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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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00:02:58,140 --> 00:03:01,820
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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00:04:18,089 --> 00:04:24,950
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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00:04:37,340 --> 00:04:40,630
and maybe I should be
writing R here now.
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00:04:40,630 --> 00:04:47,800
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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00:05:04,160 --> 00:05:06,270
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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00:05:07,110 --> 00:05:09,880
U is in reduced row
echelon form but now
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00:05:09,880 --> 00:05:15,500
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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00:05:29,150 --> 00:05:32,950
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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00:05:51,820 --> 00:05:53,921
U and z- or R --
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00:05:53,921 --> 00:05:56,170
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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00:08:14,740 --> 00:08:17,750
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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00:08:37,250 --> 00:08:40,854
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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00:08:42,020 --> 00:08:42,080
Six, right.
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00:08:42,080 --> 00:08:42,289
That has rank six, I can tell.
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00:08:42,289 --> 00:08:43,030
What was -- what's the
rank of this B, while --
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00:08:43,030 --> 00:08:43,159
while we're at on TV for a
few weeks, did you see that,
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00:08:43,159 --> 00:08:43,659
it?
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The rank of B, is
that six or three?
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00:08:56,690 --> 00:08:58,200
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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00:08:59,940 --> 00:09:02,520
Because we actually got it to
where we could just see three
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pivots.
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00:09:03,330 --> 00:09:08,230
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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00:09:15,020 --> 00:09:18,410
space, of the null space --
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00:09:18,410 --> 00:09:22,370
of the null space
of C transpose?
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00:09:22,370 --> 00:09:22,980
Oh, boy.
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00:09:22,980 --> 00:09:27,260
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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00:09:32,450 --> 00:09:38,200
It looks like it's
ten by six, is it?
169
00:09:38,200 --> 00:09:49,280
Ten by six, so C is ten by six,
so m is ten, so C has ten rows,
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00:09:49,280 --> 00:09:54,190
C transpose has ten columns,
so there are ten columns there.
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00:09:54,190 --> 00:09:57,950
So how many free variables
have I got, once I --
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00:09:57,950 --> 00:10:00,950
There's quite a few trues,
shall I take a poll,
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00:10:00,950 --> 00:10:03,950
if I start with the ten
columns in C transpose,
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that's the m for the original C.
175
00:10:06,050 --> 00:10:07,640
And what do I subtract off?
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Six.
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00:10:08,140 --> 00:10:10,240
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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00:10:12,040 --> 00:10:12,540
OK.
181
00:10:12,540 --> 00:10:16,840
If b is in the -- and what
would -- what does the exam say
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00:10:16,840 --> 00:10:19,080
So I think that's
the right answer --
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00:10:19,080 --> 00:10:23,580
the dimension of the null space
of C transpose would be four.
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00:10:23,580 --> 00:10:24,870
Right.
185
00:10:24,870 --> 00:10:25,870
OK.
186
00:10:25,870 --> 00:10:27,570
Yeah.
187
00:10:27,570 --> 00:10:28,650
OK.
188
00:10:28,650 --> 00:10:34,710
So that's one question, at
least it brought in some --
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00:10:34,710 --> 00:10:37,130
some of the dimension counts.
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00:10:37,130 --> 00:10:37,630
OK.
191
00:10:37,630 --> 00:10:40,088
Here's another type of question.
think about, what's the --
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00:10:40,088 --> 00:10:44,520
what's the shape of the matrix,
193
00:10:44,520 --> 00:10:52,870
I give you an equation,
Ax equals two four two.
194
00:10:52,870 --> 00:10:54,590
And I give you the
complete solution.
195
00:10:54,590 --> 00:10:57,992
And what's its rank?
196
00:10:57,992 --> 00:10:59,325
But I don't give you the matrix.
197
00:11:05,390 --> 00:11:10,810
And another -- there's another
vector, zero, zero, one.
198
00:11:10,810 --> 00:11:11,310
OK.
199
00:11:15,951 --> 00:11:16,450
All right.
200
00:11:16,450 --> 00:11:22,660
My first question is what's
the dimension of the row space?
201
00:11:22,660 --> 00:11:24,160
Of the matrix A?
202
00:11:24,160 --> 00:11:31,890
So the main thing that you
want to get from this question
203
00:11:31,890 --> 00:11:36,130
is that a question
could start this way.
204
00:11:36,130 --> 00:11:38,770
exam, don't just tell me.
205
00:11:38,770 --> 00:11:40,890
Sort of backward way.
206
00:11:40,890 --> 00:11:45,130
so I guess I'm
asking you what is
207
00:11:45,130 --> 00:11:50,420
the column space for this
basis for the null space.
208
00:11:50,420 --> 00:11:56,770
By giving you the answer and
not telling you what the problem
209
00:11:56,770 --> 00:12:02,590
homework, so I was watching
it, so there were three --
210
00:12:02,590 --> 00:12:03,650
the is.
211
00:12:03,650 --> 00:12:07,890
But we can get a
lot of information,
212
00:12:07,890 --> 00:12:11,590
and sometimes we can
get complete information
213
00:12:11,590 --> 00:12:13,710
about that matrix A.
214
00:12:13,710 --> 00:12:14,240
OK.
215
00:12:14,240 --> 00:12:19,530
So what's the dimension
of the row space of A?
216
00:12:19,530 --> 00:12:21,120
What's the rank?
217
00:12:21,120 --> 00:12:27,470
Tell me about what's the size
of the matrix, yeah, just --
218
00:12:27,470 --> 00:12:32,230
These are the things
we want to matrix b
219
00:12:32,230 --> 00:12:36,470
and I'll answer them
without multiplying it out.2
220
00:12:36,470 --> 00:12:41,760
Its rank -- tell me something
about its null space,
221
00:12:41,760 --> 00:12:46,000
I heard the right
answer for the rank,
222
00:12:46,000 --> 00:12:49,700
the rank is one in this case.
223
00:12:49,700 --> 00:12:50,230
Why?
224
00:12:50,230 --> 00:12:53,940
Because the dimension
of the null space,
225
00:12:53,940 --> 00:12:59,760
so the dimension of this
instant?3 the null space of A
226
00:12:59,760 --> 00:13:05,050
is from knowing that that's the
complete solution, it's two.
227
00:13:05,050 --> 00:13:08,760
I'm seeing two vectors
here, and they're
228
00:13:08,760 --> 00:13:12,460
independent in the
null space of A,
229
00:13:12,460 --> 00:13:16,170
because they have to be in the
230
00:13:16,170 --> 00:13:20,400
OK, which -- did
you watch that quiz,
231
00:13:20,400 --> 00:13:26,750
there was a quiz program null
space of A if I'm allowed
232
00:13:26,750 --> 00:13:31,650
to throw into the solution
any amount of those vectors,
233
00:13:31,650 --> 00:13:35,620
that tells me that's the
null space part then.
234
00:13:35,620 --> 00:13:38,400
So the dimension of the null
space is two, and then I --
235
00:13:38,400 --> 00:13:44,010
of course I know the dimensions
of all the -- four subspaces.
236
00:13:44,010 --> 00:13:46,090
Now actually it asks
what's the matrix?
237
00:13:46,090 --> 00:13:52,440
Well, what's the
matrix in this case?
238
00:13:52,440 --> 00:14:01,210
Do we want to -- shall I
try to figure that out?
239
00:14:01,210 --> 00:14:01,790
Sure.
240
00:14:01,790 --> 00:14:03,290
Let's -- you'd like
me to do it, OK.
241
00:14:03,290 --> 00:14:04,164
Well, what's the ----
242
00:14:04,164 --> 00:14:05,590
I actually say in this in the
243
00:14:05,590 --> 00:14:09,630
So what about the matrix,
or let me at least start it,
244
00:14:09,630 --> 00:14:10,140
OK.
245
00:14:10,140 --> 00:14:14,720
If A times this x
gives two, four, two,
246
00:14:14,720 --> 00:14:19,300
what does that tell
me about the matrix A?
247
00:14:19,300 --> 00:14:26,210
If A times that x solves that
equation then it tells me that
248
00:14:26,210 --> 00:14:28,900
the first column is --
249
00:14:28,900 --> 00:14:35,560
the first column of A is
-- one, two, one, right.
250
00:14:35,560 --> 00:14:37,580
The first column of A
has to be one, two, one,
251
00:14:37,580 --> 00:14:37,650
because if I
multiply by x, that's
252
00:14:37,650 --> 00:14:37,730
going to multiply just
matrix? the first column,
253
00:14:37,730 --> 00:14:37,790
and give me two, four, two.
254
00:14:37,790 --> 00:14:37,900
And then I've got two
more columns to find,
255
00:14:37,900 --> 00:14:38,000
and what information have
I got to find them with?
256
00:14:38,000 --> 00:14:38,070
A basis for the null space.
257
00:14:38,070 --> 00:14:38,130
I've got the null space.
258
00:14:38,130 --> 00:14:38,230
So the fact that this
is in the null space,
259
00:14:38,230 --> 00:14:38,370
what does that tell So what
is if b has the form --
260
00:14:38,370 --> 00:14:40,411
so I guess I'm asking
what's me about the matrix?
261
00:14:40,411 --> 00:14:59,047
A matrix that has zero,
zero, one in its null space?
262
00:15:05,510 --> 00:15:10,840
That tells me that the last
column of the matrix is zeroes.
263
00:15:10,840 --> 00:15:11,880
so how many think true?
264
00:15:11,880 --> 00:15:13,820
Because this is
in the null space,
265
00:15:13,820 --> 00:15:14,897
the last column has to be
266
00:15:14,897 --> 00:15:16,230
When could it be solved? zeroes.
267
00:15:16,230 --> 00:15:19,510
And because this is in
the null space, what's
268
00:15:19,510 --> 00:15:23,850
the second column?
269
00:15:23,850 --> 00:15:25,880
Well, this in the
null space means
270
00:15:25,880 --> 00:15:27,910
that if I multiply
A by that vector
271
00:15:27,910 --> 00:15:32,530
I must be getting zeroes, so I
think that better be minus one,
272
00:15:32,530 --> 00:15:44,370
minus two, and minus one.
273
00:15:44,370 --> 00:15:45,300
OK.
274
00:15:45,300 --> 00:15:50,340
That's a type of question
that just brings out
275
00:15:50,340 --> 00:15:53,050
the information that's in
that complete solution.
276
00:15:53,050 --> 00:15:53,180
And then actually I go on to
ask what vectors -- for what
277
00:15:53,180 --> 00:15:53,250
OK. vectors B can
Ax=b be solved?
278
00:15:53,250 --> 00:15:53,380
Ax=b can be solved if what --
so I'm looking for a condition
279
00:15:53,380 --> 00:15:53,460
Definitely not. on b,
if any. interesting --
280
00:15:53,460 --> 00:15:53,550
the novel point was there
were three ways that
281
00:15:53,550 --> 00:15:53,640
Can it be solved for
every right-hand side b?
282
00:15:53,640 --> 00:15:54,140
No.
283
00:15:54,140 --> 00:16:07,315
And the answer is --
284
00:16:11,410 --> 00:16:25,505
yes or no --
285
00:16:29,555 --> 00:16:30,054
right.
286
00:16:34,097 --> 00:16:34,596
solvable?
287
00:16:38,700 --> 00:16:44,920
the column space of this
matrix, and what is it?
288
00:16:44,920 --> 00:16:50,810
It's so the column space of
that matrix is all multiples b
289
00:16:50,810 --> 00:16:54,490
Can you tell me something --
290
00:16:54,490 --> 00:16:59,170
so I'll ask questions
about this --
291
00:16:59,170 --> 00:17:04,609
b is a multiple
of one, two, one.
292
00:17:04,609 --> 00:17:05,290
Right?
293
00:17:05,290 --> 00:17:14,140
I can solve the thing if it's
a multiple of one, two, one,
294
00:17:14,140 --> 00:17:18,220
and of course sure enough --
295
00:17:18,220 --> 00:17:24,140
yeah, that was a multiple
of one, two, one,
296
00:17:24,140 --> 00:17:26,829
and so I had a solution.
297
00:17:26,829 --> 00:17:34,640
So this is a case where
we've got lots of null space.
298
00:17:34,640 --> 00:17:44,760
Let me just recall rank is
big, don't forget those cases,
299
00:17:44,760 --> 00:17:57,390
don't forget the other cases
when r is as big as it can be,
300
00:17:57,390 --> 00:18:00,580
OK. r equal m or r equal n.
301
00:18:00,580 --> 00:18:04,970
Those are -- we had a full
lecture on that, the full rank,
302
00:18:04,970 --> 00:18:09,480
full lecture, and important
-- important case.
303
00:18:09,480 --> 00:18:13,471
I gave you every chance
to think about that.
304
00:18:13,471 --> 00:18:13,970
OK.
305
00:18:13,970 --> 00:18:15,760
I'll just move on.
306
00:18:15,760 --> 00:18:19,800
I think this is the
best type of review.
307
00:18:19,800 --> 00:18:23,380
So I'm going to
solve Bx equals zero.
308
00:18:23,380 --> 00:18:26,080
It's just brings
these ideas out.
309
00:18:26,080 --> 00:18:30,511
Apologies to the camera while
I recover glasses and exam.
310
00:18:30,511 --> 00:18:31,010
OK.
311
00:18:31,010 --> 00:18:33,700
How about a few true-false ones?
312
00:18:33,700 --> 00:18:37,100
Actually there won't be
a true-false on the quiz.
313
00:18:37,100 --> 00:18:40,980
But it gives us a
moment of quick review.
314
00:18:40,980 --> 00:18:44,500
True or false, how do
you feel about it at
315
00:18:44,500 --> 00:18:45,190
Here's one.
316
00:18:45,190 --> 00:18:46,930
If the null space --
317
00:18:46,930 --> 00:18:49,400
I have a square matrix.
318
00:18:49,400 --> 00:18:52,740
If its null space is
just the zero vector,
319
00:18:52,740 --> 00:18:57,750
what about the null
space of A transpose?
320
00:18:57,750 --> 00:19:00,840
If the null space of A
is just the zero vector,
321
00:19:00,840 --> 00:19:03,580
and the matrix is
square, what do I
322
00:19:03,580 --> 00:19:06,430
know about the null
space of A transpose?
323
00:19:06,430 --> 00:19:07,420
Also the zero vector.
324
00:19:07,420 --> 00:19:07,960
Good.
325
00:19:07,960 --> 00:19:10,270
And that's a very
very important fact.
326
00:19:10,270 --> 00:19:11,630
OK.
327
00:19:11,630 --> 00:19:15,000
How about this?
328
00:19:15,000 --> 00:19:20,450
These -- look at the space
of five by five matrices
329
00:19:20,450 --> 00:19:23,280
as a vector space.
330
00:19:23,280 --> 00:19:25,690
So it's actually a
twenty-five-dimensional vector
331
00:19:25,690 --> 00:19:26,190
space.
332
00:19:26,190 --> 00:19:28,990
All five by five matrices.
333
00:19:28,990 --> 00:19:31,770
Look at the invertible matrices.
334
00:19:31,770 --> 00:19:35,430
Do they form a subspace?
335
00:19:35,430 --> 00:19:40,430
So I have this five by --
a space of all five by five
336
00:19:40,430 --> 00:19:41,580
matrices.
337
00:19:41,580 --> 00:19:44,180
I can add them, I can
multiply by numbers.
338
00:19:44,180 --> 00:19:48,110
But now I narrow down
to the invertible ones.
339
00:19:48,110 --> 00:19:50,700
And I ask are they a subspace?
340
00:19:50,700 --> 00:19:54,600
And you -- your answer is --
341
00:19:54,600 --> 00:19:57,418
quiet, but nevertheless
definite, no.
342
00:20:00,130 --> 00:20:00,800
Right?
343
00:20:00,800 --> 00:20:04,040
Because if I add two invertible
matrices I have no idea if the
344
00:20:04,040 --> 00:20:06,570
No. answer is invertible.
345
00:20:06,570 --> 00:20:09,340
If I multiply that
invertible -- well,
346
00:20:09,340 --> 00:20:11,600
it doesn't even have
the zero matrix in it,
347
00:20:11,600 --> 00:20:13,630
it couldn't be a subspace.
348
00:20:13,630 --> 00:20:16,020
I have to be able to
multiply by zero --
349
00:20:16,020 --> 00:20:21,010
and stay in my subspace, and the
invertible ones wouldn't work.
350
00:20:21,010 --> 00:20:24,210
Well, the singular ones
wouldn't work either.
351
00:20:24,210 --> 00:20:27,540
They have zero -- the zero
matrix is in the singular
352
00:20:27,540 --> 00:20:31,720
matrices, but if I add two
singular matrices I don't know
353
00:20:31,720 --> 00:20:33,410
if the answer is
singular or not.
354
00:20:33,410 --> 00:20:34,340
OK.
355
00:20:34,340 --> 00:20:36,420
So another true-false.
356
00:20:36,420 --> 00:20:36,510
If b squared equals
zero then b equals zero.
357
00:20:36,510 --> 00:20:36,580
columns then the
question is does Ax=b,
358
00:20:36,580 --> 00:20:37,904
is it always True or false?
359
00:20:37,904 --> 00:20:40,320
If b squared equals zero, true,
false? you could get help,
360
00:20:40,320 --> 00:20:40,820
right --
361
00:20:40,820 --> 00:20:54,590
b squared equals zero, b has to
be a square -- square matrix,
362
00:20:54,590 --> 00:20:57,960
so that I can
multiply it by itself,
363
00:20:57,960 --> 00:21:01,910
does that imply that B is zero?
364
00:21:06,450 --> 00:21:10,470
Are there matrices whose square
could be the zero matrix?
365
00:21:14,720 --> 00:21:15,630
Yes or no?
366
00:21:15,630 --> 00:21:17,830
Yes there are.
367
00:21:17,830 --> 00:21:20,350
There are matrices whose
square is the zero matrix.
368
00:21:20,350 --> 00:21:21,725
So this statement is false.
369
00:21:24,280 --> 00:21:26,980
If b squared is zero, we
don't know that b is zero.
370
00:21:26,980 --> 00:21:31,290
For example -- the best
example is that matrix.
371
00:21:31,290 --> 00:21:35,510
That matrix is a
dangerous matrix.
372
00:21:35,510 --> 00:21:42,580
It will come up in later parts
of this course as an example
373
00:21:42,580 --> 00:21:44,890
of what can go wrong.
374
00:21:44,890 --> 00:21:47,720
And here is a real
simple -- so this --
375
00:21:47,720 --> 00:21:52,010
so if I square that without
doing the multiplication
376
00:21:52,010 --> 00:21:55,940
and finding the
matrix b. matrix,
377
00:21:55,940 --> 00:22:04,770
I do get the zero
matrix,and it shows --
378
00:22:04,770 --> 00:22:05,270
OK.
379
00:22:05,270 --> 00:22:16,220
A system of m
equations in m unknowns
380
00:22:16,220 --> 00:22:30,620
is solvable for
every right-hand side
381
00:22:30,620 --> 00:22:36,490
if the columns are independent.
382
00:22:36,490 --> 00:22:39,870
So can I say that again, I'm --
383
00:22:39,870 --> 00:22:41,750
I'll write it down then
384
00:22:41,750 --> 00:22:42,440
OK.
385
00:22:42,440 --> 00:22:43,820
for short.
386
00:22:43,820 --> 00:22:47,270
m by m matrix independent
387
00:22:47,270 --> 00:22:52,110
So give me a basis for --
388
00:22:52,110 --> 00:22:56,250
for the null space of B.
389
00:22:56,250 --> 00:22:57,630
Let's see.
390
00:22:57,630 --> 00:23:03,150
but you could only
use each way once,
391
00:23:03,150 --> 00:23:09,370
so you couldn't like
use them all the time.
392
00:23:09,370 --> 00:23:11,440
So remember that?
393
00:23:11,440 --> 00:23:17,650
You could -- so you
could poll the audience,
394
00:23:17,650 --> 00:23:21,800
and that was a very --
395
00:23:21,800 --> 00:23:26,630
that was a hundred
percent successful way,
396
00:23:26,630 --> 00:23:31,460
so I'll poll the
audience on this.
397
00:23:31,460 --> 00:23:34,920
If the other possibility --
398
00:23:34,920 --> 00:23:40,440
another possibility you could
call your friend, right,
399
00:23:40,440 --> 00:23:48,030
or he's your friend until he
gives you the wrong answer,
400
00:23:48,030 --> 00:23:49,410
which --
401
00:23:49,410 --> 00:23:57,010
that turned out subspaces,
then A is some multiple of B.
402
00:23:57,010 --> 00:24:01,840
to be very unreliable,
you know, you'd
403
00:24:01,840 --> 00:24:05,990
call up your
brother or something
404
00:24:05,990 --> 00:24:12,200
and ask him for the capital
of whatever, Bosnia.
405
00:24:12,200 --> 00:24:15,940
What does he know,
he makes some guess,
406
00:24:15,940 --> 00:24:29,560
So what -- I would just take
extreme cases if it was me,
407
00:24:29,560 --> 00:24:32,602
matrix, independent columns,
is Ax=b always solvable?
408
00:24:32,602 --> 00:24:33,810
Maybe just hands up for that?
409
00:24:33,810 --> 00:24:34,310
A few.
410
00:24:39,360 --> 00:24:42,420
And who says no?
411
00:24:42,420 --> 00:24:46,730
Oh, gosh, this audience
is not reliable.
412
00:24:46,730 --> 00:24:47,230
Fifty fifty.
413
00:24:47,230 --> 00:24:51,070
I guess I'd say, I'd vote yes.
414
00:24:51,070 --> 00:24:54,120
Because independent
columns, that
415
00:24:54,120 --> 00:24:57,190
means that the rank
is the full size m,
416
00:24:57,190 --> 00:24:59,890
I have a matrix of rank m.
417
00:24:59,890 --> 00:25:01,530
That means it's --
418
00:25:01,530 --> 00:25:04,710
I mean it's square, so
it's an invertible matrix,
419
00:25:04,710 --> 00:25:07,330
and nothing could go wrong.
420
00:25:07,330 --> 00:25:08,090
Yes.
421
00:25:08,090 --> 00:25:12,070
So that's the good case and
we always expect it in chapter
422
00:25:12,070 --> 00:25:18,130
two, but of course
chapter three is --
423
00:25:18,130 --> 00:25:24,990
only one of the possibilities.
424
00:25:24,990 --> 00:25:26,360
OK.
425
00:25:26,360 --> 00:25:41,460
Let me move on to another
question from an old quiz.
426
00:25:41,460 --> 00:25:42,840
OK.
427
00:25:42,840 --> 00:25:44,210
OK.
428
00:25:44,210 --> 00:25:46,960
Let's see.
429
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
430
00:26:07,550 --> 00:26:08,920
OK.
431
00:26:08,920 --> 00:26:19,910
as a product of a
couple of matrices,
432
00:26:19,910 --> 00:26:32,260
one, one, zero, zero,
one, zero, one, zero, one,
433
00:26:32,260 --> 00:26:39,130
times another matrix, one, zero,
434
00:26:39,130 --> 00:26:52,860
I would like to ask you
questions about that matrix
435
00:26:52,860 --> 00:27:07,960
minus one, two, zero, one,
one, minus one, and all zeroes.
436
00:27:07,960 --> 00:27:09,330
OK.
437
00:27:09,330 --> 00:27:11,070
Let's see, what
dimension I in --
438
00:27:11,070 --> 00:27:17,210
the null space of B
is a subspace of R.
439
00:27:17,210 --> 00:27:20,110
What size vectors
I looking for here?
440
00:27:20,110 --> 00:27:24,124
Because if we don't
know the size,
441
00:27:24,124 --> 00:27:26,040
we aren't going to find
it, right? the null --
442
00:27:26,040 --> 00:27:33,530
this matrix is three
by four obviously.
443
00:27:33,530 --> 00:27:37,950
So if we're looking for the null
space we're looking for those
444
00:27:37,950 --> 00:27:39,870
vectors x in R^4.
445
00:27:39,870 --> 00:27:40,830
OK.
446
00:27:40,830 --> 00:27:45,490
So the null space of B is
certainly a subspace of R^4.
447
00:27:45,490 --> 00:27:48,490
What do you think
its dimension is?
448
00:27:48,490 --> 00:27:51,930
Of course once we
find the basis we
449
00:27:51,930 --> 00:27:55,010
would know the dimension
immediately, but let's
450
00:27:55,010 --> 00:28:00,590
stop first, what's the
rank of this matrix B?
451
00:28:00,590 --> 00:28:05,010
Let's see, what -- is
that matrix invertible,
452
00:28:05,010 --> 00:28:06,306
that square one there?
453
00:28:10,890 --> 00:28:15,110
Let's say sure, I
think it is, yes,
454
00:28:15,110 --> 00:28:17,330
that matrix B looks invertible.
455
00:28:17,330 --> 00:28:21,230
Is that pretty clear?
456
00:28:21,230 --> 00:28:23,090
Yeah.
457
00:28:23,090 --> 00:28:24,180
Yeah.
458
00:28:24,180 --> 00:28:28,450
So I've gone wrong in
this course already,
459
00:28:28,450 --> 00:28:33,210
but I'll still hope that
that matrix is invertible.
460
00:28:33,210 --> 00:28:36,650
Yeah, yeah, because if I look
for a combination of those
461
00:28:36,650 --> 00:28:41,600
three columns -- well, I
couldn't use this middle column
462
00:28:41,600 --> 00:28:46,890
because it would have a one
and in a position that I --
463
00:28:46,890 --> 00:28:51,370
column is otherwise all zero,
so a combination that gives zero
464
00:28:51,370 --> 00:28:56,180
can't give us that problem, and
then the other two are clearly
465
00:28:56,180 --> 00:29:00,090
independent sets -- so
that matrix is invertible.
466
00:29:00,090 --> 00:29:03,600
Later we could take a
determinant or other things.
467
00:29:03,600 --> 00:29:04,100
OK.
468
00:29:04,100 --> 00:29:06,500
What's the setup?
469
00:29:06,500 --> 00:29:11,850
If I have an invertible matrix,
a nice invertible square
470
00:29:11,850 --> 00:29:16,620
matrix, times this guy,
times this second factor,
471
00:29:16,620 --> 00:29:18,840
and I'm looking
for the null space,
472
00:29:18,840 --> 00:29:20,305
does this have any effect?
473
00:29:22,960 --> 00:29:26,490
Is the null -- so what I'm
asking is is the null space
474
00:29:26,490 --> 00:29:33,500
of B the same as the null
space of just this part?
475
00:29:33,500 --> 00:29:35,950
I think so.
476
00:29:35,950 --> 00:29:36,820
I think so.
477
00:29:36,820 --> 00:29:41,470
If Bx is zero, then
multiplying by that guy
478
00:29:41,470 --> 00:29:42,870
I'll still have zero.
479
00:29:42,870 --> 00:29:47,970
But also if this times
some x give zero,
480
00:29:47,970 --> 00:29:51,550
I could always multiply on the
left by the inverse of that,
481
00:29:51,550 --> 00:29:55,100
because it is invertible,
and I would discover
482
00:29:55,100 --> 00:29:59,040
that this kind of Bx is zero.
483
00:29:59,040 --> 00:30:02,700
You want me to write
some of that down --
484
00:30:02,700 --> 00:30:06,730
if I have a product
here, C times --
485
00:30:06,730 --> 00:30:14,040
times D, say, and
if C is invertible,
486
00:30:14,040 --> 00:30:18,660
the null space of CD,
well, it will the same
487
00:30:18,660 --> 00:30:20,880
as the null space of D.
488
00:30:20,880 --> 00:30:23,770
If C is invertible.
489
00:30:26,600 --> 00:30:29,570
Multiplying by an invertible
matrix on the left
490
00:30:29,570 --> 00:30:32,520
can't change the null space.
491
00:30:32,520 --> 00:30:33,050
OK.
492
00:30:33,050 --> 00:30:35,550
So basically I'm asking you
for the null space of this.
493
00:30:35,550 --> 00:30:38,920
so when do I know -- well, I
would say suppose the matrix
494
00:30:38,920 --> 00:30:40,540
I don't have to do
the multiplication
495
00:30:40,540 --> 00:30:43,210
because I have C is invertible.
496
00:30:43,210 --> 00:30:46,220
That first factor
C is invertible.
497
00:30:46,220 --> 00:30:48,550
It's not going to
change the null space.
498
00:30:48,550 --> 00:30:49,320
OK.
499
00:30:49,320 --> 00:30:55,430
So can we just write down a
basis now for the null space?
500
00:30:55,430 --> 00:30:59,060
So what's the basis for the
null space of -- of that?
501
00:30:59,060 --> 00:31:06,080
So basis for the null space
I'm looking for the two --
502
00:31:06,080 --> 00:31:08,330
there are two pivot
columns obviously.
503
00:31:08,330 --> 00:31:10,200
It clearly has rank two.
504
00:31:10,200 --> 00:31:15,450
If -- so true or false, if
A and B have the same four
505
00:31:15,450 --> 00:31:18,070
I'm looking for the
two special solutions.
506
00:31:18,070 --> 00:31:21,070
They'll come from the
third and the fourth.
507
00:31:21,070 --> 00:31:22,200
The free variables.
508
00:31:22,200 --> 00:31:25,940
OK. so if the third
free variable is a one,
509
00:31:25,940 --> 00:31:29,690
then I think probably I
need a minus one there
510
00:31:29,690 --> 00:31:32,320
and a one there, it looks like.
511
00:31:32,320 --> 00:31:36,690
Do you agree that if I then do
that multiplication I'll get
512
00:31:36,690 --> 00:31:37,190
zero?
513
00:31:37,190 --> 00:31:40,940
And if I have one in the
fourth variable, then
514
00:31:40,940 --> 00:31:46,190
maybe I need a one in the second
variable and maybe a minus two
515
00:31:46,190 --> 00:31:47,970
in the third.
516
00:31:47,970 --> 00:31:51,310
So I just reasoned that through
and then if I look back I see
517
00:31:51,310 --> 00:31:55,450
sure enough that the free
variable part that I sometimes
518
00:31:55,450 --> 00:31:59,410
call F, that up --
that two by two corner,
519
00:31:59,410 --> 00:32:06,250
is sitting here with
all its signs reversed.
520
00:32:06,250 --> 00:32:08,720
So that's -- here
I'm seeing minus F,
521
00:32:08,720 --> 00:32:13,110
and here I'm seeing the identity
in the null space matrix.
522
00:32:13,110 --> 00:32:15,460
OK, so that's the null space.
523
00:32:15,460 --> 00:32:23,980
Another question is solve
Bx equal one, zero, one.
524
00:32:23,980 --> 00:32:25,190
OK.
525
00:32:25,190 --> 00:32:34,890
So that's one question, now
solve complete solutions.
526
00:32:34,890 --> 00:32:39,090
To Bx equal one, zero, one.
527
00:32:42,360 --> 00:32:45,200
OK.
528
00:32:45,200 --> 00:32:49,020
Yeah, so I guess I'm
seeing if I wanted
529
00:32:49,020 --> 00:32:54,790
to get one, zero, one - What's
our particular solution?
530
00:32:54,790 --> 00:32:56,460
So I'm looking for a
particular solution
531
00:32:56,460 --> 00:32:57,720
and then the null space
532
00:32:57,720 --> 00:32:59,390
part.
533
00:32:59,390 --> 00:33:00,090
OK.
534
00:33:00,090 --> 00:33:05,360
I-- actually the
first column of B,
535
00:33:05,360 --> 00:33:08,840
so what's the first
column of our matrix B?
536
00:33:08,840 --> 00:33:10,770
It's the vector one, zero, one.
537
00:33:10,770 --> 00:33:14,140
The first column of
our matrix agrees
538
00:33:14,140 --> 00:33:15,430
with the right-hand side.
539
00:33:15,430 --> 00:33:18,670
So I guess I'm
thinking x particular
540
00:33:18,670 --> 00:33:22,570
plus x null space will be
the particular solution,
541
00:33:22,570 --> 00:33:27,814
since the first column of B is
exactly right, that's great.
542
00:33:27,814 --> 00:33:29,980
And then I have C times
that first null space vector
543
00:33:29,980 --> 00:33:37,881
and D times the other
null space vector.
544
00:33:37,881 --> 00:33:38,380
Right?
545
00:33:38,380 --> 00:33:42,700
The two -- the null space
part of the solution,
546
00:33:42,700 --> 00:33:48,060
as always has the
arbitrary constants,
547
00:33:48,060 --> 00:33:51,380
the particular solution doesn't
have any arbitrary constants,
548
00:33:51,380 --> 00:33:54,840
it's one particular solution,
and in this case it'll --
549
00:33:54,840 --> 00:33:56,160
that one would do.
550
00:33:56,160 --> 00:33:57,070
OK.
551
00:33:57,070 --> 00:33:57,750
Fine.
552
00:33:57,750 --> 00:34:03,030
so those are questions
taken from old quizzes,
553
00:34:03,030 --> 00:34:09,781
any questions coming to mind?
554
00:34:09,781 --> 00:34:10,280
Yeah.
555
00:34:10,280 --> 00:34:10,780
Q: value.
556
00:34:10,780 --> 00:34:11,300
OK.
557
00:34:11,300 --> 00:34:15,000
Well, so that
particular x particular,
558
00:34:15,000 --> 00:34:22,330
it says that let's see,
when I multiply by this guy,
559
00:34:22,330 --> 00:34:26,850
I'm going to get the
first column of B.
560
00:34:26,850 --> 00:34:30,050
That -- if that's a
solution, I multiply B,
561
00:34:30,050 --> 00:34:34,060
B times this x will be
the first column of B,
562
00:34:34,060 --> 00:34:39,480
and so I'm saying that the first
column of this B agrees with
563
00:34:39,480 --> 00:34:41,300
the right-hand side.
564
00:34:41,300 --> 00:34:47,340
So I'm saying that look at the
first column of that matrix B.
565
00:34:47,340 --> 00:34:50,780
If you do the
multiplication, it's --
566
00:34:50,780 --> 00:34:52,659
so what's the first
column of that matrix?
567
00:34:52,659 --> 00:34:54,370
Is that how you do
that multiplication?
568
00:34:54,370 --> 00:34:56,203
I multiply that matrix
by that first column.
569
00:34:56,203 --> 00:34:59,400
And it picks out one, zero, one.
570
00:34:59,400 --> 00:35:11,020
So the first column
of B is exactly that.
571
00:35:11,020 --> 00:35:18,650
And therefore a particular
solution will be this guy.
572
00:35:18,650 --> 00:35:19,960
So I'll repeat that question.
573
00:35:19,960 --> 00:35:21,150
Yeah.
574
00:35:21,150 --> 00:35:21,650
OK.
575
00:35:21,650 --> 00:35:22,150
Yes.
576
00:35:22,150 --> 00:35:23,170
Q: particular solution.
577
00:35:23,170 --> 00:35:26,320
Any of the solutions can
be the particular one
578
00:35:26,320 --> 00:35:28,180
that we pick out.
579
00:35:28,180 --> 00:35:31,940
So like this plus -- plus this
would be another particular
580
00:35:31,940 --> 00:35:32,610
OK.
581
00:35:32,610 --> 00:35:33,110
solution.
582
00:35:43,290 --> 00:35:48,260
It would be another solution.
583
00:35:48,260 --> 00:35:53,040
The particular is just
telling us only take one.
584
00:35:53,040 --> 00:35:57,370
But it's not telling us
which one we have to take.
585
00:35:57,370 --> 00:35:58,850
We take the most convenient one.
586
00:35:58,850 --> 00:36:02,560
I guess in this -- in this
problem that was that one.
587
00:36:02,560 --> 00:36:03,480
Good.
588
00:36:03,480 --> 00:36:05,750
Other questions?
589
00:36:05,750 --> 00:36:06,250
Yes.
590
00:36:06,250 --> 00:36:10,550
And this pattern
of particular plus
591
00:36:10,550 --> 00:36:15,370
null space, of course, that's
going throughout mathematics
592
00:36:15,370 --> 00:36:17,770
of linear systems.
593
00:36:17,770 --> 00:36:20,970
We're really doing mathematics
of linear systems here.
594
00:36:20,970 --> 00:36:24,070
Our systems are discrete and
they're finite-dimensional --
595
00:36:24,070 --> 00:36:31,860
and so it's linear algebra, but
this particular plus null space
596
00:36:31,860 --> 00:36:37,850
goes -- that doesn't depend
on having finite matrices --
597
00:36:37,850 --> 00:36:42,990
that spreads much --
that spreads everywhere.
598
00:36:42,990 --> 00:36:46,280
OK, I'm going to just
like to encourage
599
00:36:46,280 --> 00:36:48,370
you to take problems
out of the book,
600
00:36:48,370 --> 00:36:50,250
let me do the same myself.
601
00:36:50,250 --> 00:36:54,160
OK well here's some
easy true or falses.
602
00:36:54,160 --> 00:36:58,440
I don't know why the
author put these in here.
603
00:36:58,440 --> 00:36:59,530
OK.
604
00:36:59,530 --> 00:37:03,720
If m=n, then the row space
equals the column space.
605
00:37:03,720 --> 00:37:07,390
is invertible -- suppose
A is an invertible
606
00:37:07,390 --> 00:37:09,790
So these are true or falses.
607
00:37:09,790 --> 00:37:13,230
If m equals n, so that
means the matrix is square,
608
00:37:13,230 --> 00:37:17,410
then the row space
equals the column space?
609
00:37:17,410 --> 00:37:18,760
False, good.
610
00:37:18,760 --> 00:37:20,260
Good, what is equal there?
611
00:37:20,260 --> 00:37:25,830
What can I say is
equal, if M -- well,
612
00:37:25,830 --> 00:37:27,440
yeah.
613
00:37:27,440 --> 00:37:29,610
Yeah it -- so that's
definitely false --
614
00:37:29,610 --> 00:37:31,160
the row space and
the column space,
615
00:37:31,160 --> 00:37:37,550
and this matrix is like always
a good example to consider.
616
00:37:37,550 --> 00:37:40,530
So there's a square
matrix but it's row
617
00:37:40,530 --> 00:37:44,200
space is the multiples
of zero, one,
618
00:37:44,200 --> 00:37:48,310
and its column space is
the multiples of one, zero.
619
00:37:48,310 --> 00:37:50,240
Very different.
620
00:37:50,240 --> 00:37:51,680
The row space and
the column space
621
00:37:51,680 --> 00:37:54,230
are totally different
for that matrix.
622
00:37:54,230 --> 00:37:57,730
Now of course if the
matrix was symmetric,
623
00:37:57,730 --> 00:38:01,780
well, then clearly the row
space equals the column space.
624
00:38:01,780 --> 00:38:02,280
OK.
625
00:38:02,280 --> 00:38:03,930
How about this question?
626
00:38:03,930 --> 00:38:08,910
The matrices A and minus A
share the same four subspaces?
627
00:38:11,490 --> 00:38:15,230
Do the matrices A and minus
A have the same column space,
628
00:38:15,230 --> 00:38:17,100
do they have the
same null space,
629
00:38:17,100 --> 00:38:18,480
do they have the same row
630
00:38:18,480 --> 00:38:19,050
space?
631
00:38:19,050 --> 00:38:20,960
What's the answer on that?
632
00:38:20,960 --> 00:38:31,120
Yes or no.
633
00:38:31,120 --> 00:38:31,620
Yes.
634
00:38:31,620 --> 00:38:32,310
Good.
635
00:38:32,310 --> 00:38:33,100
OK.
636
00:38:33,100 --> 00:38:33,900
How about this?
637
00:38:33,900 --> 00:38:40,020
If A and B have the
same four subspaces,
638
00:38:40,020 --> 00:38:41,470
then A is a multiple of B.
639
00:38:41,470 --> 00:38:43,261
If -- suppose those
subspaces are the same.
640
00:38:43,261 --> 00:38:44,680
Then is A a multiple of B?
641
00:38:44,680 --> 00:38:45,510
OK.
642
00:38:45,510 --> 00:38:53,000
How how do you answer
a question like that?
643
00:38:53,000 --> 00:38:56,500
Of course if you want to
answer it yes, then I would --
644
00:38:56,500 --> 00:38:59,460
then they'd have to
think of a reason why.
645
00:38:59,460 --> 00:39:02,050
If you want to answer no
way, then you would --
646
00:39:02,050 --> 00:39:06,410
and I would sort of like
first I would try to think no,
647
00:39:06,410 --> 00:39:10,540
I would say can I come up with
an example where it isn't true?
648
00:39:10,540 --> 00:39:17,680
Let me repeat the question.
649
00:39:17,680 --> 00:39:24,810
And then write the answer.
650
00:39:24,810 --> 00:39:30,520
matrix, then what --
651
00:39:30,520 --> 00:39:40,510
suppose it's six by
six invertible matrix,
652
00:39:40,510 --> 00:39:59,070
then what's its row space, and
its column space is all of R^6,
653
00:39:59,070 --> 00:40:17,620
and the null space, and the null
space of A transpose would be
654
00:40:17,620 --> 00:40:21,900
the zero vector.
655
00:40:21,900 --> 00:40:36,170
So every invertible matrix
is going to give that answer.
656
00:40:36,170 --> 00:40:49,020
If I have a six by
six invertible matrix,
657
00:40:49,020 --> 00:40:51,602
I know what those subspaces are.
658
00:40:51,602 --> 00:40:53,060
Heck, that was back
in chapter two,
659
00:40:53,060 --> 00:40:55,720
when I didn't even know
what subspaces were.
660
00:40:55,720 --> 00:41:01,890
The row space and column space
are both all six-dimensional
661
00:41:01,890 --> 00:41:05,470
space -- the whole space,
and the rank is six,
662
00:41:05,470 --> 00:41:10,280
in other words, and the null
spaces have zero dimension.
663
00:41:10,280 --> 00:41:11,870
So do you see now the answer?
664
00:41:11,870 --> 00:41:13,060
So A and B could be.
665
00:41:13,060 --> 00:41:16,970
So A and B could be for example
any -- so I'm going to say
666
00:41:16,970 --> 00:41:17,470
false.
667
00:41:17,470 --> 00:41:21,070
Because A and B for example --
668
00:41:21,070 --> 00:41:34,230
So an example: A and B any
invertible six by six, six
669
00:41:34,230 --> 00:41:34,730
by six.
670
00:41:34,730 --> 00:41:39,230
So those would have
the same four subspaces
671
00:41:39,230 --> 00:41:40,480
but they wouldn't be the same.
672
00:41:40,480 --> 00:41:44,890
Of course th- there should be
something about those matrices
673
00:41:44,890 --> 00:41:45,930
that would be the same.
674
00:41:45,930 --> 00:41:50,070
It's sort of a natural
problem, so now actually we're
675
00:41:50,070 --> 00:41:52,120
getting to a math question.
676
00:41:52,120 --> 00:41:54,390
The answer is this is not true.
677
00:41:54,390 --> 00:41:59,300
One matrix doesn't have to
be a multiple of the other.
678
00:41:59,300 --> 00:42:02,420
But there must be
something that's true.
679
00:42:02,420 --> 00:42:07,770
And that would be sort of like
a natural question to ask.
680
00:42:07,770 --> 00:42:15,850
If they have the same
subspaces, same four subspaces,
681
00:42:15,850 --> 00:42:29,700
then what -- what could you --
682
00:42:29,700 --> 00:42:32,480
instinct wasn't
necessarily right.
683
00:42:32,480 --> 00:42:37,700
But I hope you now see that
the correct answer is false.
684
00:42:37,700 --> 00:42:40,550
And then you might think
OK, well, they certainly
685
00:42:40,550 --> 00:42:43,530
do have the same rank.
686
00:42:43,530 --> 00:42:51,860
But do -- obviously if they
have the same four subspaces,
687
00:42:51,860 --> 00:42:54,460
they have the same rank.
688
00:42:54,460 --> 00:42:58,620
I might say if
they have the well,
689
00:42:58,620 --> 00:43:03,820
I could extend that question and
think about other possibilities
690
00:43:03,820 --> 00:43:08,500
and finally come up with
something that was true.
691
00:43:08,500 --> 00:43:11,620
But I won't press that one.
692
00:43:11,620 --> 00:43:15,270
Let me keep going with
practice questions.
693
00:43:15,270 --> 00:43:19,430
And these practice questions
are quite appropriate I
694
00:43:19,430 --> 00:43:21,510
think for the exam.
695
00:43:21,510 --> 00:43:22,030
OK.
696
00:43:22,030 --> 00:43:23,070
let's see.
697
00:43:23,070 --> 00:43:29,310
If I exchange two rows of A
which subspaces stay the same?
698
00:43:29,310 --> 00:43:32,950
So I'm trying to
take out questions
699
00:43:32,950 --> 00:43:38,670
that we can answer without you
know we can answer quickly.
700
00:43:38,670 --> 00:43:45,560
If I have a matrix A, and I
exchange two of its rows, which
701
00:43:45,560 --> 00:43:47,570
subspaces stay the same?
702
00:43:47,570 --> 00:43:50,560
The row space does
stay the same.
703
00:43:50,560 --> 00:43:53,440
And the null space
stays the same.
704
00:43:53,440 --> 00:43:54,270
Good.
705
00:43:54,270 --> 00:43:55,200
Good.
706
00:43:55,200 --> 00:43:56,690
Correct.
707
00:43:56,690 --> 00:43:59,170
Column space would
be a wrong answer.
708
00:43:59,170 --> 00:44:00,280
OK.
709
00:44:00,280 --> 00:44:03,140
all right, here's a question.
710
00:44:03,140 --> 00:44:05,730
Oh, this leads into
the next chapter.
711
00:44:05,730 --> 00:44:08,990
Why can the vector
one, two, three not
712
00:44:08,990 --> 00:44:12,280
be a row and also
in the null space?
713
00:44:12,280 --> 00:44:15,040
Fitting we close
with this question.
714
00:44:15,040 --> 00:44:19,860
Close is -- so V
equal this one, two,
715
00:44:19,860 --> 00:44:33,900
three can't be in the null space
of a matrix and the row space.
716
00:44:33,900 --> 00:44:40,690
And my question is why not?
717
00:44:40,690 --> 00:44:41,378
Why not?
718
00:44:45,030 --> 00:44:47,950
So this is a
question that we can
719
00:44:47,950 --> 00:44:51,360
because it's sort of asked
in a straightforward way,
720
00:44:51,360 --> 00:44:55,410
we can figure out an answer.
721
00:44:55,410 --> 00:44:57,460
Well, actually yeah --
722
00:44:57,460 --> 00:45:00,550
I'll even pin it down, it
can't be in the null space --
723
00:45:00,550 --> 00:45:04,420
and be a row.
724
00:45:04,420 --> 00:45:07,050
I'll even pin it down further.
725
00:45:07,050 --> 00:45:12,550
Ask it to be a row of A.
726
00:45:12,550 --> 00:45:15,000
Why not?
727
00:45:15,000 --> 00:45:18,820
So I'm -- we know the
dimensions of these spaces.
728
00:45:18,820 --> 00:45:26,330
But now I'm asking you sort
of like the overlap between --
729
00:45:26,330 --> 00:45:30,320
so the null space
and the row space,
730
00:45:30,320 --> 00:45:33,210
those are in the same
n-dimensional space.
731
00:45:33,210 --> 00:45:41,410
Those are -- well, those are
both subspaces of n-dimensional
732
00:45:41,410 --> 00:45:45,060
space, and I'm basically
saying they can't overlap.
733
00:45:45,060 --> 00:45:49,040
I can't have a vector like
this, a typical vector, that's
734
00:45:49,040 --> 00:45:53,110
in the null space and it's
also a row of the matrix.
735
00:45:53,110 --> 00:45:54,580
Why is that?
736
00:45:54,580 --> 00:45:56,350
So that's a new sort of idea.
737
00:45:56,350 --> 00:45:58,320
Let's just see
what it would mean.
738
00:45:58,320 --> 00:46:04,860
I mean that A times this V,
why can this A times this
739
00:46:04,860 --> 00:46:07,770
V it can't be zero.
740
00:46:07,770 --> 00:46:12,410
Oh well, if it's
zero, so this is --
741
00:46:12,410 --> 00:46:16,830
I'm getting it into
the null space here.
742
00:46:16,830 --> 00:46:20,750
So this is -- now let's put
that vector's in the null space,
743
00:46:20,750 --> 00:46:27,600
why can't the first row of
a matrix be one, two, three?
744
00:46:27,600 --> 00:46:35,470
I can fill out the
matrix as I like.
745
00:46:35,470 --> 00:46:37,010
Why is that impossible?
746
00:46:37,010 --> 00:46:39,310
Well, you're seeing
it's impossible, right?
747
00:46:39,310 --> 00:46:44,690
That if that was a row of the
matrix and in the null space,
748
00:46:44,690 --> 00:46:48,421
that number would not be
zero, it would be fourteen.
749
00:46:48,421 --> 00:46:48,920
Right.
750
00:46:48,920 --> 00:46:52,960
So now we actually are beginning
to get a more complete picture
751
00:46:52,960 --> 00:46:55,170
of these four subspaces.
752
00:46:55,170 --> 00:47:00,220
The two that are over
in n-dimensional space,
753
00:47:00,220 --> 00:47:04,450
they actually only
share the zero vector.
754
00:47:04,450 --> 00:47:08,110
The intersection of the
null space and the row space
755
00:47:08,110 --> 00:47:09,310
is only the zero vector.
756
00:47:09,310 --> 00:47:14,970
And in fact the null space is
perpendicular to the row space.
757
00:47:14,970 --> 00:47:18,920
That'll be the first
topic let's see,
758
00:47:18,920 --> 00:47:23,940
we have a holiday Monday --
759
00:47:23,940 --> 00:47:29,800
and I'll see you Wednesday
with perpendiculars.
760
00:47:29,800 --> 00:47:33,980
And I'll see you Friday.
761
00:47:33,980 --> 00:47:39,710
So good luck on the quiz.