WEBVTT
00:00:05.870 --> 00:00:08.750
OK, when the camera
says, we'll start.
00:00:08.750 --> 00:00:13.320
You want to give me a signal?
00:00:13.320 --> 00:00:17.290
OK, this is lecture
eight in linear algebra,
00:00:17.290 --> 00:00:20.020
and this is the lecture
where we completely
00:00:20.020 --> 00:00:22.820
solve linear equations.
00:00:22.820 --> 00:00:23.810
So Ax=b.
00:00:27.080 --> 00:00:29.040
That's our goal.
00:00:29.040 --> 00:00:32.159
If it has a solution.
00:00:32.159 --> 00:00:35.990
It certainly can happen
that there is no solution.
00:00:35.990 --> 00:00:40.370
We have to identify that
possibility by elimination.
00:00:40.370 --> 00:00:45.110
And then if there is a solution
we want to find out is there
00:00:45.110 --> 00:00:48.650
only one solution or are
-- is there a whole family
00:00:48.650 --> 00:00:51.190
of solutions, and
then find them all.
00:00:51.190 --> 00:00:52.370
OK.
00:00:52.370 --> 00:00:57.420
Can I use as an
example the same matrix
00:00:57.420 --> 00:01:01.010
that I had last
time when we were
00:01:01.010 --> 00:01:02.890
looking for the null space.
00:01:02.890 --> 00:01:10.060
So the, the matrix has
rows 1 2 2 2, 2 4 6 8,
00:01:10.060 --> 00:01:13.150
and the third row -- you
remember the main point was
00:01:13.150 --> 00:01:21.890
the third row, 3 6 8 10, is the
sum of row one plus row two.
00:01:21.890 --> 00:01:25.710
In other words, if I add
those left-hand sides,
00:01:25.710 --> 00:01:28.710
I get the third left-hand side.
00:01:28.710 --> 00:01:31.530
So you can tell
me right away what
00:01:31.530 --> 00:01:35.460
elimination is going to discover
about the right-hand sides.
00:01:35.460 --> 00:01:41.570
What's -- there is a
condition on b1, b2,
00:01:41.570 --> 00:01:44.690
and b3 for this system
to have a solution.
00:01:44.690 --> 00:01:50.240
Most cases -- if I took
these numbers to be one --
00:01:50.240 --> 00:01:53.950
5, and 17, there would
not be a solution.
00:01:53.950 --> 00:01:58.700
In fact, if I took those
first numbers to be 1 and 5,
00:01:58.700 --> 00:02:03.600
what is the only b3
that would be OK?
00:02:03.600 --> 00:02:05.560
Six.
00:02:05.560 --> 00:02:09.860
If the left-hand -- if these
left-hand sides add up to that,
00:02:09.860 --> 00:02:10.729
then B --
00:02:10.729 --> 00:02:15.000
I need b1 plus b2 to equal b3.
00:02:15.000 --> 00:02:19.470
Let's just see how
elimination discovers that.
00:02:19.470 --> 00:02:23.430
But we can see it coming, right?
00:02:23.430 --> 00:02:26.450
That if -- let me say
it in other words.
00:02:26.450 --> 00:02:29.300
If some combination
on the left-hand side
00:02:29.300 --> 00:02:33.090
gives all 0s then
the same combination
00:02:33.090 --> 00:02:35.580
on the right-hand
side must give 0.
00:02:35.580 --> 00:02:36.080
OK.
00:02:36.080 --> 00:02:43.670
So let me take that
example and write down
00:02:43.670 --> 00:02:46.670
instead of copying out
all the plus signs,
00:02:46.670 --> 00:02:49.830
let me write down the matrix.
00:02:49.830 --> 00:02:59.130
1 2 2 2, 2 4 6 8,
and that 6 3 8 10,
00:02:59.130 --> 00:03:02.930
where the third row is the
sum of the first two rows.
00:03:02.930 --> 00:03:06.700
Now how do we deal with
the right-hand side?
00:03:06.700 --> 00:03:09.840
That's -- we want to do the same
thing to the right-hand side
00:03:09.840 --> 00:03:12.700
that we're doing to these
rows on the left side,
00:03:12.700 --> 00:03:17.920
so we just tack on the
right-hand side as another
00:03:17.920 --> 00:03:20.660
vector, another column.
00:03:20.660 --> 00:03:26.220
So this is the augmented matrix.
00:03:29.640 --> 00:03:35.840
It's, it's the matrix A
with the vector b tacked on.
00:03:35.840 --> 00:03:38.440
In Matlab, that's all
you would need to type.
00:03:38.440 --> 00:03:39.050
OK.
00:03:39.050 --> 00:03:41.470
So we do elimination on that.
00:03:41.470 --> 00:03:43.580
Can we just do
elimination quickly?
00:03:43.580 --> 00:03:46.790
The first pivot is fine,
I subtract two of this
00:03:46.790 --> 00:03:49.470
away from this, three
of this away from this,
00:03:49.470 --> 00:03:54.870
so I have 1 2 2 2 b1.
00:03:54.870 --> 00:04:00.840
Two of those away will
give me 0 0 2 and 4,
00:04:00.840 --> 00:04:03.570
and that was b2 minus two b1.
00:04:03.570 --> 00:04:07.130
I, I have to do the same
thing to that third,
00:04:07.130 --> 00:04:08.630
that last column.
00:04:08.630 --> 00:04:10.700
And then three of
these away from this
00:04:10.700 --> 00:04:17.980
gave me 0 0 2 4 b3
minus three b1s.
00:04:17.980 --> 00:04:21.660
So that's the,
that's elimination
00:04:21.660 --> 00:04:25.830
with the first column completed.
00:04:25.830 --> 00:04:26.750
We move on.
00:04:26.750 --> 00:04:29.570
There's the first pivot still.
00:04:29.570 --> 00:04:31.500
Here is the second pivot.
00:04:31.500 --> 00:04:34.660
We're always remembering,
now, these are then
00:04:34.660 --> 00:04:36.650
going to be the pivot columns.
00:04:41.810 --> 00:04:47.640
And let me get the final
result -- well, let me --
00:04:47.640 --> 00:04:51.615
can I do it by eraser?
00:04:55.550 --> 00:05:00.680
We're capable of subtracting
this row from this row,
00:05:00.680 --> 00:05:05.280
just by -- that'll knock this
out completely and give me
00:05:05.280 --> 00:05:08.090
the row of 0s, and on
the right-hand side,
00:05:08.090 --> 00:05:12.660
when I subtract this away
from this, what do I have?
00:05:16.260 --> 00:05:22.550
I think I have b3 minus a b2,
and I had minus three b1s.
00:05:22.550 --> 00:05:25.380
This is going to, it's
going to be a minus a b1.
00:05:25.380 --> 00:05:28.000
Oh yeah that's
exactly what I expect.
00:05:31.490 --> 00:05:34.430
So now the -- what's
the last equation?
00:05:34.430 --> 00:05:38.930
The last equation, this
represented by this zero row,
00:05:38.930 --> 00:05:45.750
that last equation is, says 0
equals b3 minus b2 minus b1.
00:05:45.750 --> 00:05:51.519
So that's the condition
for solvability.
00:05:51.519 --> 00:05:53.310
That's the condition
on the right-hand side
00:05:53.310 --> 00:05:54.450
that we expected.
00:05:54.450 --> 00:05:57.900
It says that b1+b2
has to match b3,
00:05:57.900 --> 00:06:02.350
and if our numbers happen
to have been 1, 5, and 6 --
00:06:02.350 --> 00:06:07.220
so let me take,
suppose b is 1 5 6.
00:06:07.220 --> 00:06:09.920
That's an OK b.
00:06:09.920 --> 00:06:13.120
And when I do this
elimination, what will I have?
00:06:13.120 --> 00:06:16.390
The b1 will still be a 1.
00:06:16.390 --> 00:06:18.950
b2 would be 5 minus
2, this would be a 3.
00:06:18.950 --> 00:06:24.860
5 -- my 6 minus 5 minus
1, this will be --
00:06:24.860 --> 00:06:29.800
this is the main point --
this will be a 0, thanks.
00:06:29.800 --> 00:06:30.300
OK.
00:06:30.300 --> 00:06:32.720
So the last equation is OK now.
00:06:37.080 --> 00:06:41.880
And I can proceed to solve the
two equations that are really
00:06:41.880 --> 00:06:44.160
there with four unknowns.
00:06:44.160 --> 00:06:48.660
OK, I, I, I want to do
that, so this, this b is OK.
00:06:48.660 --> 00:06:51.700
It allows a solution.
00:06:51.700 --> 00:06:56.850
We're going to be,
naturally, interested
00:06:56.850 --> 00:07:04.200
to keep track what are
the conditions on b that
00:07:04.200 --> 00:07:06.550
make the equation solvable.
00:07:06.550 --> 00:07:11.020
So let me write down
what we already see
00:07:11.020 --> 00:07:14.490
before I continue to solve it.
00:07:14.490 --> 00:07:17.286
Let me first --
solvability, solvability.
00:07:23.450 --> 00:07:31.015
So which -- so this is the
condition on the right-hand
00:07:31.015 --> 00:07:31.515
sides.
00:07:34.170 --> 00:07:36.010
And what is that condition?
00:07:36.010 --> 00:07:39.440
This is solvability
always of Ax=b.
00:07:39.440 --> 00:07:45.150
So Ax=b is solvable --
00:07:45.150 --> 00:07:50.870
well, actually, we had an answer
in the language of the column
00:07:50.870 --> 00:07:52.640
space.
00:07:52.640 --> 00:07:54.530
Can you remind me
what that answer is?
00:07:54.530 --> 00:07:58.510
That, that was like our
answer from earlier lecture.
00:07:58.510 --> 00:08:00.980
b had to be in the column space.
00:08:00.980 --> 00:08:10.140
Solvable if -- when -- exactly
when b is in the column space
00:08:10.140 --> 00:08:13.560
of A.
00:08:13.560 --> 00:08:14.110
Right?
00:08:14.110 --> 00:08:17.840
That just says that b has to be
a combination of the columns,
00:08:17.840 --> 00:08:21.890
and of course that's exactly
what the equation is looking
00:08:21.890 --> 00:08:22.660
for.
00:08:22.660 --> 00:08:25.800
So that -- now I
want to answer it --
00:08:25.800 --> 00:08:30.050
the same answer but
in different language.
00:08:30.050 --> 00:08:33.990
Another way to answer this --
00:08:33.990 --> 00:08:52.980
if a combination of the rows
of A gives the zero row,
00:08:52.980 --> 00:08:57.010
and this was an example
where it happened,
00:08:57.010 --> 00:09:00.940
some combination of the rows
of A produced the zero row --
00:09:00.940 --> 00:09:04.560
then what's the
requirement on b?
00:09:04.560 --> 00:09:07.130
Since we're going to do the
same thing to both sides of all
00:09:07.130 --> 00:09:08.520
equations --
00:09:08.520 --> 00:09:16.190
the same combination of the
components of b has to give 0.
00:09:16.190 --> 00:09:16.740
Right?
00:09:16.740 --> 00:09:19.590
That's -- so if there's a
combination of the rows that
00:09:19.590 --> 00:09:29.920
gives the zero row, then the
same combination of the entries
00:09:29.920 --> 00:09:34.700
of b must give 0.
00:09:37.950 --> 00:09:40.475
And this isn't the zero
row, that's the zero number.
00:09:43.650 --> 00:09:47.860
Tthis is another way of saying
-- and it is not immediate,
00:09:47.860 --> 00:09:54.582
OK. right, that these two
statements are equivalent.
00:09:54.582 --> 00:09:56.290
But somehow they must
be, because they're
00:09:56.290 --> 00:09:59.961
both equivalent to the
solvability of the system.
00:09:59.961 --> 00:10:00.460
OK.
00:10:00.460 --> 00:10:05.030
So we've got this, this sort
of -- like question zero is,
00:10:05.030 --> 00:10:08.180
does the system have a solution?
00:10:08.180 --> 00:10:12.670
OK, I'll come back to
discuss that further.
00:10:12.670 --> 00:10:17.310
Let's go forward when it does.
00:10:17.310 --> 00:10:19.470
When there is a solution.
00:10:19.470 --> 00:10:22.684
And so what's our job now?
00:10:22.684 --> 00:10:24.600
Abstractly we sit back
and we say, OK, there's
00:10:24.600 --> 00:10:26.660
a solution, finished.
00:10:26.660 --> 00:10:27.570
It exists.
00:10:27.570 --> 00:10:29.450
But we want to construct it.
00:10:29.450 --> 00:10:34.500
So what's the
algorithm, the sequence
00:10:34.500 --> 00:10:37.330
of steps to find the solution?
00:10:37.330 --> 00:10:38.670
That's what I --
00:10:38.670 --> 00:10:42.130
and of course the
quiz and the final,
00:10:42.130 --> 00:10:45.440
I'm going to give you a system
Ax=b and I'm going to ask you
00:10:45.440 --> 00:10:48.350
for the solution,
if there is one.
00:10:48.350 --> 00:10:54.430
And so this algorithm
that you want to follow.
00:10:54.430 --> 00:10:58.290
OK, let's see.
00:10:58.290 --> 00:11:13.190
So what's the -- so now to find
the complete solution to Ax=b.
00:11:13.190 --> 00:11:14.000
OK.
00:11:14.000 --> 00:11:17.150
Let me start by
finding one solution,
00:11:17.150 --> 00:11:19.410
one particular solution.
00:11:22.030 --> 00:11:26.060
I'm expecting that I can,
because my system of equations
00:11:26.060 --> 00:11:30.730
now, that last equation
is zero equals zero,
00:11:30.730 --> 00:11:33.850
so that's all fine.
00:11:33.850 --> 00:11:36.720
I really have two equations --
00:11:36.720 --> 00:11:38.910
actually I've got
four unknowns, so I'm
00:11:38.910 --> 00:11:41.410
expecting to find
not only a solution
00:11:41.410 --> 00:11:44.230
but a whole bunch of them.
00:11:44.230 --> 00:11:46.020
But let's just find one.
00:11:46.020 --> 00:11:50.930
So step one, a particular
solution, x particular.
00:11:54.430 --> 00:11:57.010
How do I find one
particular solution?
00:11:57.010 --> 00:12:00.740
Well, let me tell you
how I, how I find it.
00:12:00.740 --> 00:12:02.160
So this is --
00:12:02.160 --> 00:12:04.080
since there are
lots of solutions,
00:12:04.080 --> 00:12:07.100
you could have your own way
to find a particular one.
00:12:07.100 --> 00:12:10.780
But this is a
pretty natural way.
00:12:10.780 --> 00:12:20.880
Set all free variables to zero.
00:12:20.880 --> 00:12:25.810
Since those free variables are
the guys that can be anything,
00:12:25.810 --> 00:12:28.790
the most convenient
choice is zero.
00:12:28.790 --> 00:12:37.975
And then solve Ax=b for
the pivot variables.
00:12:41.170 --> 00:12:44.240
So what does that
mean in this example?
00:12:44.240 --> 00:12:46.490
Which are the free variables?
00:12:46.490 --> 00:12:49.500
Which, which are the variables
that we can assign freely
00:12:49.500 --> 00:12:52.500
and then there's
one and only one way
00:12:52.500 --> 00:12:55.070
to find the pivot variables?
00:12:55.070 --> 00:13:01.120
They're x2 and -- so x2 is
zero, because that's in a column
00:13:01.120 --> 00:13:04.280
without a pivot, the
second column has no pivot.
00:13:04.280 --> 00:13:08.010
And the -- what's the other one?
00:13:08.010 --> 00:13:11.620
The fourth, x4 is zero.
00:13:11.620 --> 00:13:16.330
Because that, those
are the, the free ones.
00:13:16.330 --> 00:13:18.870
Those are in the
columns with no pivots.
00:13:18.870 --> 00:13:21.300
So you see what my
-- so when I knock --
00:13:21.300 --> 00:13:28.660
when x2 and x4 are zero,
I'm left with the --
00:13:28.660 --> 00:13:31.010
what I left with here?
00:13:31.010 --> 00:13:33.460
I'm just left with --
00:13:33.460 --> 00:13:36.500
see, now I'm not
using the two free
00:13:36.500 --> 00:13:37.020
columns.
00:13:37.020 --> 00:13:39.380
I'm only using
the pivot columns.
00:13:39.380 --> 00:13:42.370
So I'm really left with x1 --
00:13:42.370 --> 00:13:45.360
the first equation
is just x1 and two
00:13:45.360 --> 00:13:48.720
x3s should be the
right-hand side, which
00:13:48.720 --> 00:13:50.700
we picked to be a one.
00:13:50.700 --> 00:13:54.130
And the second
equation is two x3s,
00:13:54.130 --> 00:13:57.735
as it happened, turned
out to be, three.
00:14:02.090 --> 00:14:06.680
I just write it again here
with the x2 and the x4
00:14:06.680 --> 00:14:09.420
knocked out, since
we're set them to zero.
00:14:09.420 --> 00:14:14.150
And you see that we're back in
the normal case of having back
00:14:14.150 --> 00:14:16.030
-- where back
substitution will do it.
00:14:16.030 --> 00:14:21.640
So x3 is three halves,
and then we go back up
00:14:21.640 --> 00:14:25.490
and x1 is one minus two x3.
00:14:25.490 --> 00:14:29.270
That's probably minus two.
00:14:29.270 --> 00:14:30.400
Good.
00:14:30.400 --> 00:14:34.210
So now we have the
solution, x particular
00:14:34.210 --> 00:14:41.940
is the vector minus two
zero three halves zero.
00:14:44.710 --> 00:14:46.790
OK, good.
00:14:46.790 --> 00:14:52.200
That's one particular solution,
and we should and could plug it
00:14:52.200 --> 00:14:54.600
into the original system.
00:14:54.600 --> 00:14:57.010
Really if -- on
the quiz, please,
00:14:57.010 --> 00:14:59.230
it's a good thing to do.
00:14:59.230 --> 00:15:03.650
So we did all this,
these, row operations,
00:15:03.650 --> 00:15:06.960
but this is supposed to
solve the original system,
00:15:06.960 --> 00:15:09.430
and I think it does.
00:15:09.430 --> 00:15:10.120
OK.
00:15:10.120 --> 00:15:14.810
So that's x particular
which we've got.
00:15:14.810 --> 00:15:19.320
So that's like what's new today.
00:15:19.320 --> 00:15:23.780
The particular solution comes
-- first you check that you have
00:15:23.780 --> 00:15:26.700
zero equals zero, so
you're OK on the last
00:15:26.700 --> 00:15:27.920
equations.
00:15:27.920 --> 00:15:31.980
And then you set the
free variables to zero,
00:15:31.980 --> 00:15:34.830
solve for the pivot
variables, and you've
00:15:34.830 --> 00:15:38.500
got a particular solution,
the particular solution that
00:15:38.500 --> 00:15:41.150
has zero free variables.
00:15:41.150 --> 00:15:42.040
OK.
00:15:42.040 --> 00:15:45.000
Now -- but that's
only one solution,
00:15:45.000 --> 00:15:46.270
and now I'm looking for all.
00:15:49.020 --> 00:15:51.480
So how do I find the rest?
00:15:51.480 --> 00:15:58.770
The point is I can add on x --
anything out of the null space.
00:16:03.320 --> 00:16:06.190
We know how to find the
vectors in the null space --
00:16:06.190 --> 00:16:08.950
because we did it last time,
but I'll remind you what we
00:16:08.950 --> 00:16:09.710
got.
00:16:09.710 --> 00:16:12.620
And then I'll add.
00:16:15.630 --> 00:16:20.650
So the final result will be
that the complete solution --
00:16:20.650 --> 00:16:23.620
this is now the complete guy --
00:16:23.620 --> 00:16:27.980
the complete solution is
this one particular solution
00:16:27.980 --> 00:16:34.930
plus any, any vector,
all different vectors out
00:16:34.930 --> 00:16:37.600
of the null space.
00:16:37.600 --> 00:16:39.050
xn, OK.
00:16:39.050 --> 00:16:42.630
Well why, why this pattern,
because this pattern shows up
00:16:42.630 --> 00:16:46.560
through all of mathematics,
because it shows up everywhere
00:16:46.560 --> 00:16:48.690
we have linear equations.
00:16:48.690 --> 00:16:52.050
Let me just put here
the, the reason.
00:16:52.050 --> 00:17:01.830
A xp, so that's x particular,
so what does Ax particular give?
00:17:01.830 --> 00:17:05.410
That gives the correct
right-hand side b.
00:17:05.410 --> 00:17:10.520
And what does A times an
x in the null space give?
00:17:10.520 --> 00:17:11.710
Zero.
00:17:11.710 --> 00:17:17.920
So I add, and I
put in parentheses.
00:17:17.920 --> 00:17:25.420
So xp plus xn is b
plus zero, which is b.
00:17:25.420 --> 00:17:27.540
So -- oh, what I saying?
00:17:27.540 --> 00:17:30.450
Let me just say it in words.
00:17:30.450 --> 00:17:36.800
If I have one solution,
I can add on anything
00:17:36.800 --> 00:17:40.600
in the null space, because
anything in the null space
00:17:40.600 --> 00:17:43.910
has a zero right-hand
side, and I still
00:17:43.910 --> 00:17:46.670
have the correct
right-hand side B.
00:17:46.670 --> 00:17:47.850
So that's my system.
00:17:47.850 --> 00:17:50.290
That's my complete solution.
00:17:50.290 --> 00:17:54.620
Now let me write out what
that will be for this example.
00:17:54.620 --> 00:18:02.070
So in this example, x
general, x complete,
00:18:02.070 --> 00:18:07.440
the complete solution,
is x particular,
00:18:07.440 --> 00:18:12.230
which is minus two
zero three halves zero,
00:18:12.230 --> 00:18:15.900
with those zeroes in the
free variable, plus --
00:18:15.900 --> 00:18:18.410
you remember there were the
special solutions in the null
00:18:18.410 --> 00:18:21.680
space that had a one in
the free variables --
00:18:21.680 --> 00:18:24.220
or one and zero in
the free variables,
00:18:24.220 --> 00:18:29.880
and then we filled in to find
I've forgotten what they were,
00:18:29.880 --> 00:18:32.020
but maybe it was
that. the others?
00:18:32.020 --> 00:18:34.010
That was a special
solution, and then
00:18:34.010 --> 00:18:36.950
there was another
special solution that
00:18:36.950 --> 00:18:41.820
had that free variable zero and
this free variable equal one,
00:18:41.820 --> 00:18:46.260
and I have to fill those in.
00:18:46.260 --> 00:18:48.420
Let's see, can I remember
how those fill in?
00:18:48.420 --> 00:18:51.570
Maybe this was a minus
two and this was a two,
00:18:51.570 --> 00:18:53.070
possibly?
00:18:53.070 --> 00:18:57.700
I think probably that's right.
00:18:57.700 --> 00:18:59.290
I'm not -- yeah.
00:18:59.290 --> 00:19:05.230
Does that look write to you?
00:19:05.230 --> 00:19:08.480
I would have to remember
what are my equations.
00:19:08.480 --> 00:19:11.930
Can I, rather than go
way back to that board,
00:19:11.930 --> 00:19:14.680
let me remember the
first equation was
00:19:14.680 --> 00:19:19.520
two x3 plus two x4
equaling zero now,
00:19:19.520 --> 00:19:22.450
because I'm looking for
the guys in the null space.
00:19:22.450 --> 00:19:28.510
So I set x4 to be one
and the second equation,
00:19:28.510 --> 00:19:32.850
that I didn't copy again, gave
me minus two for this and then
00:19:32.850 --> 00:19:35.090
-- yeah, so I
think that's right.
00:19:35.090 --> 00:19:40.131
Two minus four and
two gives zero, check.
00:19:40.131 --> 00:19:40.630
OK.
00:19:40.630 --> 00:19:43.830
Those were the
special solutions.
00:19:43.830 --> 00:19:46.060
What do we do to get
the complete solution?
00:19:49.860 --> 00:19:52.270
How do I get the
complete solution now?
00:19:52.270 --> 00:19:57.020
I multiply this by
anything, c1, say,
00:19:57.020 --> 00:19:58.930
and I multiply
this by anything --
00:19:58.930 --> 00:20:00.960
I take any combination.
00:20:00.960 --> 00:20:04.140
Remember that's how we
described the null space?
00:20:04.140 --> 00:20:09.060
The null space consists
of all combinations of --
00:20:09.060 --> 00:20:10.760
so this is xn --
00:20:10.760 --> 00:20:15.430
all combinations of
the special solutions.
00:20:15.430 --> 00:20:18.410
There were two special
solutions because there
00:20:18.410 --> 00:20:20.730
were two free variables.
00:20:20.730 --> 00:20:24.560
And we want to
make that count --
00:20:24.560 --> 00:20:26.660
carefully now.
00:20:26.660 --> 00:20:27.857
Just while I'm up here.
00:20:27.857 --> 00:20:30.190
So there's, that's what the
-- that's the kind of answer
00:20:30.190 --> 00:20:31.230
I'm looking for.
00:20:31.230 --> 00:20:34.810
Is there a constant
multiplying this guy?
00:20:34.810 --> 00:20:38.790
Is there a free constant
that multiplies x particular?
00:20:38.790 --> 00:20:40.180
No way.
00:20:40.180 --> 00:20:44.430
Right? x particular
solves A xp=b.
00:20:44.430 --> 00:20:47.450
I'm not allowed to
multiply that by three.
00:20:47.450 --> 00:20:51.120
But Axn, I'm allowed to
multiply xn by three,
00:20:51.120 --> 00:20:56.790
or add to another xn, because I
keep getting zero on the right.
00:20:56.790 --> 00:20:57.290
OK.
00:20:57.290 --> 00:21:02.250
So, so again, xp is
one particular guy.
00:21:02.250 --> 00:21:04.630
xn is a whole subspace.
00:21:04.630 --> 00:21:05.590
Right?
00:21:05.590 --> 00:21:09.380
It's one guy plus, plus
anything from a subspace.
00:21:09.380 --> 00:21:11.120
Let me draw it.
00:21:11.120 --> 00:21:14.740
Let me try to -- oh.
00:21:14.740 --> 00:21:19.950
I want to draw, I want
to graph all this --
00:21:19.950 --> 00:21:25.910
I want to, I want to
plot all solutions.
00:21:25.910 --> 00:21:29.020
Now x.
00:21:29.020 --> 00:21:32.850
So what dimension I in?
00:21:32.850 --> 00:21:35.950
This is a unfortunate point.
00:21:35.950 --> 00:21:38.650
How many components does x have?
00:21:38.650 --> 00:21:39.150
Four.
00:21:39.150 --> 00:21:40.280
There are four unknowns.
00:21:40.280 --> 00:21:47.150
So I have to draw a four
dimensional picture on this MIT
00:21:47.150 --> 00:21:48.450
cheap blackboard.
00:21:48.450 --> 00:21:48.950
OK.
00:21:48.950 --> 00:21:50.970
So here we go.
00:21:50.970 --> 00:21:58.080
x1 -- Einstein could do
it, but, this, this is --
00:21:58.080 --> 00:22:06.090
those are four
perpendicular axes in --
00:22:06.090 --> 00:22:08.740
representing four
dimensional space.
00:22:08.740 --> 00:22:09.720
OK.
00:22:09.720 --> 00:22:12.520
Where are my solutions?
00:22:12.520 --> 00:22:16.580
Do my solutions form a subspace?
00:22:16.580 --> 00:22:20.470
Does the set of solutions
to Ax=b form a subspace?
00:22:20.470 --> 00:22:21.900
No way.
00:22:21.900 --> 00:22:23.890
What does it actually
look like, though?
00:22:23.890 --> 00:22:26.530
A subspace is in this picture.
00:22:26.530 --> 00:22:30.250
This part is a subspace, right?
00:22:30.250 --> 00:22:33.060
That part is some,
like, two dimensional,
00:22:33.060 --> 00:22:35.860
because I've got two
parameters, so it's --
00:22:35.860 --> 00:22:41.120
I'm thinking of this null space
as a two dimensional subspace
00:22:41.120 --> 00:22:42.890
inside R^4.
00:22:42.890 --> 00:22:46.410
Now I have to tell you and
will tell you next time,
00:22:46.410 --> 00:22:49.760
what does it mean to say a
subspace, what's the dimension
00:22:49.760 --> 00:22:50.580
of a subspace.
00:22:50.580 --> 00:22:52.680
But you see what
it's going to be.
00:22:52.680 --> 00:22:58.180
It's the number of free
independent constants
00:22:58.180 --> 00:22:59.750
that we can choose.
00:22:59.750 --> 00:23:03.560
So somehow there'll be a two
dimensional subspace, not
00:23:03.560 --> 00:23:07.860
a line, and not a three
dimensional plane, but only
00:23:07.860 --> 00:23:10.110
a two dimensional guy.
00:23:10.110 --> 00:23:12.710
But it's doesn't go
through the origin
00:23:12.710 --> 00:23:15.600
because it goes
through this point.
00:23:15.600 --> 00:23:17.350
So there's x particular.
00:23:17.350 --> 00:23:19.970
x particular is somewhere here.
00:23:19.970 --> 00:23:21.580
x particular.
00:23:21.580 --> 00:23:25.750
So it's somehow a subspace --
can I try to draw it that way?
00:23:28.950 --> 00:23:36.070
It's a two dimensional subspace
that goes through x particular
00:23:36.070 --> 00:23:39.970
and then onwards by --
so there's x particular,
00:23:39.970 --> 00:23:44.090
and I added on
xn, and there's x.
00:23:44.090 --> 00:23:46.420
There's x=xp+xn.
00:23:46.420 --> 00:23:51.430
But the xn was anywhere
in this subspace,
00:23:51.430 --> 00:23:56.700
so that filled out a plane.
00:23:56.700 --> 00:23:58.930
It's a subspace --
00:23:58.930 --> 00:24:02.320
it's not a subspace,
what I saying?
00:24:02.320 --> 00:24:05.220
It's like a flat thing,
it's like a subspace,
00:24:05.220 --> 00:24:08.450
but it's been shifted,
away from the origin.
00:24:08.450 --> 00:24:11.450
It doesn't contain zero.
00:24:11.450 --> 00:24:12.200
Thanks.
00:24:12.200 --> 00:24:13.150
OK.
00:24:13.150 --> 00:24:16.340
That's the picture, and
that's the algorithm.
00:24:16.340 --> 00:24:20.940
So the algorithm is just
go through elimination
00:24:20.940 --> 00:24:25.150
and, find the
particular solution,
00:24:25.150 --> 00:24:27.290
and then find those
special solutions.
00:24:27.290 --> 00:24:30.010
You can do that.
00:24:30.010 --> 00:24:34.620
Let me take our time here
in the lecture to think,
00:24:34.620 --> 00:24:39.240
about the bigger picture.
00:24:39.240 --> 00:24:43.030
So let me think about --
00:24:43.030 --> 00:24:46.050
so this is my pattern.
00:24:46.050 --> 00:24:47.040
Now I want to think --
00:24:47.040 --> 00:24:54.410
I want to ask you
about a question --
00:24:54.410 --> 00:24:57.260
I want to ask you
some questions.
00:24:57.260 --> 00:25:01.070
So when I mean think bigger,
I mean I'll think about an m
00:25:01.070 --> 00:25:09.495
by n matrix A of rank r.
00:25:12.640 --> 00:25:13.140
OK.
00:25:15.820 --> 00:25:17.630
What's our definition of rank?
00:25:17.630 --> 00:25:22.910
Our current definition of
rank is number of pivots.
00:25:22.910 --> 00:25:23.410
OK.
00:25:23.410 --> 00:25:26.400
First of all, how are
these numbers related?
00:25:26.400 --> 00:25:31.050
Can you tell me a
relation between r and m?
00:25:31.050 --> 00:25:35.860
If I have m rows in the
matrix and R pivots, --
00:25:35.860 --> 00:25:42.240
then I certainly know, always --
00:25:42.240 --> 00:25:46.890
what relation do I
know between r and m?
00:25:46.890 --> 00:25:49.810
r is less or equal, right?
00:25:49.810 --> 00:25:53.620
Because I've got m rows, I
can't have more than m pivots,
00:25:53.620 --> 00:25:56.720
I might have m and
I might have fewer.
00:25:56.720 --> 00:26:01.980
Also, I've got n columns.
00:26:01.980 --> 00:26:04.630
So what's the relation
between r and n?
00:26:04.630 --> 00:26:10.480
It's the same, less or
equal, because a column
00:26:10.480 --> 00:26:14.150
can't have more than one pivot.
00:26:14.150 --> 00:26:17.450
So I can't have more
than n pivots altogether.
00:26:17.450 --> 00:26:18.740
OK, OK.
00:26:18.740 --> 00:26:22.360
So I have an m by
n matrix of rank r.
00:26:22.360 --> 00:26:25.420
And I always know r less than
or equal to m, r less than
00:26:25.420 --> 00:26:26.560
or equal to n.
00:26:26.560 --> 00:26:29.870
Now I'm specially
interested in the case
00:26:29.870 --> 00:26:35.540
of full rank, when the rank
r is as big as it can be.
00:26:35.540 --> 00:26:40.840
Well, I guess I've got two
separate possibilities here,
00:26:40.840 --> 00:26:44.310
depending on what these
numbers m and n are.
00:26:44.310 --> 00:26:49.970
So let me talk about the
case of full column rank.
00:26:53.200 --> 00:26:54.958
And by that I mean r=n.
00:27:02.330 --> 00:27:11.960
And I want to ask you, what does
that imply about our solutions?
00:27:11.960 --> 00:27:16.190
What does that tell us
about the null space?
00:27:16.190 --> 00:27:21.240
What does that tell us
about, the complete solution?
00:27:21.240 --> 00:27:22.860
OK, so what does that mean?
00:27:22.860 --> 00:27:28.520
So I want to ask you,
well, OK, if the rank is
00:27:28.520 --> 00:27:31.580
n, what does that mean?
00:27:31.580 --> 00:27:35.570
That means there's a
pivot in every column.
00:27:35.570 --> 00:27:39.280
So how many pivot
variables are there?
00:27:39.280 --> 00:27:41.160
n.
00:27:41.160 --> 00:27:45.150
All the columns have
pivots in this case.
00:27:45.150 --> 00:27:48.150
So how many free
variables are there?
00:27:48.150 --> 00:27:50.460
None at all.
00:27:50.460 --> 00:27:52.770
So no free variables.
00:27:52.770 --> 00:27:54.770
r=n, no free variables.
00:27:57.820 --> 00:28:00.270
So what does that
tell us about what's
00:28:00.270 --> 00:28:04.610
going to happen then in our,
in our little algorithms?
00:28:04.610 --> 00:28:07.440
What will be in the null space?
00:28:07.440 --> 00:28:13.570
The null space of A
has got what in it?
00:28:13.570 --> 00:28:15.990
Only the zero vector.
00:28:15.990 --> 00:28:20.590
There are no free variables
to give other values to.
00:28:20.590 --> 00:28:23.780
So the null space is
only the zero vector.
00:28:29.770 --> 00:28:33.580
And what about our
solution to Ax=b?
00:28:33.580 --> 00:28:38.510
Solution to Ax=b?
00:28:38.510 --> 00:28:41.790
What, what's the
story on that one?
00:28:41.790 --> 00:28:43.950
So now that's coming
from today's lecture.
00:28:47.270 --> 00:28:51.020
The solution x is --
00:28:51.020 --> 00:28:52.290
what's the complete solution?
00:28:55.920 --> 00:28:59.570
It's just x particular, right?
00:28:59.570 --> 00:29:02.830
If, if, if there is an x,
if there is a solution.
00:29:02.830 --> 00:29:05.190
It's x equal x particular.
00:29:05.190 --> 00:29:08.900
There's nothing -- you know,
there's just one solution.
00:29:08.900 --> 00:29:11.050
If there's one at all.
00:29:11.050 --> 00:29:13.920
So it's unique solution --
00:29:13.920 --> 00:29:16.900
unique means only one --
00:29:16.900 --> 00:29:22.945
unique solution if it
exists, if it exists.
00:29:26.430 --> 00:29:29.880
In other words, I would say --
let me put it a different way.
00:29:29.880 --> 00:29:32.910
There're either zero
or one solutions.
00:29:38.940 --> 00:29:40.920
This is all in this case r=n.
00:29:45.140 --> 00:29:50.400
So I'm -- because many, many
applications in reality,
00:29:50.400 --> 00:29:55.693
the columns will be what
I'll later call independent.
00:29:58.710 --> 00:30:04.340
And we'll have, nothing to
look for in the null space,
00:30:04.340 --> 00:30:06.611
and we'll only have
particular solutions.
00:30:06.611 --> 00:30:07.110
OK.
00:30:09.970 --> 00:30:13.500
Everybody see that possibility?
00:30:13.500 --> 00:30:15.920
But I need an example, right?
00:30:15.920 --> 00:30:18.590
So let me create an example.
00:30:18.590 --> 00:30:23.170
What sort of a matrix -- what's
the shape of a matrix that has
00:30:23.170 --> 00:30:25.390
full column rank?
00:30:25.390 --> 00:30:30.140
So can I squeeze in
an, an example here?
00:30:30.140 --> 00:30:35.080
If it exists.
00:30:35.080 --> 00:30:38.150
Let me put in an example,
and it's just the right space
00:30:38.150 --> 00:30:41.140
to put in an example.
00:30:41.140 --> 00:30:45.330
Because the example will
be like tall and thin.
00:30:45.330 --> 00:30:47.940
It will have --
00:30:47.940 --> 00:30:54.140
well, I mean, here's an example,
one two six five, three one
00:30:54.140 --> 00:30:55.130
one one.
00:30:55.130 --> 00:30:56.610
Brilliant example.
00:30:56.610 --> 00:30:57.590
OK.
00:30:57.590 --> 00:31:06.039
So there's a matrix A,
and what's its rank?
00:31:06.039 --> 00:31:07.330
What's the rank of that matrix?
00:31:10.000 --> 00:31:12.235
How many pivots will I
find if I do elimination?
00:31:14.810 --> 00:31:15.810
Two, right?
00:31:15.810 --> 00:31:16.990
Two.
00:31:16.990 --> 00:31:20.320
I see a pivot there --
00:31:20.320 --> 00:31:23.730
oh certainly those two
columns are headed off
00:31:23.730 --> 00:31:26.810
in different directions.
00:31:26.810 --> 00:31:29.870
When I do elimination, I'll
certainly get another pivot
00:31:29.870 --> 00:31:35.270
here, fine, and I can use those
to clean out below and above.
00:31:35.270 --> 00:31:43.960
So the -- actually, tell me
what its row reduced row echelon
00:31:43.960 --> 00:31:45.360
form would be.
00:31:45.360 --> 00:31:49.410
Can you carry that,
that elimination
00:31:49.410 --> 00:31:52.360
process to the bitter end?
00:31:52.360 --> 00:31:54.590
So what do, what does that mean?
00:31:54.590 --> 00:31:57.610
I subtract a multiple of
this row from these rows.
00:31:57.610 --> 00:32:00.910
So I clean up, all zeros there.
00:32:00.910 --> 00:32:02.540
Then I've got some pivot here.
00:32:02.540 --> 00:32:04.140
What do I do with that?
00:32:04.140 --> 00:32:07.330
I go subtract it
below and above,
00:32:07.330 --> 00:32:11.860
and then I divide through,
and what's R for that example?
00:32:11.860 --> 00:32:14.450
Maybe I can -- you'll allow
me to put that just here
00:32:14.450 --> 00:32:16.580
in the next board.
00:32:16.580 --> 00:32:21.150
What's the row reduced echelon
form, just out of practice,
00:32:21.150 --> 00:32:25.300
for that matrix?
00:32:25.300 --> 00:32:28.800
It's got ones in the pivots.
00:32:28.800 --> 00:32:31.750
It's got the identity matrix,
a little two by two identity
00:32:31.750 --> 00:32:34.270
matrix, and below it all zeros.
00:32:37.850 --> 00:32:43.940
That's a matrix that really
has two independent rows,
00:32:43.940 --> 00:32:45.510
and they're the
first two, actually.
00:32:45.510 --> 00:32:47.280
The first two rows
are independent.
00:32:47.280 --> 00:32:49.190
They're not in the
same direction.
00:32:49.190 --> 00:32:52.660
But the other rows are
combinations of the first two,
00:32:52.660 --> 00:32:55.970
so --
00:32:55.970 --> 00:32:59.850
is there always a
solution to Ax=b?
00:32:59.850 --> 00:33:02.050
Tell me what's the picture here?
00:33:02.050 --> 00:33:06.880
For this matrix A, this is
a case of full column rank.
00:33:06.880 --> 00:33:11.320
The two columns are
-- give two pivots.
00:33:11.320 --> 00:33:13.090
There's nothing
in the null space.
00:33:13.090 --> 00:33:15.540
There's no combination
of those columns
00:33:15.540 --> 00:33:19.001
that gives the zero column
except the zero zero
00:33:19.001 --> 00:33:19.500
combination.
00:33:22.430 --> 00:33:25.400
So there's nothing
in the null space.
00:33:25.400 --> 00:33:29.830
But is there always a
solution to A X equal B?
00:33:29.830 --> 00:33:31.530
What's up with A X equal B?
00:33:34.390 --> 00:33:38.220
I've got four, four equations
here, and only two Xs.
00:33:40.840 --> 00:33:42.460
So the answer is certainly no.
00:33:42.460 --> 00:33:45.240
There's not always a solution.
00:33:45.240 --> 00:33:49.660
I may have zero solutions,
and if I make a random choice,
00:33:49.660 --> 00:33:51.590
I'll have zero solutions.
00:33:51.590 --> 00:33:55.690
Or if I make a great particular
choice of the right-hand side,
00:33:55.690 --> 00:33:59.160
which just happens to be a
combination of those two guys
00:33:59.160 --> 00:34:01.500
-- like tell me one right-hand
side that would have
00:34:01.500 --> 00:34:03.540
a solution.
00:34:03.540 --> 00:34:07.190
Tell me a right-hand side
that would have a solution.
00:34:07.190 --> 00:34:09.800
Well, 0 0 0 0, OK.
00:34:09.800 --> 00:34:12.880
No prize for that one.
00:34:12.880 --> 00:34:14.250
Tell me another one.
00:34:14.250 --> 00:34:18.850
Another right-hand side that
has a solution would be 4 3 7 6.
00:34:18.850 --> 00:34:21.900
I could add the two columns.
00:34:21.900 --> 00:34:25.070
What would be the total
complete solution if the
00:34:25.070 --> 00:34:28.360
Right? right-hand
side was 4 3 7 6?
00:34:28.360 --> 00:34:31.489
There would be the
particular solution one
00:34:31.489 --> 00:34:34.067
one, one of that column
plus one of that,
00:34:34.067 --> 00:34:35.150
and that would be the only
00:34:35.150 --> 00:34:36.429
solution.
00:34:36.429 --> 00:34:39.770
So there would be -- x
particular would be one one
00:34:39.770 --> 00:34:43.560
in the case when the right
side is the sum of those two
00:34:43.560 --> 00:34:46.850
columns, and that's it.
00:34:46.850 --> 00:34:50.670
So that would be a
case with one solution.
00:34:50.670 --> 00:34:51.250
OK.
00:34:51.250 --> 00:34:55.469
That, this is the typical
setup with full column rank.
00:34:55.469 --> 00:35:00.000
Now I go to full row rank.
00:35:00.000 --> 00:35:04.260
You see the sort of natural
symmetry of this discussion.
00:35:04.260 --> 00:35:14.523
Full row rank means r=m.
00:35:17.390 --> 00:35:19.940
So this is what I'm
interested in now, r=m.
00:35:23.420 --> 00:35:24.710
OK, what's up with that?
00:35:29.830 --> 00:35:31.010
How many pivots?
00:35:31.010 --> 00:35:33.000
m.
00:35:33.000 --> 00:35:40.060
So what happens when we do
elimination in that case?
00:35:40.060 --> 00:35:42.520
I'm going to get m pivots.
00:35:42.520 --> 00:35:47.520
So every row has a pivot, right?
00:35:47.520 --> 00:35:48.855
Every row has a pivot.
00:35:52.120 --> 00:35:55.950
Then what about solvability?
00:35:55.950 --> 00:35:59.880
What about this business of --
for which right-hand sides can
00:35:59.880 --> 00:36:01.120
I solve it?
00:36:01.120 --> 00:36:02.970
So that's my question.
00:36:02.970 --> 00:36:14.180
I can solve Ax=b for
which right-hand sides?
00:36:14.180 --> 00:36:18.450
Do you see what's coming?
00:36:18.450 --> 00:36:23.990
I do elimination, I
don't get any zero rows.
00:36:23.990 --> 00:36:26.890
So there aren't any
requirements on b.
00:36:26.890 --> 00:36:29.730
I can solve Ax=b for every b.
00:36:36.450 --> 00:36:39.990
I can solve Ax=b for
every right-hand side.
00:36:39.990 --> 00:36:46.705
So this is the existence,
exists a solution.
00:36:49.260 --> 00:36:57.180
Now tell me, so the, u- u- so
every row has a pivot in it.
00:36:57.180 --> 00:37:00.820
So how many free
variables are there?
00:37:00.820 --> 00:37:04.780
How many free
variables in this case?
00:37:04.780 --> 00:37:07.560
If I had n variables
to start with,
00:37:07.560 --> 00:37:11.180
how many are used up
by pivot variables?
00:37:11.180 --> 00:37:13.890
r, which is m.
00:37:13.890 --> 00:37:25.600
So I'm left with, left
with n-r free variables.
00:37:31.050 --> 00:37:31.910
OK.
00:37:31.910 --> 00:37:37.220
So this case of full row
rank I can always solve,
00:37:37.220 --> 00:37:41.440
and then this tells me how
many variables are free,
00:37:41.440 --> 00:37:43.400
and this is of course n-m.
00:37:43.400 --> 00:37:48.040
This is n-m free variables.
00:37:48.040 --> 00:37:48.980
Can I do an example?
00:37:52.310 --> 00:37:54.750
You know, the best way for
me to do an example is just
00:37:54.750 --> 00:37:58.140
to transpose that example.
00:37:58.140 --> 00:38:01.950
So let me take, let me take
that matrix that had column one
00:38:01.950 --> 00:38:05.970
two six five and make it a row.
00:38:05.970 --> 00:38:11.470
And let me take three one
one one as the second row.
00:38:11.470 --> 00:38:18.700
And let me ask you, this is
my matrix A, what's its rank?
00:38:18.700 --> 00:38:20.230
What's the rank of that matrix?
00:38:20.230 --> 00:38:24.560
Sorry to ask, but
not sorry really,
00:38:24.560 --> 00:38:27.130
because we're just
getting the idea of rank.
00:38:27.130 --> 00:38:29.770
What's the rank of that matrix?
00:38:29.770 --> 00:38:32.260
Two, exactly, two.
00:38:32.260 --> 00:38:33.770
There will be two pivots.
00:38:33.770 --> 00:38:36.770
What will the row
reduced echelon form be?
00:38:36.770 --> 00:38:38.850
Anybody know that one?
00:38:38.850 --> 00:38:42.230
Actually, tell me not only --
you have to tell me not only
00:38:42.230 --> 00:38:45.110
the, there'll be two pivots
but which will be the pivot
00:38:45.110 --> 00:38:46.550
columns.
00:38:46.550 --> 00:38:50.060
Which columns of this matrix
will be pivot columns?
00:38:50.060 --> 00:38:53.140
So the first column
is fine, and then
00:38:53.140 --> 00:38:55.720
I go on to the next
column, and what do I get?
00:38:55.720 --> 00:38:57.640
Do I get a second
pivot out of --
00:38:57.640 --> 00:39:00.410
will I get a second
pivot in this position?
00:39:00.410 --> 00:39:01.300
Yes.
00:39:01.300 --> 00:39:07.200
So the pivots, when I get all
the way to R, will be there.
00:39:07.200 --> 00:39:13.860
And here will be some numbers.
00:39:13.860 --> 00:39:18.670
This is the part that
I previously called F.
00:39:18.670 --> 00:39:23.720
This is the part that -- the
pivot columns in R will be
00:39:23.720 --> 00:39:25.680
the identity matrix.
00:39:25.680 --> 00:39:31.300
There are no zero rows, no zero
rows, because the rank is two.
00:39:31.300 --> 00:39:34.630
But there'll be stuff over here.
00:39:34.630 --> 00:39:42.530
And that will, enter the special
solutions and the null space.
00:39:42.530 --> 00:39:43.230
OK.
00:39:43.230 --> 00:39:51.840
So this is a typical matrix
with r=m smaller than n.
00:39:51.840 --> 00:39:56.135
Now finally I've got a
space here for r=m=n.
00:40:01.540 --> 00:40:06.190
I'm off in the corner here with
the most important case of all.
00:40:06.190 --> 00:40:08.910
So what's up with this matrix?
00:40:08.910 --> 00:40:10.970
So let me give an example.
00:40:10.970 --> 00:40:15.065
OK, brilliant example, 1 2 3 1.
00:40:19.860 --> 00:40:23.628
Tell me what -- how do I
describe a matrix that has rank
00:40:23.628 --> 00:40:24.127
r=m=n?
00:40:26.930 --> 00:40:32.560
So the matrix is square,
right, it's a square matrix.
00:40:32.560 --> 00:40:36.350
And if I know its rank is
-- it's full rank, now.
00:40:36.350 --> 00:40:39.290
I don't have to say full
column rank or full row rank --
00:40:39.290 --> 00:40:43.800
I just say full rank, because
the count, column count
00:40:43.800 --> 00:40:47.130
and the row count are
the same, and the rank
00:40:47.130 --> 00:40:49.040
is as big as it can be.
00:40:49.040 --> 00:40:51.090
And what kind of a
matrix have I got?
00:40:53.920 --> 00:40:56.670
It's invertible.
00:40:56.670 --> 00:41:01.510
So that's exactly the
invertible matrices.
00:41:01.510 --> 00:41:06.310
r=m=n means the -- what's
the row echelon form,
00:41:06.310 --> 00:41:10.920
the reduced row echelon form,
for an invertible matrix?
00:41:10.920 --> 00:41:14.630
For a square, nice,
square, invertible matrix?
00:41:14.630 --> 00:41:17.320
It's I.
00:41:17.320 --> 00:41:18.880
Right.
00:41:18.880 --> 00:41:25.530
So you see that the,
the good matrices
00:41:25.530 --> 00:41:31.580
are the ones that kind of
come out trivially in R.
00:41:31.580 --> 00:41:34.270
You reduce them all the
way to the identity matrix.
00:41:34.270 --> 00:41:37.900
What's the null space for
this, for this matrix?
00:41:37.900 --> 00:41:41.170
Can I just hammer
away with questions?
00:41:41.170 --> 00:41:43.160
What's the null space
for this matrix?
00:41:45.910 --> 00:41:51.570
The null space of that matrix
is the zero vector only.
00:41:51.570 --> 00:41:54.660
The zero vector only.
00:41:54.660 --> 00:41:58.530
What are the conditions
to solve Ax=b?
00:41:58.530 --> 00:42:01.950
Which right-hand sides b are OK?
00:42:01.950 --> 00:42:07.560
If I want to solve Ax=b for
this example, so A is this,
00:42:07.560 --> 00:42:14.140
b is b1 b2, what are the
conditions on b1 and b2?
00:42:14.140 --> 00:42:17.170
None at all, right.
00:42:17.170 --> 00:42:21.850
So this is the case, this is
the case where I can solve --
00:42:21.850 --> 00:42:25.660
so I've coming back here, I
can -- since the rank equals m,
00:42:25.660 --> 00:42:28.460
I can solve for every b.
00:42:28.460 --> 00:42:33.840
And since the rank is also
n, there's a unique solution.
00:42:33.840 --> 00:42:36.640
Let me summarize the
whole picture here.
00:42:39.210 --> 00:42:41.210
Here's the whole picture.
00:42:41.210 --> 00:42:44.780
I could have r=m=n.
00:42:44.780 --> 00:42:51.190
This is the case where this
is the identity matrix.
00:42:51.190 --> 00:42:53.595
And this is the case where
there is one solution.
00:42:56.790 --> 00:43:02.300
That's the square
invertible chapter two case.
00:43:02.300 --> 00:43:04.420
Now we're into chapter three.
00:43:04.420 --> 00:43:08.330
We could have r=m
smaller than n.
00:43:11.920 --> 00:43:13.730
Now that's what
we had over there,
00:43:13.730 --> 00:43:17.610
and the row echelon form
looked like the identity
00:43:17.610 --> 00:43:18.960
with some zero rows.
00:43:21.960 --> 00:43:27.565
And that was the case where
there are zero or one solution.
00:43:31.420 --> 00:43:34.420
When I say solution
I mean to Ax=b.
00:43:37.540 --> 00:43:39.590
So this case,
there's always one.
00:43:39.590 --> 00:43:42.130
This case there's zero or one.
00:43:42.130 --> 00:43:45.640
And now let me take the
case of full column rank,
00:43:45.640 --> 00:43:55.860
but some, extra rows.
00:43:55.860 --> 00:44:00.000
So now R has --
00:44:00.000 --> 00:44:04.700
well, the identity --
00:44:04.700 --> 00:44:08.280
I'm almost tempted to write
the identity matrix and then F,
00:44:08.280 --> 00:44:10.130
but that isn't
necessarily right.
00:44:15.970 --> 00:44:20.360
I have -- is that right?
00:44:20.360 --> 00:44:24.310
Am I getting this correct here?
00:44:24.310 --> 00:44:25.430
Oh, I'm not!
00:44:25.430 --> 00:44:26.390
My God!
00:44:26.390 --> 00:44:33.330
This is the case R equals n,
the columns, the columns are,
00:44:33.330 --> 00:44:34.390
are OK.
00:44:34.390 --> 00:44:38.770
That's the case that was on that
board, r=n, full column rank.
00:44:38.770 --> 00:44:43.240
Now I want the case
where m is smaller than n
00:44:43.240 --> 00:44:46.550
and I've got extra columns.
00:44:46.550 --> 00:44:47.050
OK.
00:44:47.050 --> 00:44:47.810
There we go.
00:44:52.030 --> 00:44:57.140
So this is now the
case of full row rank,
00:44:57.140 --> 00:45:02.070
and it looks like
I F except that I
00:45:02.070 --> 00:45:08.430
can't be sure that the pivot
columns are the first columns.
00:45:08.430 --> 00:45:14.690
So the I and the F, could
be partly mixed into the I.
00:45:14.690 --> 00:45:18.670
Can I write that
with just like that?
00:45:18.670 --> 00:45:23.830
So the F could be sort
of partly into the I
00:45:23.830 --> 00:45:28.450
if the first columns
weren't the pivot columns.
00:45:28.450 --> 00:45:31.760
Now how many solutions
in this case?
00:45:31.760 --> 00:45:34.330
There's always a solution.
00:45:34.330 --> 00:45:35.850
This is the existence case.
00:45:35.850 --> 00:45:37.020
There's always a solution.
00:45:37.020 --> 00:45:39.160
We're not getting any zero rows.
00:45:39.160 --> 00:45:41.430
There are no zero rows here.
00:45:41.430 --> 00:45:45.635
So there's always either one
or infinitely many solutions.
00:45:50.061 --> 00:45:50.560
OK.
00:45:53.230 --> 00:45:55.600
Actually, I guess there's
always an infinite number,
00:45:55.600 --> 00:46:02.760
because we always have some
null space to deal with.
00:46:02.760 --> 00:46:06.150
Then the final case is
where r is smaller than m
00:46:06.150 --> 00:46:08.880
and smaller than n.
00:46:08.880 --> 00:46:09.380
OK.
00:46:09.380 --> 00:46:14.830
Now that's the case
where R is the identity
00:46:14.830 --> 00:46:19.920
with some free stuff but
with some zero rows too.
00:46:19.920 --> 00:46:23.580
And that's the case where
there's either no solution --
00:46:23.580 --> 00:46:29.200
because we didn't get a zero
equals zero for some bs,
00:46:29.200 --> 00:46:32.330
or infinitely many solutions.
00:46:37.100 --> 00:46:38.010
OK.
00:46:38.010 --> 00:46:44.310
Do you -- this board really
summarizes the lecture,
00:46:44.310 --> 00:46:47.370
and this sentence
summarizes the lecture.
00:46:47.370 --> 00:46:55.070
The rank tells you everything
about the number of solutions.
00:46:55.070 --> 00:46:57.010
That number, the
rank r, tells you
00:46:57.010 --> 00:47:01.620
all the information except the
exact entries in the solutions.
00:47:01.620 --> 00:47:04.140
For that you go to the matrix.
00:47:04.140 --> 00:47:05.150
OK, good.
00:47:05.150 --> 00:47:08.770
Have a great weekend, and
I'll see you on Monday.