WEBVTT
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OK.
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Uh this is the review
lecture for the first part
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of the 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
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three was about the
rectangular matrices
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where we had null spaces and
null spaces of A transpose,
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and ranks, and all
the things that
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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
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those four subspaces.
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OK, I'm thinking to
start at least I'll
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just look at old exams,
read out questions, write
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on the board what
I need to and we
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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,
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what are the possible dimensions
of the subspace spanned
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by u, 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 --
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because I told you the
vectors 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 subspace would
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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
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U, so this matrix is five by
three, and I find it helpful to
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just see
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visually what five by three
means, what that shape is.
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Three columns.
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Three columns in U,
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then five rows,
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three pivots,
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and what's the null space?
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The null space of U is --
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and it asks for a
spec-of course I'm
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looking for an answer
that isn't just
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the definition of
the null space,
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but is the null space of this
matrix, with this 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 --
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so the only thing in this
null space is the zero vector,
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and I --
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I could even say what that
vector is, zero, columns also?
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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 question has multiple
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parts.
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What's the -- oh now it asks you
about a ten by three matrix, B,
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which is the matrix U and two U.
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It actually -- I would
probably be writing R --
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and maybe I should be
writing R here now.
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This exam goes back a few
years when I emphasized U more
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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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Yes. 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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So I'm seeing the same length
in b, three, and also in x.
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these are --
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U is in reduced row
echelon form but now
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I've doubled the
height of the matrix,
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So the (x)-s that it multiplies
have three components,
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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 --
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let's -- I'll stay with this
letter U but I'm what was
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the name of that --2 winning
a million dollars or really
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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 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
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through what would
row elimination,
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what would row reduction do.
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Have I thought this through,
so what -- what are we --
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if we start doing elimination,
basically we're going
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to subtract these
rows from these --
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In the column
space, because I do
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know that it can so it's going
to take us to U, U, zero,
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and minus U, For example, I'm
going to ask you for a I guess,
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right?
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But is it a- is
it three by three?
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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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doing linear
algebra of course --
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but I didn't have
to do the 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 to --
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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. there?1 And now what
more would I do if I'm really
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shooting for R, 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
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there and there, and there's
just one little twist
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still to go.
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Do you see what that
final twist might be?
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It certainly has three rows.
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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 take,
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this isn't like a big
deal at all, but --
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but if I really want this
to be in reduced form,
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what would I still --
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might I still have to do?
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I might have some
zero rows here,
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I might have some zero
rows here that strictly
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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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first of all?
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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 --
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while we're at on TV for a
few weeks, did you see that,
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it?
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The rank of B, is
that six or three?
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So A is a three by three matrix.
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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,
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what's the dimension be solved
exactly when B is in the column
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space, of the null space --
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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, so C has ten rows,
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C transpose has ten columns,
so there are ten columns there.
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So how many free variables
have I got, once I --
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There's quite a few trues,
shall I take a poll,
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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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If b is in the -- and what
would -- what does the exam say
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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.
think about, what's the --
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what's the shape of the matrix,
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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.
00:10:54.590 --> 00:10:57.992
And what's its rank?
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But I don't give you the matrix.
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And another -- there's another
vector, zero, 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
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is that a question
could start this way.
00:11:36.130 --> 00:11:38.770
exam, don't just tell me.
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Sort of backward way.
00:11:40.890 --> 00:11:45.130
so I guess I'm
asking you what is
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the column space for this
basis for the null space.
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By giving you the answer and
not telling you what the problem
00:11:56.770 --> 00:12:02.590
homework, so I was watching
it, so there were three --
00:12:02.590 --> 00:12:03.650
the is.
00:12:03.650 --> 00:12:07.890
But we can get a
lot of information,
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and sometimes we can
get complete information
00:12:11.590 --> 00:12:13.710
about that matrix A.
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OK.
00:12:14.240 --> 00:12:19.530
So what's the dimension
of the row space of A?
00:12:19.530 --> 00:12:21.120
What's the rank?
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Tell me about what's the size
of the matrix, yeah, just --
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These are the things
we want to matrix b
00:12:32.230 --> 00:12:36.470
and I'll answer them
without multiplying it out.2
00:12:36.470 --> 00:12:41.760
Its rank -- tell me something
about its null space,
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I heard the right
answer for the rank,
00:12:46.000 --> 00:12:49.700
the rank is one in this case.
00:12:49.700 --> 00:12:50.230
Why?
00:12:50.230 --> 00:12:53.940
Because the dimension
of the null space,
00:12:53.940 --> 00:12:59.760
so the dimension of this
instant?3 the null space of A
00:12:59.760 --> 00:13:05.050
is from knowing that that's the
complete solution, it's two.
00:13:05.050 --> 00:13:08.760
I'm seeing two vectors
here, and they're
00:13:08.760 --> 00:13:12.460
independent in the
null space of A,
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because they have to be in the
00:13:16.170 --> 00:13:20.400
OK, which -- did
you watch that quiz,
00:13:20.400 --> 00:13:26.750
there was a quiz program null
space of A if I'm allowed
00:13:26.750 --> 00:13:31.650
to throw into the solution
any amount of those vectors,
00:13:31.650 --> 00:13:35.620
that tells me that's the
null space part then.
00:13:35.620 --> 00:13:38.400
So the dimension of the null
space is two, and then I --
00:13:38.400 --> 00:13:44.010
of course I know the dimensions
of all the -- four subspaces.
00:13:44.010 --> 00:13:46.090
Now actually it asks
what's the matrix?
00:13:46.090 --> 00:13:52.440
Well, what's the
matrix in this case?
00:13:52.440 --> 00:14:01.210
Do we want to -- shall I
try to figure that out?
00:14:01.210 --> 00:14:01.790
Sure.
00:14:01.790 --> 00:14:03.290
Let's -- you'd like
me to do it, OK.
00:14:03.290 --> 00:14:04.164
Well, what's the ----
00:14:04.164 --> 00:14:05.590
I actually say in this in the
00:14:05.590 --> 00:14:09.630
So what about the matrix,
or let me at least start it,
00:14:09.630 --> 00:14:10.140
OK.
00:14:10.140 --> 00:14:14.720
If A times this x
gives two, four, two,
00:14:14.720 --> 00:14:19.300
what does that tell
me about the matrix A?
00:14:19.300 --> 00:14:26.210
If A times that x solves that
equation then it tells me that
00:14:26.210 --> 00:14:28.900
the first column is --
00:14:28.900 --> 00:14:35.560
the first column of A is
-- one, two, one, right.
00:14:35.560 --> 00:14:37.580
The first column of A
has to be one, two, one,
00:14:37.580 --> 00:14:37.650
because if I
multiply by x, that's
00:14:37.650 --> 00:14:37.730
going to multiply just
matrix? the first column,
00:14:37.730 --> 00:14:37.790
and give me two, four, two.
00:14:37.790 --> 00:14:37.900
And then I've got two
more columns to find,
00:14:37.900 --> 00:14:38.000
and what information have
I got to find them with?
00:14:38.000 --> 00:14:38.070
A basis for the null space.
00:14:38.070 --> 00:14:38.130
I've got the null space.
00:14:38.130 --> 00:14:38.230
So the fact that this
is in the null space,
00:14:38.230 --> 00:14:38.370
what does that tell So what
is if b has the form --
00:14:38.370 --> 00:14:40.411
so I guess I'm asking
what's me about the matrix?
00:14:40.411 --> 00:14:59.047
A matrix that has zero,
zero, one in its null space?
00:15:05.510 --> 00:15:10.840
That tells me that the last
column of the matrix is zeroes.
00:15:10.840 --> 00:15:11.880
so how many think true?
00:15:11.880 --> 00:15:13.820
Because this is
in the null space,
00:15:13.820 --> 00:15:14.897
the last column has to be
00:15:14.897 --> 00:15:16.230
When could it be solved? zeroes.
00:15:16.230 --> 00:15:19.510
And because this is in
the null space, what's
00:15:19.510 --> 00:15:23.850
the second column?
00:15:23.850 --> 00:15:25.880
Well, this in the
null space means
00:15:25.880 --> 00:15:27.910
that if I multiply
A by that vector
00:15:27.910 --> 00:15:32.530
I must be getting zeroes, so I
think that better be minus one,
00:15:32.530 --> 00:15:44.370
minus two, and minus one.
00:15:44.370 --> 00:15:45.300
OK.
00:15:45.300 --> 00:15:50.340
That's a type of question
that just brings out
00:15:50.340 --> 00:15:53.050
the information that's in
that complete solution.
00:15:53.050 --> 00:15:53.180
And then actually I go on to
ask what vectors -- for what
00:15:53.180 --> 00:15:53.250
OK. vectors B can
Ax=b be solved?
00:15:53.250 --> 00:15:53.380
Ax=b can be solved if what --
so I'm looking for a condition
00:15:53.380 --> 00:15:53.460
Definitely not. on b,
if any. interesting --
00:15:53.460 --> 00:15:53.550
the novel point was there
were three ways that
00:15:53.550 --> 00:15:53.640
Can it be solved for
every right-hand side b?
00:15:53.640 --> 00:15:54.140
No.
00:15:54.140 --> 00:16:07.315
And the answer is --
00:16:11.410 --> 00:16:25.505
yes or no --
00:16:29.555 --> 00:16:30.054
right.
00:16:34.097 --> 00:16:34.596
solvable?
00:16:38.700 --> 00:16:44.920
the column space of this
matrix, and what is it?
00:16:44.920 --> 00:16:50.810
It's so the column space of
that matrix is all multiples b
00:16:50.810 --> 00:16:54.490
Can you tell me something --
00:16:54.490 --> 00:16:59.170
so I'll ask questions
about this --
00:16:59.170 --> 00:17:04.609
b is a multiple
of one, two, one.
00:17:04.609 --> 00:17:05.290
Right?
00:17:05.290 --> 00:17:14.140
I can solve the thing if it's
a multiple of one, two, one,
00:17:14.140 --> 00:17:18.220
and of course sure enough --
00:17:18.220 --> 00:17:24.140
yeah, that was a multiple
of one, two, one,
00:17:24.140 --> 00:17:26.829
and so I had a solution.
00:17:26.829 --> 00:17:34.640
So this is a case where
we've got lots of null space.
00:17:34.640 --> 00:17:44.760
Let me just recall rank is
big, don't forget those cases,
00:17:44.760 --> 00:17:57.390
don't forget the other cases
when r is as big as it can be,
00:17:57.390 --> 00:18:00.580
OK. r equal m or r equal n.
00:18:00.580 --> 00:18:04.970
Those are -- we had a full
lecture on that, the full rank,
00:18:04.970 --> 00:18:09.480
full lecture, and important
-- important case.
00:18:09.480 --> 00:18:13.471
I gave you every chance
to think about that.
00:18:13.471 --> 00:18:13.970
OK.
00:18:13.970 --> 00:18:15.760
I'll just move on.
00:18:15.760 --> 00:18:19.800
I think this is the
best type of review.
00:18:19.800 --> 00:18:23.380
So I'm going to
solve Bx equals zero.
00:18:23.380 --> 00:18:26.080
It's just brings
these ideas out.
00:18:26.080 --> 00:18:30.511
Apologies to the camera while
I recover glasses and exam.
00:18:30.511 --> 00:18:31.010
OK.
00:18:31.010 --> 00:18:33.700
How about a few true-false ones?
00:18:33.700 --> 00:18:37.100
Actually there won't be
a true-false on the quiz.
00:18:37.100 --> 00:18:40.980
But it gives us a
moment of quick review.
00:18:40.980 --> 00:18:44.500
True or false, how do
you feel about it at
00:18:44.500 --> 00:18:45.190
Here's one.
00:18:45.190 --> 00:18:46.930
If the null space --
00:18:46.930 --> 00:18:49.400
I have a square matrix.
00:18:49.400 --> 00:18:52.740
If its null space is
just the zero vector,
00:18:52.740 --> 00:18:57.750
what about the null
space of A transpose?
00:18:57.750 --> 00:19:00.840
If the null space of A
is just the zero vector,
00:19:00.840 --> 00:19:03.580
and the matrix is
square, what do I
00:19:03.580 --> 00:19:06.430
know about the null
space of A transpose?
00:19:06.430 --> 00:19:07.420
Also the zero vector.
00:19:07.420 --> 00:19:07.960
Good.
00:19:07.960 --> 00:19:10.270
And that's a very
very important fact.
00:19:10.270 --> 00:19:11.630
OK.
00:19:11.630 --> 00:19:15.000
How about this?
00:19:15.000 --> 00:19:20.450
These -- look at the space
of five by five matrices
00:19:20.450 --> 00:19:23.280
as a vector space.
00:19:23.280 --> 00:19:25.690
So it's actually a
twenty-five-dimensional vector
00:19:25.690 --> 00:19:26.190
space.
00:19:26.190 --> 00:19:28.990
All five by five matrices.
00:19:28.990 --> 00:19:31.770
Look at the invertible matrices.
00:19:31.770 --> 00:19:35.430
Do they form a subspace?
00:19:35.430 --> 00:19:40.430
So I have this five by --
a space of all five by five
00:19:40.430 --> 00:19:41.580
matrices.
00:19:41.580 --> 00:19:44.180
I can add them, I can
multiply by numbers.
00:19:44.180 --> 00:19:48.110
But now I narrow down
to the invertible ones.
00:19:48.110 --> 00:19:50.700
And I ask are they a subspace?
00:19:50.700 --> 00:19:54.600
And you -- your answer is --
00:19:54.600 --> 00:19:57.418
quiet, but nevertheless
definite, no.
00:20:00.130 --> 00:20:00.800
Right?
00:20:00.800 --> 00:20:04.040
Because if I add two invertible
matrices I have no idea if the
00:20:04.040 --> 00:20:06.570
No. answer is invertible.
00:20:06.570 --> 00:20:09.340
If I multiply that
invertible -- well,
00:20:09.340 --> 00:20:11.600
it doesn't even have
the zero matrix in it,
00:20:11.600 --> 00:20:13.630
it couldn't be a subspace.
00:20:13.630 --> 00:20:16.020
I have to be able to
multiply by zero --
00:20:16.020 --> 00:20:21.010
and stay in my subspace, and the
invertible ones wouldn't work.
00:20:21.010 --> 00:20:24.210
Well, the singular ones
wouldn't work either.
00:20:24.210 --> 00:20:27.540
They have zero -- the zero
matrix is in the singular
00:20:27.540 --> 00:20:31.720
matrices, but if I add two
singular matrices I don't know
00:20:31.720 --> 00:20:33.410
if the answer is
singular or not.
00:20:33.410 --> 00:20:34.340
OK.
00:20:34.340 --> 00:20:36.420
So another true-false.
00:20:36.420 --> 00:20:36.510
If b squared equals
zero then b equals zero.
00:20:36.510 --> 00:20:36.580
columns then the
question is does Ax=b,
00:20:36.580 --> 00:20:37.904
is it always True or false?
00:20:37.904 --> 00:20:40.320
If b squared equals zero, true,
false? you could get help,
00:20:40.320 --> 00:20:40.820
right --
00:20:40.820 --> 00:20:54.590
b squared equals zero, b has to
be a square -- square matrix,
00:20:54.590 --> 00:20:57.960
so that I can
multiply it by itself,
00:20:57.960 --> 00:21:01.910
does that imply that B is zero?
00:21:06.450 --> 00:21:10.470
Are there matrices whose square
could be the zero matrix?
00:21:14.720 --> 00:21:15.630
Yes or no?
00:21:15.630 --> 00:21:17.830
Yes there are.
00:21:17.830 --> 00:21:20.350
There are matrices whose
square is the zero matrix.
00:21:20.350 --> 00:21:21.725
So this statement is false.
00:21:24.280 --> 00:21:26.980
If b squared is zero, we
don't know that b is zero.
00:21:26.980 --> 00:21:31.290
For example -- the best
example is that matrix.
00:21:31.290 --> 00:21:35.510
That matrix is a
dangerous matrix.
00:21:35.510 --> 00:21:42.580
It will come up in later parts
of this course as an example
00:21:42.580 --> 00:21:44.890
of what can go wrong.
00:21:44.890 --> 00:21:47.720
And here is a real
simple -- so this --
00:21:47.720 --> 00:21:52.010
so if I square that without
doing the multiplication
00:21:52.010 --> 00:21:55.940
and finding the
matrix b. matrix,
00:21:55.940 --> 00:22:04.770
I do get the zero
matrix,and it shows --
00:22:04.770 --> 00:22:05.270
OK.
00:22:05.270 --> 00:22:16.220
A system of m
equations in m unknowns
00:22:16.220 --> 00:22:30.620
is solvable for
every right-hand side
00:22:30.620 --> 00:22:36.490
if the columns are independent.
00:22:36.490 --> 00:22:39.870
So can I say that again, I'm --
00:22:39.870 --> 00:22:41.750
I'll write it down then
00:22:41.750 --> 00:22:42.440
OK.
00:22:42.440 --> 00:22:43.820
for short.
00:22:43.820 --> 00:22:47.270
m by m matrix independent
00:22:47.270 --> 00:22:52.110
So give me a basis for --
00:22:52.110 --> 00:22:56.250
for the null space of B.
00:22:56.250 --> 00:22:57.630
Let's see.
00:22:57.630 --> 00:23:03.150
but you could only
use each way once,
00:23:03.150 --> 00:23:09.370
so you couldn't like
use them all the time.
00:23:09.370 --> 00:23:11.440
So remember that?
00:23:11.440 --> 00:23:17.650
You could -- so you
could poll the audience,
00:23:17.650 --> 00:23:21.800
and that was a very --
00:23:21.800 --> 00:23:26.630
that was a hundred
percent successful way,
00:23:26.630 --> 00:23:31.460
so I'll poll the
audience on this.
00:23:31.460 --> 00:23:34.920
If the other possibility --
00:23:34.920 --> 00:23:40.440
another possibility you could
call your friend, right,
00:23:40.440 --> 00:23:48.030
or he's your friend until he
gives you the wrong answer,
00:23:48.030 --> 00:23:49.410
which --
00:23:49.410 --> 00:23:57.010
that turned out subspaces,
then A is some multiple of B.
00:23:57.010 --> 00:24:01.840
to be very unreliable,
you know, you'd
00:24:01.840 --> 00:24:05.990
call up your
brother or something
00:24:05.990 --> 00:24:12.200
and ask him for the capital
of whatever, Bosnia.
00:24:12.200 --> 00:24:15.940
What does he know,
he makes some guess,
00:24:15.940 --> 00:24:29.560
So what -- I would just take
extreme cases if it was me,
00:24:29.560 --> 00:24:32.602
matrix, independent columns,
is Ax=b always solvable?
00:24:32.602 --> 00:24:33.810
Maybe just hands up for that?
00:24:33.810 --> 00:24:34.310
A few.
00:24:39.360 --> 00:24:42.420
And who says no?
00:24:42.420 --> 00:24:46.730
Oh, gosh, this audience
is not reliable.
00:24:46.730 --> 00:24:47.230
Fifty fifty.
00:24:47.230 --> 00:24:51.070
I guess I'd say, I'd vote yes.
00:24:51.070 --> 00:24:54.120
Because independent
columns, that
00:24:54.120 --> 00:24:57.190
means that the rank
is the full size m,
00:24:57.190 --> 00:24:59.890
I have a matrix of rank m.
00:24:59.890 --> 00:25:01.530
That means it's --
00:25:01.530 --> 00:25:04.710
I mean it's square, so
it's an invertible matrix,
00:25:04.710 --> 00:25:07.330
and nothing could go wrong.
00:25:07.330 --> 00:25:08.090
Yes.
00:25:08.090 --> 00:25:12.070
So that's the good case and
we always expect it in chapter
00:25:12.070 --> 00:25:18.130
two, but of course
chapter three is --
00:25:18.130 --> 00:25:24.990
only one of the possibilities.
00:25:24.990 --> 00:25:26.360
OK.
00:25:26.360 --> 00:25:41.460
Let me move on to another
question from an old quiz.
00:25:41.460 --> 00:25:42.840
OK.
00:25:42.840 --> 00:25:44.210
OK.
00:25:44.210 --> 00:25:46.960
Let's see.
00:25:46.960 --> 00:26:07.550
I'm going to give you a matrix,
but I'm going to give it to you
00:26:07.550 --> 00:26:08.920
OK.
00:26:08.920 --> 00:26:19.910
as a product of a
couple of matrices,
00:26:19.910 --> 00:26:32.260
one, one, zero, zero,
one, zero, one, zero, one,
00:26:32.260 --> 00:26:39.130
times another matrix, one, zero,
00:26:39.130 --> 00:26:52.860
I would like to ask you
questions about that matrix
00:26:52.860 --> 00:27:07.960
minus one, two, zero, one,
one, minus one, and all zeroes.
00:27:07.960 --> 00:27:09.330
OK.
00:27:09.330 --> 00:27:11.070
Let's see, what
dimension I in --
00:27:11.070 --> 00:27:17.210
the null space of B
is a subspace of R.
00:27:17.210 --> 00:27:20.110
What size vectors
I looking for here?
00:27:20.110 --> 00:27:24.124
Because if we don't
know the size,
00:27:24.124 --> 00:27:26.040
we aren't going to find
it, right? the null --
00:27:26.040 --> 00:27:33.530
this matrix is three
by four obviously.
00:27:33.530 --> 00:27:37.950
So if we're looking for the null
space we're looking for those
00:27:37.950 --> 00:27:39.870
vectors x in R^4.
00:27:39.870 --> 00:27:40.830
OK.
00:27:40.830 --> 00:27:45.490
So the null space of B is
certainly a subspace of R^4.
00:27:45.490 --> 00:27:48.490
What do you think
its dimension is?
00:27:48.490 --> 00:27:51.930
Of course once we
find the basis we
00:27:51.930 --> 00:27:55.010
would know the dimension
immediately, but let's
00:27:55.010 --> 00:28:00.590
stop first, what's the
rank of this matrix B?
00:28:00.590 --> 00:28:05.010
Let's see, what -- is
that matrix invertible,
00:28:05.010 --> 00:28:06.306
that square one there?
00:28:10.890 --> 00:28:15.110
Let's say sure, I
think it is, yes,
00:28:15.110 --> 00:28:17.330
that matrix B looks invertible.
00:28:17.330 --> 00:28:21.230
Is that pretty clear?
00:28:21.230 --> 00:28:23.090
Yeah.
00:28:23.090 --> 00:28:24.180
Yeah.
00:28:24.180 --> 00:28:28.450
So I've gone wrong in
this course already,
00:28:28.450 --> 00:28:33.210
but I'll still hope that
that matrix is invertible.
00:28:33.210 --> 00:28:36.650
Yeah, yeah, because if I look
for a combination of those
00:28:36.650 --> 00:28:41.600
three columns -- well, I
couldn't use this middle column
00:28:41.600 --> 00:28:46.890
because it would have a one
and in a position that I --
00:28:46.890 --> 00:28:51.370
column is otherwise all zero,
so a combination that gives zero
00:28:51.370 --> 00:28:56.180
can't give us that problem, and
then the other two are clearly
00:28:56.180 --> 00:29:00.090
independent sets -- so
that matrix is invertible.
00:29:00.090 --> 00:29:03.600
Later we could take a
determinant or other things.
00:29:03.600 --> 00:29:04.100
OK.
00:29:04.100 --> 00:29:06.500
What's the setup?
00:29:06.500 --> 00:29:11.850
If I have an invertible matrix,
a nice invertible square
00:29:11.850 --> 00:29:16.620
matrix, times this guy,
times this second factor,
00:29:16.620 --> 00:29:18.840
and I'm looking
for the null space,
00:29:18.840 --> 00:29:20.305
does this have any effect?
00:29:22.960 --> 00:29:26.490
Is the null -- so what I'm
asking is is the null space
00:29:26.490 --> 00:29:33.500
of B the same as the null
space of just this part?
00:29:33.500 --> 00:29:35.950
I think so.
00:29:35.950 --> 00:29:36.820
I think so.
00:29:36.820 --> 00:29:41.470
If Bx is zero, then
multiplying by that guy
00:29:41.470 --> 00:29:42.870
I'll still have zero.
00:29:42.870 --> 00:29:47.970
But also if this times
some x give zero,
00:29:47.970 --> 00:29:51.550
I could always multiply on the
left by the inverse of that,
00:29:51.550 --> 00:29:55.100
because it is invertible,
and I would discover
00:29:55.100 --> 00:29:59.040
that this kind of Bx is zero.
00:29:59.040 --> 00:30:02.700
You want me to write
some of that down --
00:30:02.700 --> 00:30:06.730
if I have a product
here, C times --
00:30:06.730 --> 00:30:14.040
times D, say, and
if C is invertible,
00:30:14.040 --> 00:30:18.660
the null space of CD,
well, it will the same
00:30:18.660 --> 00:30:20.880
as the null space of D.
00:30:20.880 --> 00:30:23.770
If C is invertible.
00:30:26.600 --> 00:30:29.570
Multiplying by an invertible
matrix on the left
00:30:29.570 --> 00:30:32.520
can't change the null space.
00:30:32.520 --> 00:30:33.050
OK.
00:30:33.050 --> 00:30:35.550
So basically I'm asking you
for the null space of this.
00:30:35.550 --> 00:30:38.920
so when do I know -- well, I
would say suppose the matrix
00:30:38.920 --> 00:30:40.540
I don't have to do
the multiplication
00:30:40.540 --> 00:30:43.210
because I have C is invertible.
00:30:43.210 --> 00:30:46.220
That first factor
C is invertible.
00:30:46.220 --> 00:30:48.550
It's not going to
change the null space.
00:30:48.550 --> 00:30:49.320
OK.
00:30:49.320 --> 00:30:55.430
So can we just write down a
basis now for the null space?
00:30:55.430 --> 00:30:59.060
So what's the basis for the
null space of -- of that?
00:30:59.060 --> 00:31:06.080
So basis for the null space
I'm looking for the two --
00:31:06.080 --> 00:31:08.330
there are two pivot
columns obviously.
00:31:08.330 --> 00:31:10.200
It clearly has rank two.
00:31:10.200 --> 00:31:15.450
If -- so true or false, if
A and B have the same four
00:31:15.450 --> 00:31:18.070
I'm looking for the
two special solutions.
00:31:18.070 --> 00:31:21.070
They'll come from the
third and the fourth.
00:31:21.070 --> 00:31:22.200
The free variables.
00:31:22.200 --> 00:31:25.940
OK. so if the third
free variable is a one,
00:31:25.940 --> 00:31:29.690
then I think probably I
need a minus one there
00:31:29.690 --> 00:31:32.320
and a one there, it looks like.
00:31:32.320 --> 00:31:36.690
Do you agree that if I then do
that multiplication I'll get
00:31:36.690 --> 00:31:37.190
zero?
00:31:37.190 --> 00:31:40.940
And if I have one in the
fourth variable, then
00:31:40.940 --> 00:31:46.190
maybe I need a one in the second
variable and maybe a minus two
00:31:46.190 --> 00:31:47.970
in the third.
00:31:47.970 --> 00:31:51.310
So I just reasoned that through
and then if I look back I see
00:31:51.310 --> 00:31:55.450
sure enough that the free
variable part that I sometimes
00:31:55.450 --> 00:31:59.410
call F, that up --
that two by two corner,
00:31:59.410 --> 00:32:06.250
is sitting here with
all its signs reversed.
00:32:06.250 --> 00:32:08.720
So that's -- here
I'm seeing minus F,
00:32:08.720 --> 00:32:13.110
and here I'm seeing the identity
in the null space matrix.
00:32:13.110 --> 00:32:15.460
OK, so that's the null space.
00:32:15.460 --> 00:32:23.980
Another question is solve
Bx equal one, zero, one.
00:32:23.980 --> 00:32:25.190
OK.
00:32:25.190 --> 00:32:34.890
So that's one question, now
solve complete solutions.
00:32:34.890 --> 00:32:39.090
To Bx equal one, zero, one.
00:32:42.360 --> 00:32:45.200
OK.
00:32:45.200 --> 00:32:49.020
Yeah, so I guess I'm
seeing if I wanted
00:32:49.020 --> 00:32:54.790
to get one, zero, one - What's
our particular solution?
00:32:54.790 --> 00:32:56.460
So I'm looking for a
particular solution
00:32:56.460 --> 00:32:57.720
and then the null space
00:32:57.720 --> 00:32:59.390
part.
00:32:59.390 --> 00:33:00.090
OK.
00:33:00.090 --> 00:33:05.360
I-- actually the
first column of B,
00:33:05.360 --> 00:33:08.840
so what's the first
column of our matrix B?
00:33:08.840 --> 00:33:10.770
It's the vector one, zero, one.
00:33:10.770 --> 00:33:14.140
The first column of
our matrix agrees
00:33:14.140 --> 00:33:15.430
with the right-hand side.
00:33:15.430 --> 00:33:18.670
So I guess I'm
thinking x particular
00:33:18.670 --> 00:33:22.570
plus x null space will be
the particular solution,
00:33:22.570 --> 00:33:27.814
since the first column of B is
exactly right, that's great.
00:33:27.814 --> 00:33:29.980
And then I have C times
that first null space vector
00:33:29.980 --> 00:33:37.881
and D times the other
null space vector.
00:33:37.881 --> 00:33:38.380
Right?
00:33:38.380 --> 00:33:42.700
The two -- the null space
part of the solution,
00:33:42.700 --> 00:33:48.060
as always has the
arbitrary constants,
00:33:48.060 --> 00:33:51.380
the particular solution doesn't
have any arbitrary constants,
00:33:51.380 --> 00:33:54.840
it's one particular solution,
and in this case it'll --
00:33:54.840 --> 00:33:56.160
that one would do.
00:33:56.160 --> 00:33:57.070
OK.
00:33:57.070 --> 00:33:57.750
Fine.
00:33:57.750 --> 00:34:03.030
so those are questions
taken from old quizzes,
00:34:03.030 --> 00:34:09.781
any questions coming to mind?
00:34:09.781 --> 00:34:10.280
Yeah.
00:34:10.280 --> 00:34:10.780
Q: value.
00:34:10.780 --> 00:34:11.300
OK.
00:34:11.300 --> 00:34:15.000
Well, so that
particular x particular,
00:34:15.000 --> 00:34:22.330
it says that let's see,
when I multiply by this guy,
00:34:22.330 --> 00:34:26.850
I'm going to get the
first column of B.
00:34:26.850 --> 00:34:30.050
That -- if that's a
solution, I multiply B,
00:34:30.050 --> 00:34:34.060
B times this x will be
the first column of B,
00:34:34.060 --> 00:34:39.480
and so I'm saying that the first
column of this B agrees with
00:34:39.480 --> 00:34:41.300
the right-hand side.
00:34:41.300 --> 00:34:47.340
So I'm saying that look at the
first column of that matrix B.
00:34:47.340 --> 00:34:50.780
If you do the
multiplication, it's --
00:34:50.780 --> 00:34:52.659
so what's the first
column of that matrix?
00:34:52.659 --> 00:34:54.370
Is that how you do
that multiplication?
00:34:54.370 --> 00:34:56.203
I multiply that matrix
by that first column.
00:34:56.203 --> 00:34:59.400
And it picks out one, zero, one.
00:34:59.400 --> 00:35:11.020
So the first column
of B is exactly that.
00:35:11.020 --> 00:35:18.650
And therefore a particular
solution will be this guy.
00:35:18.650 --> 00:35:19.960
So I'll repeat that question.
00:35:19.960 --> 00:35:21.150
Yeah.
00:35:21.150 --> 00:35:21.650
OK.
00:35:21.650 --> 00:35:22.150
Yes.
00:35:22.150 --> 00:35:23.170
Q: particular solution.
00:35:23.170 --> 00:35:26.320
Any of the solutions can
be the particular one
00:35:26.320 --> 00:35:28.180
that we pick out.
00:35:28.180 --> 00:35:31.940
So like this plus -- plus this
would be another particular
00:35:31.940 --> 00:35:32.610
OK.
00:35:32.610 --> 00:35:33.110
solution.
00:35:43.290 --> 00:35:48.260
It would be another solution.
00:35:48.260 --> 00:35:53.040
The particular is just
telling us only take one.
00:35:53.040 --> 00:35:57.370
But it's not telling us
which one we have to take.
00:35:57.370 --> 00:35:58.850
We take the most convenient one.
00:35:58.850 --> 00:36:02.560
I guess in this -- in this
problem that was that one.
00:36:02.560 --> 00:36:03.480
Good.
00:36:03.480 --> 00:36:05.750
Other questions?
00:36:05.750 --> 00:36:06.250
Yes.
00:36:06.250 --> 00:36:10.550
And this pattern
of particular plus
00:36:10.550 --> 00:36:15.370
null space, of course, that's
going throughout mathematics
00:36:15.370 --> 00:36:17.770
of linear systems.
00:36:17.770 --> 00:36:20.970
We're really doing mathematics
of linear systems here.
00:36:20.970 --> 00:36:24.070
Our systems are discrete and
they're finite-dimensional --
00:36:24.070 --> 00:36:31.860
and so it's linear algebra, but
this particular plus null space
00:36:31.860 --> 00:36:37.850
goes -- that doesn't depend
on having finite matrices --
00:36:37.850 --> 00:36:42.990
that spreads much --
that spreads everywhere.
00:36:42.990 --> 00:36:46.280
OK, I'm going to just
like to encourage
00:36:46.280 --> 00:36:48.370
you to take problems
out of the book,
00:36:48.370 --> 00:36:50.250
let me do the same myself.
00:36:50.250 --> 00:36:54.160
OK well here's some
easy true or falses.
00:36:54.160 --> 00:36:58.440
I don't know why the
author put these in here.
00:36:58.440 --> 00:36:59.530
OK.
00:36:59.530 --> 00:37:03.720
If m=n, then the row space
equals the column space.
00:37:03.720 --> 00:37:07.390
is invertible -- suppose
A is an invertible
00:37:07.390 --> 00:37:09.790
So these are true or falses.
00:37:09.790 --> 00:37:13.230
If m equals n, so that
means the matrix is square,
00:37:13.230 --> 00:37:17.410
then the row space
equals the column space?
00:37:17.410 --> 00:37:18.760
False, good.
00:37:18.760 --> 00:37:20.260
Good, what is equal there?
00:37:20.260 --> 00:37:25.830
What can I say is
equal, if M -- well,
00:37:25.830 --> 00:37:27.440
yeah.
00:37:27.440 --> 00:37:29.610
Yeah it -- so that's
definitely false --
00:37:29.610 --> 00:37:31.160
the row space and
the column space,
00:37:31.160 --> 00:37:37.550
and this matrix is like always
a good example to consider.
00:37:37.550 --> 00:37:40.530
So there's a square
matrix but it's row
00:37:40.530 --> 00:37:44.200
space is the multiples
of zero, one,
00:37:44.200 --> 00:37:48.310
and its column space is
the multiples of one, zero.
00:37:48.310 --> 00:37:50.240
Very different.
00:37:50.240 --> 00:37:51.680
The row space and
the column space
00:37:51.680 --> 00:37:54.230
are totally different
for that matrix.
00:37:54.230 --> 00:37:57.730
Now of course if the
matrix was symmetric,
00:37:57.730 --> 00:38:01.780
well, then clearly the row
space equals the column space.
00:38:01.780 --> 00:38:02.280
OK.
00:38:02.280 --> 00:38:03.930
How about this question?
00:38:03.930 --> 00:38:08.910
The matrices A and minus A
share the same four subspaces?
00:38:11.490 --> 00:38:15.230
Do the matrices A and minus
A have the same column space,
00:38:15.230 --> 00:38:17.100
do they have the
same null space,
00:38:17.100 --> 00:38:18.480
do they have the same row
00:38:18.480 --> 00:38:19.050
space?
00:38:19.050 --> 00:38:20.960
What's the answer on that?
00:38:20.960 --> 00:38:31.120
Yes or no.
00:38:31.120 --> 00:38:31.620
Yes.
00:38:31.620 --> 00:38:32.310
Good.
00:38:32.310 --> 00:38:33.100
OK.
00:38:33.100 --> 00:38:33.900
How about this?
00:38:33.900 --> 00:38:40.020
If A and B have the
same four subspaces,
00:38:40.020 --> 00:38:41.470
then A is a multiple of B.
00:38:41.470 --> 00:38:43.261
If -- suppose those
subspaces are the same.
00:38:43.261 --> 00:38:44.680
Then is A a multiple of B?
00:38:44.680 --> 00:38:45.510
OK.
00:38:45.510 --> 00:38:53.000
How how do you answer
a question like that?
00:38:53.000 --> 00:38:56.500
Of course if you want to
answer it yes, then I would --
00:38:56.500 --> 00:38:59.460
then they'd have to
think of a reason why.
00:38:59.460 --> 00:39:02.050
If you want to answer no
way, then you would --
00:39:02.050 --> 00:39:06.410
and I would sort of like
first I would try to think no,
00:39:06.410 --> 00:39:10.540
I would say can I come up with
an example where it isn't true?
00:39:10.540 --> 00:39:17.680
Let me repeat the question.
00:39:17.680 --> 00:39:24.810
And then write the answer.
00:39:24.810 --> 00:39:30.520
matrix, then what --
00:39:30.520 --> 00:39:40.510
suppose it's six by
six invertible matrix,
00:39:40.510 --> 00:39:59.070
then what's its row space, and
its column space is all of R^6,
00:39:59.070 --> 00:40:17.620
and the null space, and the null
space of A transpose would be
00:40:17.620 --> 00:40:21.900
the zero vector.
00:40:21.900 --> 00:40:36.170
So every invertible matrix
is going to give that answer.
00:40:36.170 --> 00:40:49.020
If I have a six by
six invertible matrix,
00:40:49.020 --> 00:40:51.602
I know what those subspaces are.
00:40:51.602 --> 00:40:53.060
Heck, that was back
in chapter two,
00:40:53.060 --> 00:40:55.720
when I didn't even know
what subspaces were.
00:40:55.720 --> 00:41:01.890
The row space and column space
are both all six-dimensional
00:41:01.890 --> 00:41:05.470
space -- the whole space,
and the rank is six,
00:41:05.470 --> 00:41:10.280
in other words, and the null
spaces have zero dimension.
00:41:10.280 --> 00:41:11.870
So do you see now the answer?
00:41:11.870 --> 00:41:13.060
So A and B could be.
00:41:13.060 --> 00:41:16.970
So A and B could be for example
any -- so I'm going to say
00:41:16.970 --> 00:41:17.470
false.
00:41:17.470 --> 00:41:21.070
Because A and B for example --
00:41:21.070 --> 00:41:34.230
So an example: A and B any
invertible six by six, six
00:41:34.230 --> 00:41:34.730
by six.
00:41:34.730 --> 00:41:39.230
So those would have
the same four subspaces
00:41:39.230 --> 00:41:40.480
but they wouldn't be the same.
00:41:40.480 --> 00:41:44.890
Of course th- there should be
something about those matrices
00:41:44.890 --> 00:41:45.930
that would be the same.
00:41:45.930 --> 00:41:50.070
It's sort of a natural
problem, so now actually we're
00:41:50.070 --> 00:41:52.120
getting to a math question.
00:41:52.120 --> 00:41:54.390
The answer is this is not true.
00:41:54.390 --> 00:41:59.300
One matrix doesn't have to
be a multiple of the other.
00:41:59.300 --> 00:42:02.420
But there must be
something that's true.
00:42:02.420 --> 00:42:07.770
And that would be sort of like
a natural question to ask.
00:42:07.770 --> 00:42:15.850
If they have the same
subspaces, same four subspaces,
00:42:15.850 --> 00:42:29.700
then what -- what could you --
00:42:29.700 --> 00:42:32.480
instinct wasn't
necessarily right.
00:42:32.480 --> 00:42:37.700
But I hope you now see that
the correct answer is false.
00:42:37.700 --> 00:42:40.550
And then you might think
OK, well, they certainly
00:42:40.550 --> 00:42:43.530
do have the same rank.
00:42:43.530 --> 00:42:51.860
But do -- obviously if they
have the same four subspaces,
00:42:51.860 --> 00:42:54.460
they have the same rank.
00:42:54.460 --> 00:42:58.620
I might say if
they have the well,
00:42:58.620 --> 00:43:03.820
I could extend that question and
think about other possibilities
00:43:03.820 --> 00:43:08.500
and finally come up with
something that was true.
00:43:08.500 --> 00:43:11.620
But I won't press that one.
00:43:11.620 --> 00:43:15.270
Let me keep going with
practice questions.
00:43:15.270 --> 00:43:19.430
And these practice questions
are quite appropriate I
00:43:19.430 --> 00:43:21.510
think for the exam.
00:43:21.510 --> 00:43:22.030
OK.
00:43:22.030 --> 00:43:23.070
let's see.
00:43:23.070 --> 00:43:29.310
If I exchange two rows of A
which subspaces stay the same?
00:43:29.310 --> 00:43:32.950
So I'm trying to
take out questions
00:43:32.950 --> 00:43:38.670
that we can answer without you
know we can answer quickly.
00:43:38.670 --> 00:43:45.560
If I have a matrix A, and I
exchange two of its rows, which
00:43:45.560 --> 00:43:47.570
subspaces stay the same?
00:43:47.570 --> 00:43:50.560
The row space does
stay the same.
00:43:50.560 --> 00:43:53.440
And the null space
stays the same.
00:43:53.440 --> 00:43:54.270
Good.
00:43:54.270 --> 00:43:55.200
Good.
00:43:55.200 --> 00:43:56.690
Correct.
00:43:56.690 --> 00:43:59.170
Column space would
be a wrong answer.
00:43:59.170 --> 00:44:00.280
OK.
00:44:00.280 --> 00:44:03.140
all right, here's a question.
00:44:03.140 --> 00:44:05.730
Oh, this leads into
the next chapter.
00:44:05.730 --> 00:44:08.990
Why can the vector
one, two, three not
00:44:08.990 --> 00:44:12.280
be a row and also
in the null space?
00:44:12.280 --> 00:44:15.040
Fitting we close
with this question.
00:44:15.040 --> 00:44:19.860
Close is -- so V
equal this one, two,
00:44:19.860 --> 00:44:33.900
three can't be in the null space
of a matrix and the row space.
00:44:33.900 --> 00:44:40.690
And my question is why not?
00:44:40.690 --> 00:44:41.378
Why not?
00:44:45.030 --> 00:44:47.950
So this is a
question that we can
00:44:47.950 --> 00:44:51.360
because it's sort of asked
in a straightforward way,
00:44:51.360 --> 00:44:55.410
we can figure out an answer.
00:44:55.410 --> 00:44:57.460
Well, actually yeah --
00:44:57.460 --> 00:45:00.550
I'll even pin it down, it
can't be in the null space --
00:45:00.550 --> 00:45:04.420
and be a row.
00:45:04.420 --> 00:45:07.050
I'll even pin it down further.
00:45:07.050 --> 00:45:12.550
Ask it to be a row of A.
00:45:12.550 --> 00:45:15.000
Why not?
00:45:15.000 --> 00:45:18.820
So I'm -- we know the
dimensions of these spaces.
00:45:18.820 --> 00:45:26.330
But now I'm asking you sort
of like the overlap between --
00:45:26.330 --> 00:45:30.320
so the null space
and the row space,
00:45:30.320 --> 00:45:33.210
those are in the same
n-dimensional space.
00:45:33.210 --> 00:45:41.410
Those are -- well, those are
both subspaces of n-dimensional
00:45:41.410 --> 00:45:45.060
space, and I'm basically
saying they can't overlap.
00:45:45.060 --> 00:45:49.040
I can't have a vector like
this, a typical vector, that's
00:45:49.040 --> 00:45:53.110
in the null space and it's
also a row of the matrix.
00:45:53.110 --> 00:45:54.580
Why is that?
00:45:54.580 --> 00:45:56.350
So that's a new sort of idea.
00:45:56.350 --> 00:45:58.320
Let's just see
what it would mean.
00:45:58.320 --> 00:46:04.860
I mean that A times this V,
why can this A times this
00:46:04.860 --> 00:46:07.770
V it can't be zero.
00:46:07.770 --> 00:46:12.410
Oh well, if it's
zero, so this is --
00:46:12.410 --> 00:46:16.830
I'm getting it into
the null space here.
00:46:16.830 --> 00:46:20.750
So this is -- now let's put
that vector's in the null space,
00:46:20.750 --> 00:46:27.600
why can't the first row of
a matrix be one, two, three?
00:46:27.600 --> 00:46:35.470
I can fill out the
matrix as I like.
00:46:35.470 --> 00:46:37.010
Why is that impossible?
00:46:37.010 --> 00:46:39.310
Well, you're seeing
it's impossible, right?
00:46:39.310 --> 00:46:44.690
That if that was a row of the
matrix and in the null space,
00:46:44.690 --> 00:46:48.421
that number would not be
zero, it would be fourteen.
00:46:48.421 --> 00:46:48.920
Right.
00:46:48.920 --> 00:46:52.960
So now we actually are beginning
to get a more complete picture
00:46:52.960 --> 00:46:55.170
of these four subspaces.
00:46:55.170 --> 00:47:00.220
The two that are over
in n-dimensional space,
00:47:00.220 --> 00:47:04.450
they actually only
share the zero vector.
00:47:04.450 --> 00:47:08.110
The intersection of the
null space and the row space
00:47:08.110 --> 00:47:09.310
is only the zero vector.
00:47:09.310 --> 00:47:14.970
And in fact the null space is
perpendicular to the row space.
00:47:14.970 --> 00:47:18.920
That'll be the first
topic let's see,
00:47:18.920 --> 00:47:23.940
we have a holiday Monday --
00:47:23.940 --> 00:47:29.800
and I'll see you Wednesday
with perpendiculars.
00:47:29.800 --> 00:47:33.980
And I'll see you Friday.
00:47:33.980 --> 00:47:39.710
So good luck on the quiz.