00:38:28
What's the answer on that?
602
00:38:28 --> 00:38:29
Yes or no.
603
00:38:29 --> 00:38:29
Yes.
604
00:38:29 --> 00:38:30
Good.
605
00:38:30 --> 00:38:30
OK.
606
00:38:30 --> 00:38:31
How about this?
607
00:38:31 --> 00:38:37.2
If A and B have the same four
subspaces, then A is a multiple
608
00:38:37.2 --> 00:38:38
of B.
609
00:38:38 --> 00:38:44
If -- suppose those subspaces
are the same.
610
00:38:44 --> 00:38:47
Then is A a multiple of B?
611
00:38:47 --> 00:38:48.13
OK.
612
00:38:48.13 --> 00:38:53
How how do you answer a
question like that?
613
00:38:53 --> 00:39:01
Of course if you want to answer
it yes, then I would --
614
00:39:01 --> 00:39:04
then they'd have to think of a
reason why.
615
00:39:04 --> 00:39:09
If you want to answer no way,
then you would -- and I would
616
00:39:09 --> 00:39:13
sort of like first I would try
to think no, I would say can I
617
00:39:13 --> 00:39:17
come up with an example where it
isn't true?
618
00:39:17 --> 00:39:20
Let me repeat the question.
619
00:39:20 --> 00:35:44
And then write the answer.
620
00:35:44 --> 00:35:19
OK.
621
00:35:19 --> 00:31:19
So I'll repeat that question.
622
00:31:19 --> 00:24:07
If -- so true or false,
if A and B have the same four
623
00:24:07 --> 00:18:44
subspaces, then A is some
multiple of B.
624
00:18:44 --> 00:13:04
True or false,
how do you feel about it at
625
00:13:04 --> 00:09:54
this instant?3
626
00:09:54 --> 00:15:12.39
There's quite a few trues,
shall I take a poll,
627
00:15:12.39 --> 00:17:51
so how many think true?
628
00:17:51 --> 00:18:12
OK.
629
00:18:12 --> 00:23:09
I gave you every chance to
think about that.
630
00:23:09 --> 00:24:19.19
Let's see.
631
00:24:19.19 --> 00:30:39
So what -- I would just take
extreme cases if it was me,
632
00:30:39 --> 00:37:07
so when do I know -- well,
I would say suppose the matrix
633
00:37:07 --> 00:40:23
is invertible --
suppose A is an invertible
634
00:40:23 --> 00:40:28
matrix, then what -- suppose
it's six by six invertible
635
00:40:28 --> 00:40:32
matrix, then what's its row
space, and its column space is
636
00:40:32 --> 00:40:37
all of R^6, and the null space,
and the null space of A
637
00:40:37 --> 00:40:40
transpose would be the zero
vector.
638
00:40:40 --> 00:40:46
So every invertible matrix is
going to give that answer.
639
00:40:46 --> 00:40:49
If I have a six by six
invertible matrix,
640
00:40:49 --> 00:40:52.8
I know what those subspaces
are.
641
00:40:52.8 --> 00:40:58
Heck, that was back in chapter
two, when I didn't even know
642
00:40:58 --> 00:41:00
what subspaces were.
643
00:41:00 --> 00:41:05
The row space and column space
are both all six-dimensional
644
00:41:05 --> 00:41:08
space --
the whole space,
645
00:41:08 --> 00:41:11
and the rank is six,
in other words,
646
00:41:11 --> 00:41:14
and the null spaces have zero
dimension.
647
00:41:14 --> 00:41:16
So do you see now the answer?
648
00:41:16 --> 00:41:18
So A and B could be.
649
00:41:18 --> 00:41:23
So A and B could be for example
any -- so I'm going to say
650
00:41:23 --> 00:41:23
false.
651
00:41:23 --> 00:41:28
Because A and B for example --
So an example:
652
00:41:28 --> 00:41:34
A and B any invertible six by
six, six by six.
653
00:41:34 --> 00:41:41
So those would have the same
four subspaces but they wouldn't
654
00:41:41 --> 00:41:42
be the same.
655
00:41:42 --> 00:41:50
Of course th- there should be
something about those matrices
656
00:41:50 --> 00:41:54
that would be the same.
657
00:41:54 --> 00:41:58
It's sort of a natural problem,
so now actually we're getting
658
00:41:58 --> 00:41:59
to a math question.
659
00:41:59 --> 00:42:02
The answer is this is not true.
660
00:42:02 --> 00:42:06
One matrix doesn't have to be a
multiple of the other.
661
00:42:06 --> 00:42:09
But there must be something
that's true.
662
00:42:09 --> 00:42:14
And that would be sort of like
a natural question to ask.
663
00:42:14 --> 00:42:27
If they have the same
subspaces, same four subspaces,
664
00:42:27 --> 00:42:38
then what -- what could you --
instinct wasn't necessarily
665
00:42:38 --> 00:42:39
right.
666
00:42:39 --> 00:42:43
But I hope you now see that the
correct answer is false.
667
00:42:43 --> 00:42:47
And then you might think OK,
well, they certainly do have
668
00:42:47 --> 00:42:48
the same rank.
669
00:42:48 --> 00:42:52.62
But do -- obviously if they
have the same four subspaces,
670
00:42:52.62 --> 00:42:55
they have the same rank.
671
00:42:55 --> 00:43:00
I might say if they have the
well, I could extend that
672
00:43:00 --> 00:43:06
question and think about other
possibilities and finally come
673
00:43:06 --> 00:43:10
up with something that was true.
674
00:43:10 --> 00:43:12
But I won't press that one.
675
00:43:12 --> 00:43:17
Let me keep going with practice
questions.
676
00:43:17 --> 00:43:23
And these practice questions
are quite appropriate I think
677
00:43:23 --> 00:43:24
for the exam.
678
00:43:24 --> 00:43:26.21
OK.
let's see.
679
00:43:26.21 --> 00:43:31.96
If I exchange two rows of A
which subspaces stay the same?
680
00:43:31.96 --> 00:43:37
So I'm trying to take out
questions that we can answer
681
00:43:37 --> 00:43:42.25
without you know we can answer
quickly.
682
00:43:42.25 --> 00:43:47
If I have a matrix A,
and I exchange two of its rows,
683
00:43:47 --> 00:43:50
which subspaces stay the same?
684
00:43:50 --> 00:43:53
The row space does stay the
same.
685
00:43:53 --> 00:43:57
And the null space stays the
same.
686
00:43:57 --> 00:43:57
Good.
687
00:43:57 --> 00:43:58
Good.
688
00:43:58 --> 00:43:59
Correct.
689
00:43:59 --> 00:44:03
Column space would be a wrong
answer.
690
00:44:03 --> 00:44:07.85
OK.
all right, here's a question.
691
00:44:07.85 --> 00:44:12
Oh, this leads into the next
chapter.
692
00:44:12 --> 00:44:18
Why can the vector one,
two, three not be a row and
693
00:44:18 --> 00:44:21
also in the null space?
694
00:44:21 --> 00:44:25
Fitting we close with this
question.
695
00:44:25 --> 00:44:30
Close is --
so V equal this one,
696
00:44:30 --> 00:44:38
two, three can't be in the null
space of a matrix and the row
697
00:44:38 --> 00:44:39
space.
698
00:44:39 --> 00:44:42
And my question is why not?
699
00:44:42 --> 00:44:43
Why not?
700
00:44:43 --> 00:44:51.71
So this is a question that we
can because it's sort of asked
701
00:44:51.71 --> 00:45:00
in a straightforward way,
we can figure out an answer.
702
00:45:00 --> 00:45:05
Well, actually yeah -- I'll
even pin it down,
703
00:45:05 --> 00:45:10
it can't be in the null space
-- and be a row.
704
00:45:10 --> 00:45:14
I'll even pin it down further.
705
00:45:14 --> 00:45:17
Ask it to be a row of A.
706
00:45:17 --> 00:45:18
Why not?
707
00:45:18 --> 00:45:25
So I'm -- we know the
dimensions of these spaces.
708
00:45:25 --> 00:45:30
But now I'm asking you sort of
like the overlap between -- so
709
00:45:30 --> 00:45:36
the null space and the row
space, those are in the same
710
00:45:36 --> 00:45:38
n-dimensional space.
711
00:45:38 --> 00:45:42
Those are -- well,
those are both subspaces of
712
00:45:42 --> 00:45:47
n-dimensional space,
and I'm basically saying they
713
00:45:47 --> 00:45:50
can't overlap.
714
00:45:50 --> 00:45:54
I can't have a vector like
this, a typical vector,
715
00:45:54 --> 00:45:59
that's in the null space and
it's also a row of the matrix.
716
00:45:59 --> 00:46:00
Why is that?
717
00:46:00 --> 00:46:03
So that's a new sort of idea.
718
00:46:03 --> 00:46:07
Let's just see what it would
mean.
719
00:46:07 --> 00:46:11
I mean that A times this V,
why can this A times this V it
720
00:46:11 --> 00:46:13.04
can't be zero.
721
00:46:13.04 --> 00:46:17
Oh well, if it's zero,
so this is -- I'm getting it
722
00:46:17 --> 00:46:19
into the null space here.
723
00:46:19 --> 00:46:24
So this is -- now let's put
that vector's in the null space,
724
00:46:24 --> 00:46:27
why can't the first row of a
matrix be one,
725
00:46:27 --> 00:46:29
two, three?
726
00:46:29 --> 00:46:34
I can fill out the matrix as I
like.
727
00:46:34 --> 00:46:38
Why is that impossible?
728
00:46:38 --> 00:46:45.25
Well, you're seeing it's
impossible, right?
729
00:46:45.25 --> 00:46:49
That if that was a row of the
matrix and in the null space,
730
00:46:49 --> 00:46:53
that number would not be zero,
it would be fourteen.
731
00:46:53 --> 00:46:53
Right.
732
00:46:53 --> 00:46:58
So now we actually are
beginning to get a more complete
733
00:46:58 --> 00:47:00
picture of these four subspaces.
734
00:47:00 --> 00:47:03
The two that are over in
n-dimensional space,
735
00:47:03 --> 00:47:07
they actually only share the
zero vector.
736
00:47:07 --> 00:47:12
The intersection of the null
space and the row space is only
737
00:47:12 --> 00:47:14
the zero vector.
738
00:47:14 --> 00:47:19
And in fact the null space is
perpendicular to the row space.
739
00:47:19 --> 00:47:24
That'll be the first topic
let's see, we have a holiday
740
00:47:24 --> 00:47:28
Monday --
and I'll see you Wednesday with
741
00:47:28 --> 00:47:29.66
perpendiculars.
742
00:47:29.66 --> 00:47:31
And I'll see you Friday.
743
00:47:31 --> 00:47:34
So good luck on the quiz.