WEBVTT

00:00:00.500 --> 00:00:02.620
GILBERT STRANG: OK.

00:00:02.620 --> 00:00:05.850
We're coming to the point
where we need matrices.

00:00:05.850 --> 00:00:12.420
That's the point when we
have several equations,

00:00:12.420 --> 00:00:17.520
several differential
equations instead of just one.

00:00:17.520 --> 00:00:20.060
And it's a matrix that
does that coupling.

00:00:20.060 --> 00:00:26.180
So can I-- this won't be a
full course in linear algebra.

00:00:26.180 --> 00:00:28.680
That would be
available, you may know

00:00:28.680 --> 00:00:34.650
on, open courseware for 18.06.

00:00:34.650 --> 00:00:36.780
That's the linear
algebra course.

00:00:36.780 --> 00:00:40.480
But [INAUDIBLE] facts,
and why not just

00:00:40.480 --> 00:00:43.630
say them here in a few minutes?

00:00:43.630 --> 00:00:46.760
So I have a matrix.

00:00:46.760 --> 00:00:48.605
Well there's a matrix.

00:00:48.605 --> 00:00:50.960
That's a 3 by 3 matrix.

00:00:50.960 --> 00:00:55.690
And first I want to ask how
does it multiply a vector.

00:00:55.690 --> 00:00:59.770
So there it is multiplying
a vector, v1, v2, v3.

00:00:59.770 --> 00:01:04.370
And what's the result, key idea?

00:01:04.370 --> 00:01:09.090
It takes the answer
on the right-hand side

00:01:09.090 --> 00:01:14.430
is this number v1, times that
column, plus this number,

00:01:14.430 --> 00:01:18.350
that number times the second
column, plus the third number,

00:01:18.350 --> 00:01:21.160
the third number times
the third column,

00:01:21.160 --> 00:01:24.100
combination of the columns of a.

00:01:24.100 --> 00:01:25.840
That's what a times v is.

00:01:25.840 --> 00:01:32.120
That's what the notation of
matrix multiplication produces.

00:01:32.120 --> 00:01:36.810
That's really basic to see it
as a combination of columns.

00:01:36.810 --> 00:01:38.930
Now I want to build on that.

00:01:38.930 --> 00:01:43.680
That's one particular, if
you give me v1, v2, and v3,

00:01:43.680 --> 00:01:44.970
I know how to multiply it.

00:01:44.970 --> 00:01:46.870
I take the combination.

00:01:46.870 --> 00:01:50.000
Now I would like you to
think about the result

00:01:50.000 --> 00:01:53.980
from all v1, v2, and v3.

00:01:53.980 --> 00:01:58.560
If I take all those numbers, and
I get a whole lot of answers.

00:01:58.560 --> 00:02:02.210
They're all vectors,
the result of A times v

00:02:02.210 --> 00:02:07.310
is another vector,
Av, And I want

00:02:07.310 --> 00:02:13.360
to think about Av, those
outputs, for all inputs v.

00:02:13.360 --> 00:02:16.700
So I take v1, v2, v3 to
be [AUDIO OUT] numbers.

00:02:16.700 --> 00:02:20.170
And I get all combinations
of those three columns.

00:02:20.170 --> 00:02:26.090
And usually I would get the
whole 3-dimensional space.

00:02:26.090 --> 00:02:29.300
Usually I can
produce any vector,

00:02:29.300 --> 00:02:35.120
any output b1, b2, b3
from A times v. But not

00:02:35.120 --> 00:02:37.720
for this matrix,
not for this matrix.

00:02:37.720 --> 00:02:46.060
Because this matrix is,
you could say, deficient.

00:02:46.060 --> 00:02:49.000
That third column
there, 2, 3, 3,

00:02:49.000 --> 00:02:53.640
is obviously the sum of
columns one and column two.

00:02:53.640 --> 00:02:58.220
So this v3 times
that third column

00:02:58.220 --> 00:03:01.650
just produces something
that I could already get

00:03:01.650 --> 00:03:04.290
from column one and column two.

00:03:04.290 --> 00:03:08.770
That v3 times that column
three, I could x out.

00:03:08.770 --> 00:03:12.920
That's the same as column
one, plus column two

00:03:12.920 --> 00:03:15.340
for this matrix, not usually.

00:03:15.340 --> 00:03:19.790
And then so I only really have
a combination of two columns.

00:03:19.790 --> 00:03:21.000
It's a combination of three.

00:03:21.000 --> 00:03:24.060
But the third one was
dependent on the others.

00:03:24.060 --> 00:03:27.350
And it's really a
combination of two columns.

00:03:27.350 --> 00:03:31.840
So combinations of two
columns, two vectors

00:03:31.840 --> 00:03:37.210
in 3-dimensional
space produce a plane.

00:03:37.210 --> 00:03:39.340
I only get a plane.

00:03:39.340 --> 00:03:42.630
I don't get all of 3-dimensional
space, only a plane.

00:03:42.630 --> 00:03:47.070
And I call that plane
the column space,

00:03:47.070 --> 00:03:49.490
so the column space
of the matrix.

00:03:49.490 --> 00:03:52.410
So if you gave me
a different matrix,

00:03:52.410 --> 00:03:59.630
if you change this 3 to an
11, probably the column space

00:03:59.630 --> 00:04:04.020
now changes to--
for that matrix I

00:04:04.020 --> 00:04:07.850
think the column space would be
the whole 3-dimensional space.

00:04:07.850 --> 00:04:09.010
I get everything.

00:04:09.010 --> 00:04:14.500
But when this third column is
this the sum of the first two

00:04:14.500 --> 00:04:18.570
columns, it's not
giving me anything new.

00:04:18.570 --> 00:04:20.779
And the column space
is only a plane.

00:04:20.779 --> 00:04:24.480
And you can think of a matrix
where the column space is only

00:04:24.480 --> 00:04:28.850
a line, just one
independent column.

00:04:28.850 --> 00:04:30.870
OK.

00:04:30.870 --> 00:04:34.610
So that, we thought about this.

00:04:34.610 --> 00:04:39.280
[AUDIO OUT] is all
combinations of the columns.

00:04:39.280 --> 00:04:42.070
In other words, it's
all the results,

00:04:42.070 --> 00:04:46.520
all the outputs from A times
v. It's all the outputs

00:04:46.520 --> 00:04:50.050
from A times v. Those are the
combinations of the columns.

00:04:50.050 --> 00:04:54.160
So we can answer the most basic
question of linear algebra.

00:04:54.160 --> 00:04:56.380
When does Av equal b?

00:04:56.380 --> 00:04:58.170
Have [AUDIO OUT].

00:04:58.170 --> 00:05:02.350
When is there a v so
that I can solve this?

00:05:02.350 --> 00:05:05.730
When is there a v that
solves this equation?

00:05:05.730 --> 00:05:09.080
So it's a question about b.

00:05:09.080 --> 00:05:12.730
What is it about b that must
be true if this can be solved?

00:05:12.730 --> 00:05:16.640
Well this says that
equation is saying

00:05:16.640 --> 00:05:20.600
b is a combination
of the columns of a.

00:05:20.600 --> 00:05:26.270
So this has a
solution when b must

00:05:26.270 --> 00:05:33.360
be-- shall I say must
be in the column space.

00:05:36.290 --> 00:05:41.240
For that example,
only b's that where

00:05:41.240 --> 00:05:44.120
we can get a solution on
b's that are combinations

00:05:44.120 --> 00:05:46.080
of the first two columns.

00:05:46.080 --> 00:05:49.260
Because having the third
column at our disposal

00:05:49.260 --> 00:05:50.785
gives us no help.

00:05:50.785 --> 00:05:52.793
It doesn't give us anything new.

00:05:52.793 --> 00:05:53.292
[AUDIO OUT]

00:05:56.010 --> 00:06:00.480
It will be solvable
if b equalled 1, 1, 1.

00:06:00.480 --> 00:06:04.310
That's a combination of the
column, or if b equals 1, 2, 2.

00:06:04.310 --> 00:06:07.200
That's another simple
combination of the columns.

00:06:07.200 --> 00:06:09.880
Or if b equals 2, 3, 3.

00:06:09.880 --> 00:06:12.920
But I'm only, I'm
staying on a plane there.

00:06:12.920 --> 00:06:16.150
And most b's are off that plane.

00:06:16.150 --> 00:06:18.210
Now when there is a solution.

00:06:18.210 --> 00:06:19.250
All right.

00:06:19.250 --> 00:06:26.460
Now a second key idea
of linear algebra,

00:06:26.460 --> 00:06:29.460
can we do it in
this short video?

00:06:29.460 --> 00:06:34.670
I want to know about the
equation Av equals 0.

00:06:34.670 --> 00:06:38.210
So now I'm setting the
right-hand side to be 0.

00:06:38.210 --> 00:06:41.640
That's the 0 vector, 0, 0, 0.

00:06:41.640 --> 00:06:43.430
Does it have a solution?

00:06:43.430 --> 00:06:44.800
Does it have a solution?

00:06:44.800 --> 00:06:46.850
Let's take this example.

00:06:46.850 --> 00:06:58.840
1, 1, 1; 1, 2, 2; 2, 3, 3; now
I'm looking at the solutions

00:06:58.840 --> 00:07:02.660
when the right side is all 0.

00:07:02.660 --> 00:07:04.750
Does that have a solution?

00:07:04.750 --> 00:07:10.420
Is there a combination of those
three columns that gives 0?

00:07:10.420 --> 00:07:12.910
Well there is always
one combination.

00:07:12.910 --> 00:07:16.780
I could take 0, 0, and 0.

00:07:16.780 --> 00:07:19.850
I could take nothing,
0 of everything.

00:07:19.850 --> 00:07:22.810
0 of this column, 0 of that
column, 0 of the third column,

00:07:22.810 --> 00:07:24.780
would give me to
the 0 [AUDIO OUT].

00:07:24.780 --> 00:07:26.730
That solution is
always available.

00:07:26.730 --> 00:07:30.110
The big question is, is
there another solution.

00:07:30.110 --> 00:07:37.150
And here for this deficient,
singular, non-invertible

00:07:37.150 --> 00:07:38.940
matrix, there is.

00:07:38.940 --> 00:07:40.500
There is another solution.

00:07:40.500 --> 00:07:41.940
Let me just write it down.

00:07:41.940 --> 00:07:43.870
Let me put it in there.

00:07:43.870 --> 00:07:45.985
Do you see what the solution is?

00:07:45.985 --> 00:07:49.230
The third column is
the sum of those two.

00:07:49.230 --> 00:07:54.760
So if I want one of that
column, I should take minus 1

00:07:54.760 --> 00:07:56.450
in other column.

00:07:56.450 --> 00:08:00.510
So this is minus this
column, minus this column,

00:08:00.510 --> 00:08:03.540
plus this column
gives me the 0 column.

00:08:03.540 --> 00:08:09.060
That is a vector
in the null space.

00:08:09.060 --> 00:08:15.820
That's a solution to
Avn equals [AUDIO OUT].

00:08:15.820 --> 00:08:25.200
So the null space is all
solutions to Av equals 0.

00:08:25.200 --> 00:08:28.105
It's all the v's.

00:08:28.105 --> 00:08:31.320
The null space is
a bunch of v's.

00:08:31.320 --> 00:08:34.669
The column space
was a bunch of b's.

00:08:34.669 --> 00:08:38.130
It's just going to just
emphasize that difference.

00:08:38.130 --> 00:08:42.144
I was looking at
which b [AUDIO OUT].

00:08:42.144 --> 00:08:47.320
I wasn't paying attention to
what that solution was, just

00:08:47.320 --> 00:08:48.520
is there a solution.

00:08:48.520 --> 00:08:52.030
Then that b is in
the column space.

00:08:52.030 --> 00:08:54.450
I take b equals 0.

00:08:54.450 --> 00:08:57.590
I fixed that all important b.

00:08:57.590 --> 00:09:00.930
And now I'm looking
at the solutions.

00:09:00.930 --> 00:09:02.920
And here I find one.

00:09:02.920 --> 00:09:04.620
Can you find any more solutions?

00:09:04.620 --> 00:09:10.090
I think minus 10, minus 10, and
10 would be another solution.

00:09:10.090 --> 00:09:11.820
It's 10 times as much.

00:09:11.820 --> 00:09:14.410
And 0, 0, 0 is solution.

00:09:14.410 --> 00:09:18.100
[AUDIO OUT] line of solutions.

00:09:18.100 --> 00:09:20.170
We had a plane for
the column space.

00:09:20.170 --> 00:09:22.720
But we have a line
for the null space.

00:09:22.720 --> 00:09:24.430
Isn't that neat?

00:09:24.430 --> 00:09:26.920
One's a plane, one's
a line, dimension two

00:09:26.920 --> 00:09:28.820
plus dimension one.

00:09:28.820 --> 00:09:31.100
Two for the plane,
one for the line,

00:09:31.100 --> 00:09:35.690
adds to dimension three, the
dimension of the whole space.

00:09:35.690 --> 00:09:36.250
OK.

00:09:36.250 --> 00:09:38.520
That's a little going at in.

00:09:38.520 --> 00:09:40.430
All right.

00:09:40.430 --> 00:09:45.760
Now I ask, what
our all solutions?

00:09:45.760 --> 00:09:57.900
Complete solution
to Av equals, well

00:09:57.900 --> 00:10:01.710
let me choose some right-hand
side where there is a solution.

00:10:01.710 --> 00:10:04.070
Let me choose a
right-hand side, say

00:10:04.070 --> 00:10:05.945
if I add that column
and that column,

00:10:05.945 --> 00:10:10.220
I'll get Av-- maybe I'll
take two of that column

00:10:10.220 --> 00:10:12.010
plus one of that column.

00:10:12.010 --> 00:10:14.293
Two of the first column
with one of the second

00:10:14.293 --> 00:10:18.430
would be 3, 2 plus
that would be a 4,

00:10:18.430 --> 00:10:21.490
2 plus that would be another 4.

00:10:21.490 --> 00:10:22.750
OK.

00:10:22.750 --> 00:10:23.360
That's my b.

00:10:27.394 --> 00:10:28.810
It's a combination
of the columns.

00:10:28.810 --> 00:10:32.370
You saw me create it from
the first two columns.

00:10:32.370 --> 00:10:35.770
So now I ask, what
are all the solutions?

00:10:35.770 --> 00:10:37.930
It's in the column space.

00:10:37.930 --> 00:10:42.790
It's 2 times the first column,
plus the second column.

00:10:42.790 --> 00:10:45.400
But there may be
other solutions.

00:10:45.400 --> 00:10:53.178
So all solutions, a complete
solution, v complete

00:10:53.178 --> 00:10:55.530
is here's the key idea.

00:10:55.530 --> 00:10:58.130
And the point is that
it's the same that we know

00:10:58.130 --> 00:11:01.190
from differential equations.

00:11:01.190 --> 00:11:12.500
It's particular solution
plus any null solution.

00:11:12.500 --> 00:11:18.410
Plus all, you can
say all v null.

00:11:18.410 --> 00:11:20.630
Particular plus null solution.

00:11:20.630 --> 00:11:24.670
It's such an important concept
we just want to see it again.

00:11:24.670 --> 00:11:27.170
One particular
solution with that

00:11:27.170 --> 00:11:31.100
thing would be
particular, v particular

00:11:31.100 --> 00:11:36.620
could be-- 2-- how
did we produce that?

00:11:36.620 --> 00:11:40.580
Out of two these, plus one
of these, plus zero of that.

00:11:40.580 --> 00:11:46.090
So v particular
could be 2, 1, 0.

00:11:46.090 --> 00:11:50.580
It works for that particular
b, two of the first column,

00:11:50.580 --> 00:11:52.180
one of the second.

00:11:52.180 --> 00:11:57.100
Now then we could add in
anything in the null solution.

00:11:57.100 --> 00:11:59.070
So we have infinitely
many solutions here.

00:11:59.070 --> 00:12:03.010
We've got one solution
plus added to that,

00:12:03.010 --> 00:12:05.490
a whole line of solutions.

00:12:05.490 --> 00:12:11.340
This, all the null space,
would be all vectors like that.

00:12:11.340 --> 00:12:11.890
OK.

00:12:11.890 --> 00:12:14.470
That's the picture
that we've seen

00:12:14.470 --> 00:12:16.330
for differential equations.

00:12:16.330 --> 00:12:20.380
And I just want to bring it
out again for matrix equations,

00:12:20.380 --> 00:12:23.130
using the language
of linear algebra.

00:12:23.130 --> 00:12:25.730
That's what I'm
introducing here.

00:12:25.730 --> 00:12:29.490
I have one particular
solution, plus anything

00:12:29.490 --> 00:12:35.030
in the null [AUDIO OUT]
space of vectors that

00:12:35.030 --> 00:12:37.500
is the heart of linear algebra.

00:12:37.500 --> 00:12:39.420
Thank you.