WEBVTT
00:00:01.025 --> 00:00:03.108
The following content is
provided under a Creative
00:00:03.108 --> 00:00:03.774
Commons license.
00:00:03.774 --> 00:00:06.740
Your support will help
MIT OpenCourseWare
00:00:06.740 --> 00:00:09.340
continue to offer high quality
educational resources for free.
00:00:09.340 --> 00:00:13.380
To make a donation, or to
view additional materials
00:00:13.380 --> 00:00:18.690
from hundreds of MIT courses,
visit MIT OpenCourseWare
00:00:18.690 --> 00:00:21.820
at ocw.mit.edu.
00:00:21.820 --> 00:00:27.190
PROFESSOR STRANG: This is
the one and only review, you
00:00:27.190 --> 00:00:30.740
could say, of linear algebra.
00:00:30.740 --> 00:00:33.660
I just think linear
algebra is very important.
00:00:33.660 --> 00:00:35.420
You may have got that idea.
00:00:35.420 --> 00:00:43.430
And my website even has a little
essay called Too Much Calculus.
00:00:43.430 --> 00:00:49.660
Because I think it's crazy
for all the U.S. universities
00:00:49.660 --> 00:00:51.890
do this pretty much,
you get semester
00:00:51.890 --> 00:00:55.460
after semester in differential
calculus, integral calculus,
00:00:55.460 --> 00:00:57.130
ultimately
differential equations.
00:00:57.130 --> 00:01:00.050
You run out of steam
before the good stuff,
00:01:00.050 --> 00:01:03.340
before you run out of time.
00:01:03.340 --> 00:01:08.010
And anybody who computes,
who's living in the real world,
00:01:08.010 --> 00:01:10.830
is using linear algebra.
00:01:10.830 --> 00:01:13.030
You're taking a
differential equation,
00:01:13.030 --> 00:01:16.230
you're taking your
model, making it discrete
00:01:16.230 --> 00:01:20.830
and computing with matrices.
00:01:20.830 --> 00:01:27.080
The world's digital
now, not analog.
00:01:27.080 --> 00:01:33.870
I hope it's okay to start the
course with linear algebra.
00:01:33.870 --> 00:01:38.200
But many engineering
curricula don't fully
00:01:38.200 --> 00:01:40.170
recognize that and
so if you haven't
00:01:40.170 --> 00:01:46.770
had an official course, linear
algebra, stay with 18.085.
00:01:46.770 --> 00:01:48.280
This is a good way to learn it.
00:01:48.280 --> 00:01:51.330
You're sort of learning
what's important.
00:01:51.330 --> 00:01:58.340
So my review would be-- And
then this is-- Future Wednesdays
00:01:58.340 --> 00:02:02.190
will be in our regular
room for homework, review,
00:02:02.190 --> 00:02:06.830
questions of all kinds,
and today questions too.
00:02:06.830 --> 00:02:10.990
Shall I just fire away for
the first half of the time
00:02:10.990 --> 00:02:15.350
to give you a sense of
how I see the subject,
00:02:15.350 --> 00:02:17.950
or at least within
that limited time.
00:02:17.950 --> 00:02:22.440
And then questions
are totally welcome.
00:02:22.440 --> 00:02:24.031
Always welcome, actually.
00:02:24.031 --> 00:02:24.530
Right?
00:02:24.530 --> 00:02:26.590
So I'll just start right up.
00:02:26.590 --> 00:02:31.130
So essentially linear
algebra progresses
00:02:31.130 --> 00:02:41.040
starting with vectors to
matrices and then finally
00:02:41.040 --> 00:02:42.510
to subspaces.
00:02:42.510 --> 00:02:45.240
So that's, like,
the abstraction.
00:02:45.240 --> 00:02:49.440
You could say abstraction,
but it's not difficult,
00:02:49.440 --> 00:02:50.680
that you want to see.
00:02:50.680 --> 00:02:53.290
Until you see the
idea of a subspace,
00:02:53.290 --> 00:02:56.130
you haven't really
got linear algebra.
00:02:56.130 --> 00:02:58.460
Okay, so I'll start
at the beginning.
00:02:58.460 --> 00:03:01.190
What do you do with vectors?
00:03:01.190 --> 00:03:04.190
Answer: you take their
linear combinations.
00:03:04.190 --> 00:03:06.160
That's what you can
do with a vector.
00:03:06.160 --> 00:03:11.760
You can multiply it by a number
and you can add or subtract.
00:03:11.760 --> 00:03:13.790
So that's the key operation.
00:03:13.790 --> 00:03:18.120
Suppose I have
vectors u, v and w.
00:03:18.120 --> 00:03:20.850
Let me take three of them.
00:03:20.850 --> 00:03:22.710
So I can take
their combinations.
00:03:22.710 --> 00:03:28.170
So some combination will
be, say some number times
00:03:28.170 --> 00:03:33.230
u plus some number times v
plus some number times w.
00:03:33.230 --> 00:03:36.490
So these numbers
are called scalars.
00:03:36.490 --> 00:03:39.640
So these would be
called scalars.
00:03:39.640 --> 00:03:44.270
And the whole thing is
a linear combination.
00:03:44.270 --> 00:03:48.180
Let me abbreviate those
words, linear combination.
00:03:48.180 --> 00:03:54.390
And you get some answer, say b.
00:03:54.390 --> 00:04:00.600
But let's put it down, make
this whole discussion specific.
00:04:00.600 --> 00:04:05.120
Yeah, I started a
little early, I think.
00:04:05.120 --> 00:04:08.640
I'm going to take three
vectors, u, v and w,
00:04:08.640 --> 00:04:12.640
and take their combinations.
00:04:12.640 --> 00:04:14.920
They're carefully chosen.
00:04:14.920 --> 00:04:20.440
My u is going to be [1, -1, 0].
00:04:20.440 --> 00:04:23.580
And I'll take vectors
in three dimensions.
00:04:23.580 --> 00:04:26.550
So that means their combinations
will be in three dimensions,
00:04:26.550 --> 00:04:29.920
R^3, three-dimensional space.
00:04:29.920 --> 00:04:37.900
So that'll be u and then
v, let's take 0, I think,
00:04:37.900 --> 00:04:40.530
1 and -1.
00:04:40.530 --> 00:04:41.240
Okay.
00:04:41.240 --> 00:04:45.170
Suppose I stopped there and
took their linear combinations.
00:04:45.170 --> 00:04:47.820
It's very helpful
to see a picture
00:04:47.820 --> 00:04:49.940
in three-dimensional space.
00:04:49.940 --> 00:04:52.100
I mean the great thing
about linear algebra,
00:04:52.100 --> 00:04:55.000
it moves into
n-dimensional space,
00:04:55.000 --> 00:04:58.320
10-dimensional, 100-dimensional,
where we can't visualize,
00:04:58.320 --> 00:05:02.020
but yet, our instinct is
right if we just follow.
00:05:02.020 --> 00:05:06.980
So what's your instinct if
I took those two vectors,
00:05:06.980 --> 00:05:09.290
and notice they're
not on the same line,
00:05:09.290 --> 00:05:11.180
one isn't a multiple
of the other,
00:05:11.180 --> 00:05:13.220
they go in different directions.
00:05:13.220 --> 00:05:21.340
If I took their combinations,
say x_1*u + x_2*v. Oh now,
00:05:21.340 --> 00:05:23.930
let me push, this is
a serious question.
00:05:23.930 --> 00:05:26.630
If I took all
their combinations.
00:05:26.630 --> 00:05:28.630
So let me try to
draw a little bit.
00:05:28.630 --> 00:05:29.130
Okay.
00:05:29.130 --> 00:05:33.540
I'm in three-dimensional
space and u goes somewhere,
00:05:33.540 --> 00:05:39.030
maybe there and v goes
somewhere, maybe here.
00:05:39.030 --> 00:05:41.980
Now suppose I take
all the combinations,
00:05:41.980 --> 00:05:44.040
so I could multiply
that first guy
00:05:44.040 --> 00:05:46.740
by any number, that
would fill the line.
00:05:46.740 --> 00:05:50.780
I can multiply that second
guy, v. So this was u
00:05:50.780 --> 00:05:54.360
and this was v. I can multiply
that by any number x_2, that
00:05:54.360 --> 00:05:56.290
would fill its line.
00:05:56.290 --> 00:05:58.070
Each of those
lines I would later
00:05:58.070 --> 00:06:01.500
call a one-dimensional
subspace, just a line.
00:06:01.500 --> 00:06:06.730
But now, what happens if I take
all combinations of the two?
00:06:06.730 --> 00:06:08.550
What do you think?
00:06:08.550 --> 00:06:09.880
You got a plane.
00:06:09.880 --> 00:06:11.060
Get a plane.
00:06:11.060 --> 00:06:14.460
If I take anything on this
line and anything on this line
00:06:14.460 --> 00:06:19.460
and add them up you can see
that I'm not going to fill 3-D.
00:06:19.460 --> 00:06:23.690
But I'm going to fill
a plane and that maybe
00:06:23.690 --> 00:06:25.640
takes a little thinking.
00:06:25.640 --> 00:06:28.450
It just, then it
becomes sort of,
00:06:28.450 --> 00:06:31.230
you see that that's
what it has to be.
00:06:31.230 --> 00:06:34.670
Okay, now I'm going to
have a third vector.
00:06:34.670 --> 00:06:38.680
Okay, my third vector
will be [0, 0, 1].
00:06:38.680 --> 00:06:43.210
Okay, so that [0, 0, 1]
is 0 in the x, 0 in the y
00:06:43.210 --> 00:06:44.660
and 1 in the z direction.
00:06:44.660 --> 00:06:48.740
So there's w.
00:06:48.740 --> 00:06:51.810
Now I want to take
their combinations.
00:06:51.810 --> 00:06:54.530
So let me do that
very specifically.
00:06:54.530 --> 00:06:57.310
How do I take combinations?
00:06:57.310 --> 00:06:58.540
This is important.
00:06:58.540 --> 00:07:01.550
Seems it's very
simple, but important.
00:07:01.550 --> 00:07:07.770
I like to think of taking the
combinations of some vectors,
00:07:07.770 --> 00:07:11.920
I'm always putting vectors
into the columns of a matrix.
00:07:11.920 --> 00:07:15.120
So now I'm going to move
to step two: matrix.
00:07:15.120 --> 00:07:19.090
I'm going to move to step
two and maybe I'll put it--
00:07:19.090 --> 00:07:20.770
well no, I better put it here.
00:07:20.770 --> 00:07:26.410
Okay, step two is the matrix has
those vectors in its columns.
00:07:26.410 --> 00:07:31.790
So in this case,
it's three by three.
00:07:31.790 --> 00:07:38.240
That's my matrix and
I'm going to call it A.
00:07:38.240 --> 00:07:44.870
How do I take
combinations of vectors?
00:07:44.870 --> 00:07:47.870
I should have maybe
done it in detail here,
00:07:47.870 --> 00:07:52.790
but I'll just do it
with a matrix here.
00:07:52.790 --> 00:07:54.850
Watch this now.
00:07:54.850 --> 00:08:05.030
If I multiply A by the vector
of x's, what that does,
00:08:05.030 --> 00:08:08.080
so this is now A times
x, so very important,
00:08:08.080 --> 00:08:10.090
a matrix times a vector.
00:08:10.090 --> 00:08:12.510
What does it do?
00:08:12.510 --> 00:08:15.730
The output is just what I want.
00:08:15.730 --> 00:08:17.840
This is the output.
00:08:17.840 --> 00:08:21.510
It takes x_1 times
the first column
00:08:21.510 --> 00:08:24.460
plus x_2 times the second
plus x_3 times the third.
00:08:24.460 --> 00:08:27.390
That's the way matrix
multiplication works,
00:08:27.390 --> 00:08:28.870
by columns.
00:08:28.870 --> 00:08:30.870
And you don't always see that.
00:08:30.870 --> 00:08:32.140
Because what do you see?
00:08:32.140 --> 00:08:34.870
You probably know how
to multiply that matrix
00:08:34.870 --> 00:08:35.950
by that vector.
00:08:35.950 --> 00:08:38.700
Let me ask you to do it.
00:08:38.700 --> 00:08:40.520
What do you get?
00:08:40.520 --> 00:08:43.770
So everybody does it
a component at a time.
00:08:43.770 --> 00:08:48.730
So what's the first component
of the answer? x_1, yeah.
00:08:48.730 --> 00:08:49.800
How do you get that?
00:08:49.800 --> 00:08:52.850
It's row times the vector.
00:08:52.850 --> 00:08:56.610
And when I say "times", I
really mean that dot product.
00:08:56.610 --> 00:09:00.000
This plus this plus this is x_1.
00:09:00.000 --> 00:09:05.130
And what about the second row?
00:09:05.130 --> 00:09:06.520
-x_1 + x_2.
00:09:06.520 --> 00:09:09.730
Or I'll just say x_2 - x_1.
00:09:09.730 --> 00:09:12.480
And the third guy,
the third component
00:09:12.480 --> 00:09:22.050
would be x_3 - x_2, right?
00:09:22.050 --> 00:09:25.050
So right away I'm
going to say, I'm
00:09:25.050 --> 00:09:28.100
going to call this matrix
A a difference matrix.
00:09:28.100 --> 00:09:31.370
It always helps to
give names to things.
00:09:31.370 --> 00:09:35.880
So this A is a difference matrix
because it takes differences
00:09:35.880 --> 00:09:37.430
of the x's.
00:09:37.430 --> 00:09:40.440
And I would even say a
first difference matrix
00:09:40.440 --> 00:09:43.850
because it's just the
straightforward difference
00:09:43.850 --> 00:09:48.770
and we'll see second
differences in class Friday.
00:09:48.770 --> 00:09:50.120
So that's what A does.
00:09:50.120 --> 00:09:52.520
But you remember
my first point was
00:09:52.520 --> 00:09:55.610
that when a matrix
multiplies a vector,
00:09:55.610 --> 00:10:01.560
the result is a
combination of the columns.
00:10:01.560 --> 00:10:05.250
And that's not always, because
see, I'm looking at the picture
00:10:05.250 --> 00:10:07.040
not just by numbers.
00:10:07.040 --> 00:10:10.880
You know, with numbers
I'm just doing this stuff.
00:10:10.880 --> 00:10:12.750
But now I'm stepping
back a little bit
00:10:12.750 --> 00:10:14.830
and saying I'm combining.
00:10:14.830 --> 00:10:18.120
It's this vector times x_1.
00:10:18.120 --> 00:10:22.030
That vector times x_1 plus
this vector times x_2 plus that
00:10:22.030 --> 00:10:25.790
one times x_3 added
together gives me this.
00:10:25.790 --> 00:10:28.780
Saying nothing complicated here.
00:10:28.780 --> 00:10:36.960
It's just look at
it by vectors, also.
00:10:36.960 --> 00:10:39.080
It's a little
interesting, already.
00:10:39.080 --> 00:10:47.400
Here we multiplied these vectors
by numbers. x_1, x_2, x_3.
00:10:47.400 --> 00:10:49.130
That was our thinking here.
00:10:49.130 --> 00:10:50.770
Now our thinking
here is a little--
00:10:50.770 --> 00:10:52.420
we've switched slightly.
00:10:52.420 --> 00:10:57.870
Now I'm multiplying the
matrix times the numbers in x.
00:10:57.870 --> 00:11:01.220
Just a slight switch, multiply
the matrix times the number.
00:11:01.220 --> 00:11:03.200
And I get some answer, b.
00:11:03.200 --> 00:11:07.880
Which is this, this is b.
00:11:07.880 --> 00:11:11.960
And of course, I can do
a specific example like,
00:11:11.960 --> 00:11:19.850
suppose I take, well, I could
take the squares to be in x.
00:11:19.850 --> 00:11:27.820
So suppose I take A times the
first three squares, [1, 4, 9].
00:11:27.820 --> 00:11:30.330
What answer would I get?
00:11:30.330 --> 00:11:35.130
Just to keep it clear that
we're very specific here.
00:11:35.130 --> 00:11:38.650
So what would be the output be?
00:11:38.650 --> 00:11:42.260
I think of this as the input,
the [1, 4, 9], the x's.
00:11:42.260 --> 00:11:48.140
Now the machine is multiply
by A and here's the output.
00:11:48.140 --> 00:11:49.680
And what would be the output?
00:11:49.680 --> 00:11:51.270
What numbers am I
going to get there?
00:11:51.270 --> 00:11:52.710
Yeah?
00:11:52.710 --> 00:11:57.700
One, three,
something? [1, 3, 5].
00:11:57.700 --> 00:12:01.020
Which is actually a
little neat that you
00:12:01.020 --> 00:12:06.000
find the differences of the
squares are the odd numbers.
00:12:06.000 --> 00:12:11.280
That appealed to me
in school somehow.
00:12:11.280 --> 00:12:13.430
That was already
a bad sign, right?
00:12:13.430 --> 00:12:18.790
This dumb kid notices that you
take differences of squares
00:12:18.790 --> 00:12:21.780
and get odd numbers, whatever.
00:12:21.780 --> 00:12:25.170
So now is a big step.
00:12:25.170 --> 00:12:29.310
This was the forward
direction, right?
00:12:29.310 --> 00:12:31.960
Input, and there's the output.
00:12:31.960 --> 00:12:36.460
But now the real reality--
That's easy and important,
00:12:36.460 --> 00:12:43.590
but the more deep
problem is, what
00:12:43.590 --> 00:12:48.250
if I give you b and ask for x?
00:12:48.250 --> 00:12:52.450
So again, we're switching
the direction here.
00:12:52.450 --> 00:12:55.990
We're solving an equation now,
or three equations and three
00:12:55.990 --> 00:12:58.080
unknowns, Ax=b.
00:12:58.080 --> 00:13:02.960
So if I give you this
b, can you get x?
00:13:02.960 --> 00:13:06.270
How do you solve
three equations?
00:13:06.270 --> 00:13:07.960
We're looking backwards.
00:13:07.960 --> 00:13:14.760
Now that won't be too hard
for this particular matrix
00:13:14.760 --> 00:13:18.350
that I chose; because
it's triangular,
00:13:18.350 --> 00:13:23.180
we'll be able to go backwards.
00:13:23.180 --> 00:13:25.930
So let me do that.
00:13:25.930 --> 00:13:29.780
Let me take b to
be-- It's a vector,
00:13:29.780 --> 00:13:31.630
it's got three components.
00:13:31.630 --> 00:13:36.190
And now I'm going to
go backwards to find x.
00:13:36.190 --> 00:13:39.400
Or we will.
00:13:39.400 --> 00:13:41.950
So do you see the
three equations?
00:13:41.950 --> 00:13:47.890
Here they are: x_1 is b_1, this
is b_2, that difference is b_3.
00:13:47.890 --> 00:13:49.290
Those are my three equations.
00:13:49.290 --> 00:13:53.180
Three unknown x's, three
known right-hand sides.
00:13:53.180 --> 00:13:57.230
Or I think of it as A
times x, as a matrix
00:13:57.230 --> 00:14:00.500
times x giving a vector b.
00:14:00.500 --> 00:14:02.430
What's the answer?
00:14:02.430 --> 00:14:05.660
As I said, we're going
to be able to do this.
00:14:05.660 --> 00:14:09.300
We're going to be able to solve
this system easily because it's
00:14:09.300 --> 00:14:11.810
already triangular.
00:14:11.810 --> 00:14:14.290
And it's actually
lower triangular
00:14:14.290 --> 00:14:17.890
so that means we'll
start from the top.
00:14:17.890 --> 00:14:24.750
So the answers, the
solution will be what?
00:14:24.750 --> 00:14:30.490
Let's make room for it. x_1,
x_2, and x_3 I want to find.
00:14:30.490 --> 00:14:33.860
And what's the answer?
00:14:33.860 --> 00:14:37.450
Can we just go from
top to bottom now?
00:14:37.450 --> 00:14:42.120
What's x_1? b_1, great.
00:14:42.120 --> 00:14:44.790
What's x_2?
00:14:44.790 --> 00:14:47.010
So x_2 - x_1.
00:14:47.010 --> 00:14:48.280
These are my equations.
00:14:48.280 --> 00:14:50.620
So what's x_2 - x_1?
00:14:50.620 --> 00:14:59.720
Well, it's b_2, so what
is x_2? b_1 + b_2, right?
00:14:59.720 --> 00:15:03.030
And what's x_3?
00:15:03.030 --> 00:15:05.870
What do we need there for x_3?
00:15:05.870 --> 00:15:08.550
So I'm looking at
the third equation.
00:15:08.550 --> 00:15:11.090
That'll determine x_3.
00:15:11.090 --> 00:15:13.670
When I see it this
way, I see those ones
00:15:13.670 --> 00:15:16.190
and I see it multiplying x_3.
00:15:16.190 --> 00:15:20.750
And what do I get?
00:15:20.750 --> 00:15:25.010
Yeah, so x_3 minus
this guy is b_3,
00:15:25.010 --> 00:15:28.040
so I have to add in
another b_3, right?
00:15:28.040 --> 00:15:31.570
I'm doing sort of
substitution down as I go.
00:15:31.570 --> 00:15:38.240
Once I learned that x_1 was b_1
I used it there to find x_2.
00:15:38.240 --> 00:15:40.700
And now I'll use
x_2 to find x_3.
00:15:40.700 --> 00:15:48.520
And what do I get again? x_3
is, I'll put the x_2 over there.
00:15:48.520 --> 00:15:50.810
I think you've got
it. b_1 + b_2 + b_3.
00:15:55.410 --> 00:15:59.760
So that's the solution.
00:15:59.760 --> 00:16:02.570
Not difficult because the
matrix was triangular.
00:16:02.570 --> 00:16:06.700
But let's think
about that solution.
00:16:06.700 --> 00:16:13.520
That solution is
a matrix times b.
00:16:13.520 --> 00:16:18.540
When you look at that-- So
this is like a good early step
00:16:18.540 --> 00:16:19.700
in linear algebra.
00:16:19.700 --> 00:16:26.090
When I look at that I see
a matrix multiplying b.
00:16:26.090 --> 00:16:29.000
You take that step up
to seeing a matrix.
00:16:29.000 --> 00:16:30.560
And you can just read it off.
00:16:30.560 --> 00:16:34.930
So let me say, what's the
matrix there that's multiplying
00:16:34.930 --> 00:16:41.550
b to give that answer?
00:16:41.550 --> 00:16:46.050
Remember the columns
of this matrix-- well,
00:16:46.050 --> 00:16:47.960
I don't know how you
want to read it off.
00:16:47.960 --> 00:16:52.810
But one way is, the think
the columns of that matrix
00:16:52.810 --> 00:16:57.250
are multiplying b_1, b_2,
and b_3 to give this.
00:16:57.250 --> 00:17:01.630
So what's the first
column of the matrix?
00:17:01.630 --> 00:17:05.550
It's whatever I'm reading
off, the coefficients, really,
00:17:05.550 --> 00:17:09.150
of b_1 here: [1, 1, 1].
00:17:09.150 --> 00:17:13.101
And what's the second column
of the matrix? [0, 1, 1].
00:17:13.101 --> 00:17:13.600
Good.
00:17:13.600 --> 00:17:16.640
Zero b_2's, one, one.
00:17:16.640 --> 00:17:19.830
And the third is? [0, 0, 1].
00:17:19.830 --> 00:17:23.200
Good.
00:17:23.200 --> 00:17:27.780
Now, so lots of things
to comment here.
00:17:27.780 --> 00:17:30.700
Let me write up again
here, this is x.
00:17:30.700 --> 00:17:34.390
That was the answer.
00:17:34.390 --> 00:17:38.630
It's a matrix times b.
00:17:38.630 --> 00:17:42.580
And what's the name
of that matrix?
00:17:42.580 --> 00:17:44.470
It's the inverse matrix.
00:17:44.470 --> 00:17:48.120
If Ax gives b,
the inverse matrix
00:17:48.120 --> 00:17:50.930
does it the other way
around, x is A inverse b.
00:17:50.930 --> 00:17:53.550
Let me just put that over here.
00:17:53.550 --> 00:18:01.070
If Ax is b, then x
should be A inverse b.
00:18:01.070 --> 00:18:04.680
So we had inverse, I wrote
down inverse this morning
00:18:04.680 --> 00:18:10.150
but without saying the point,
but so you see how that comes?
00:18:10.150 --> 00:18:11.690
I mean, if I want
to go formally,
00:18:11.690 --> 00:18:15.100
I multiply both
sides by A inverse.
00:18:15.100 --> 00:18:18.010
If there is an A inverse.
00:18:18.010 --> 00:18:20.180
That's a critical
thing as we saw.
00:18:20.180 --> 00:18:21.740
Is the matrix invertible?
00:18:21.740 --> 00:18:24.430
The answer here is, yes,
there is an inverse.
00:18:24.430 --> 00:18:26.410
And what does that really mean?
00:18:26.410 --> 00:18:32.140
The inverse is the thing that
takes us from b back to x.
00:18:32.140 --> 00:18:34.820
Think of A as kind of
a-- multiplying by A
00:18:34.820 --> 00:18:39.150
is kind of a mapping,
mathematicians use the word,
00:18:39.150 --> 00:18:40.930
or transform.
00:18:40.930 --> 00:18:42.670
Transform would be good.
00:18:42.670 --> 00:18:45.850
Transform from x to b.
00:18:45.850 --> 00:18:49.100
And this is the
inverse transform.
00:18:49.100 --> 00:18:52.490
So it doesn't happen to
be the discrete Fourier
00:18:52.490 --> 00:18:55.940
transform or a wavelet
transform, it's a-- well,
00:18:55.940 --> 00:18:57.830
actually we could
give it a name.
00:18:57.830 --> 00:19:00.740
This is kind of a
difference transform, right?
00:19:00.740 --> 00:19:03.570
That's what A did,
took differences.
00:19:03.570 --> 00:19:06.780
So what does A inverse do?
00:19:06.780 --> 00:19:09.090
It takes sums.
00:19:09.090 --> 00:19:10.030
It takes sums.
00:19:10.030 --> 00:19:15.550
That's why you see 1, 1
and 1, 1, 1 along the rows
00:19:15.550 --> 00:19:18.160
because it's just
adding, and you see it
00:19:18.160 --> 00:19:20.130
here in fully display.
00:19:20.130 --> 00:19:21.730
It's a sum matrix.
00:19:21.730 --> 00:19:25.170
I might as well
call it S for sum.
00:19:25.170 --> 00:19:28.610
So that matrix, that sum
matrix is the inverse
00:19:28.610 --> 00:19:33.750
of the different matrix.
00:19:33.750 --> 00:19:40.220
And maybe, since I hit
on calculus earlier,
00:19:40.220 --> 00:19:43.590
you could say that
calculus is all about one
00:19:43.590 --> 00:19:45.420
thing and its inverse.
00:19:45.420 --> 00:19:50.170
The derivative is
A, and what's S?
00:19:50.170 --> 00:19:51.990
In calculus.
00:19:51.990 --> 00:19:53.510
The integral.
00:19:53.510 --> 00:19:57.880
The whole subject is
about one operation,
00:19:57.880 --> 00:20:00.330
now admittedly
it's not a matrix,
00:20:00.330 --> 00:20:04.810
it operates on functions
instead of just little vectors,
00:20:04.810 --> 00:20:06.870
but that's the main point.
00:20:06.870 --> 00:20:09.390
The fundamental
theorem of calculus
00:20:09.390 --> 00:20:14.380
is telling us that integration's
the inverse of differentiation.
00:20:14.380 --> 00:20:20.390
So this is good and if
I put in b = [1, 3, 5]
00:20:20.390 --> 00:20:24.660
for example just to
put in some numbers,
00:20:24.660 --> 00:20:31.340
if I put in b = [1, 3, 5], what
would the x that comes out be?
00:20:31.340 --> 00:20:33.170
[1, 4, 9].
00:20:33.170 --> 00:20:33.870
Right?
00:20:33.870 --> 00:20:35.580
Because it takes us back.
00:20:35.580 --> 00:20:39.080
Here, previously we started, we
took differences of [1, 4, 9],
00:20:39.080 --> 00:20:40.690
got [1, 3, 5].
00:20:40.690 --> 00:20:46.300
Now if we take sums of
[1, 3, 5], we get [1, 4, 9].
00:20:46.300 --> 00:20:51.240
Now we have a system
of linear equations.
00:20:51.240 --> 00:20:53.760
Now I want to step
back and see what
00:20:53.760 --> 00:20:56.570
was good about this matrix.
00:20:56.570 --> 00:20:59.520
Somehow it has an inverse.
00:20:59.520 --> 00:21:03.040
Ax=b has a solution,
in other words.
00:21:03.040 --> 00:21:05.510
And it has only one
solution, right?
00:21:05.510 --> 00:21:07.190
Because we worked it out.
00:21:07.190 --> 00:21:08.120
We had no choice.
00:21:08.120 --> 00:21:10.340
That was it.
00:21:10.340 --> 00:21:12.300
So there's just one solution.
00:21:12.300 --> 00:21:14.830
There's always one
and only one solution.
00:21:14.830 --> 00:21:18.430
It's like a perfect
transform from the x's
00:21:18.430 --> 00:21:20.400
to the b's and back again.
00:21:20.400 --> 00:21:24.200
Yeah so that's what an
invertible matrix is.
00:21:24.200 --> 00:21:29.810
It's a perfect map from one
set of x's to the x's and you
00:21:29.810 --> 00:21:33.580
can get back again.
00:21:33.580 --> 00:21:36.120
Questions always.
00:21:36.120 --> 00:21:39.850
Now I think I'm ready
for another example.
00:21:39.850 --> 00:21:41.140
There are only two examples.
00:21:41.140 --> 00:21:47.550
And actually these two examples
are on the 18.06 web page.
00:21:47.550 --> 00:21:51.480
If some people asked
after class how
00:21:51.480 --> 00:21:55.650
to get sort of a review
of linear algebra,
00:21:55.650 --> 00:22:06.220
well the 18.06 website would
be definitely a possibility.
00:22:06.220 --> 00:22:12.070
Well, I'll put down the
OpenCourseWare website, mit.edu
00:22:12.070 --> 00:22:14.780
and then you would look
at the linear algebra
00:22:14.780 --> 00:22:20.640
course or the math one.
00:22:20.640 --> 00:22:26.130
What is it?
web.math.edu, is that it?
00:22:26.130 --> 00:22:34.930
No, maybe that's an
MIT-- so is it math?
00:22:34.930 --> 00:22:39.190
I can't live without
edu at the end, right?
00:22:39.190 --> 00:22:41.700
Is it just edu?
00:22:41.700 --> 00:22:49.690
Whatever!
00:22:49.690 --> 00:22:54.070
So that website has, well, all
the old exams you could ever
00:22:54.070 --> 00:22:55.730
want if you wanted any.
00:22:55.730 --> 00:23:03.630
And it has this
example and you click
00:23:03.630 --> 00:23:06.720
on Starting With Two Matrices.
00:23:06.720 --> 00:23:09.050
And this is one of them.
00:23:09.050 --> 00:23:11.240
Okay, ready for the other.
00:23:11.240 --> 00:23:14.100
So here comes the second
matrix, second example
00:23:14.100 --> 00:23:16.560
that you can contrast.
00:23:16.560 --> 00:23:20.350
Second example is going
to have the same u.
00:23:20.350 --> 00:23:24.996
Let me put-- our matrix,
I'm going to call it,
00:23:24.996 --> 00:23:26.120
what am I going to call it?
00:23:26.120 --> 00:23:33.080
Maybe C. So it'll
have the same u.
00:23:33.080 --> 00:23:39.580
And the same v. But
I'm going to change w.
00:23:39.580 --> 00:23:41.430
And that's going to
make all the difference.
00:23:41.430 --> 00:23:48.630
My w, I'm going to
make that into w.
00:23:48.630 --> 00:23:52.700
So now I have three vectors.
00:23:52.700 --> 00:23:55.260
I can take their combinations.
00:23:55.260 --> 00:23:58.920
I can look at the
equation Cx=b .
00:23:58.920 --> 00:24:00.710
I can try to solve it.
00:24:00.710 --> 00:24:06.020
All the normal stuff
with those combinations
00:24:06.020 --> 00:24:09.200
of those three vectors.
00:24:09.200 --> 00:24:12.770
And we'll see a difference.
00:24:12.770 --> 00:24:15.880
So now, what happens
if I do-- Could
00:24:15.880 --> 00:24:24.160
I even like do just a little
erase to deal with C now?
00:24:24.160 --> 00:24:26.010
How does C differ?
00:24:26.010 --> 00:24:31.620
If I change this
multiplication from A
00:24:31.620 --> 00:24:34.420
to C, to this new matrix.
00:24:34.420 --> 00:24:38.690
Then what we've done is to put
in a minus one there, right?
00:24:38.690 --> 00:24:41.390
That's the only change we made.
00:24:41.390 --> 00:24:49.710
And what's the change in Cx?
00:24:49.710 --> 00:24:51.350
I've changed the
first row, so I'm
00:24:51.350 --> 00:24:56.720
going to change the first row of
the answer to what? x_1 - x_3.
00:25:04.110 --> 00:25:06.270
You could say again,
as I said this morning,
00:25:06.270 --> 00:25:09.050
you've sort of changed the
boundary condition maybe.
00:25:09.050 --> 00:25:14.540
You've made this difference
equation somehow circular.
00:25:14.540 --> 00:25:23.370
That's why I'm
using that letter C.
00:25:23.370 --> 00:25:25.560
Is it different?
00:25:25.560 --> 00:25:26.920
Ah, yes!
00:25:26.920 --> 00:25:29.240
I didn't get it right here.
00:25:29.240 --> 00:25:33.820
Thank you, thank you very much.
00:25:33.820 --> 00:25:35.147
Absolutely.
00:25:35.147 --> 00:25:37.480
I mean that would have been
another matrix that we could
00:25:37.480 --> 00:25:39.021
think about but it
wouldn't have made
00:25:39.021 --> 00:25:42.560
the point I wanted, so thanks,
that's absolutely great.
00:25:42.560 --> 00:25:48.260
So now it's correct
here and this is correct
00:25:48.260 --> 00:25:53.970
and I can look at equations
but can I solve them?
00:25:53.970 --> 00:25:56.320
Can I solve them?
00:25:56.320 --> 00:26:00.170
And you're guessing
already, no we can't do it.
00:26:00.170 --> 00:26:02.380
Right?
00:26:02.380 --> 00:26:07.480
So now let me maybe
go to a board, work
00:26:07.480 --> 00:26:11.440
below, because I'd hate to
erase, that was so great,
00:26:11.440 --> 00:26:14.490
that being able to solve
it in a nice clear solution
00:26:14.490 --> 00:26:17.010
and some matrix coming in.
00:26:17.010 --> 00:26:19.090
But now, how about this one?
00:26:19.090 --> 00:26:24.370
Okay.
00:26:24.370 --> 00:26:27.480
One comment I should
have made here.
00:26:27.480 --> 00:26:30.640
Suppose the b's were zero.
00:26:30.640 --> 00:26:32.610
Suppose I was looking
at, originally
00:26:32.610 --> 00:26:38.290
at A times x equal
all zeroes, What's x?
00:26:38.290 --> 00:26:41.600
If all the b's
were zero in this,
00:26:41.600 --> 00:26:46.220
this was the one that dealt with
the matrix A. If all the b's
00:26:46.220 --> 00:26:50.280
are zero then the x's are zero.
00:26:50.280 --> 00:26:53.930
The only way to get
zero right-hand sides,
00:26:53.930 --> 00:26:58.630
b's, was to have zero x's.
00:26:58.630 --> 00:27:01.670
Right?
00:27:01.670 --> 00:27:06.080
If you wanted to get zero
out, you had to put zero in.
00:27:06.080 --> 00:27:08.360
Well, you can always put
zero in and get zero out,
00:27:08.360 --> 00:27:12.250
but here you can put other
vectors in and get zero out.
00:27:12.250 --> 00:27:17.320
So I want to say there's
a solution with zeroes
00:27:17.320 --> 00:27:22.230
out, coming out of C, but
some non-zeroes going in.
00:27:22.230 --> 00:27:25.630
And of course we know
from this morning
00:27:25.630 --> 00:27:29.900
that that's a signal that it's
a different sort of matrix,
00:27:29.900 --> 00:27:36.510
there won't be an inverse,
we've got questions.
00:27:36.510 --> 00:27:39.260
Tell me all the solutions.
00:27:39.260 --> 00:27:42.780
All the solutions, so actually
not just one, well you could
00:27:42.780 --> 00:27:44.177
tell me one, tell me one first.
00:27:44.177 --> 00:27:45.260
AUDIENCE: [UNINTELLIGIBLE]
00:27:45.260 --> 00:27:46.426
PROFESSOR STRANG: [1, 1, 1].
00:27:46.426 --> 00:27:47.260
Okay.
00:27:47.260 --> 00:27:48.290
Now tell me all.
00:27:48.290 --> 00:27:50.754
AUDIENCE: C, C, C.
00:27:50.754 --> 00:27:51.920
PROFESSOR STRANG: [C, C, C].
00:27:51.920 --> 00:27:52.450
Yeah.
00:27:52.450 --> 00:27:55.230
That whole line
through [1, 1, 1].
00:27:55.230 --> 00:27:57.910
And that would be normal.
00:27:57.910 --> 00:28:01.380
So this is a line of solutions.
00:28:01.380 --> 00:28:01.880
Right.
00:28:01.880 --> 00:28:03.110
A line of a solutions.
00:28:03.110 --> 00:28:08.190
I think of [1, 1, 1] as in some
solution space, and then all
00:28:08.190 --> 00:28:08.710
multiples.
00:28:08.710 --> 00:28:10.550
That whole line.
00:28:10.550 --> 00:28:13.470
Later I would say
it's a subspace.
00:28:13.470 --> 00:28:16.570
When I say what that
word subspace means
00:28:16.570 --> 00:28:20.780
it's just this--
linear algebra's
00:28:20.780 --> 00:28:23.810
done its job beyond
just [1, 1, 1].
00:28:23.810 --> 00:28:25.030
Okay.
00:28:25.030 --> 00:28:33.510
So, again, it's this
fact of-- if we only
00:28:33.510 --> 00:28:38.290
know the differences-- Yeah.
00:28:38.290 --> 00:28:42.040
You can see different ways
that this has got problems.
00:28:42.040 --> 00:28:43.980
So that's C times x.
00:28:43.980 --> 00:28:48.520
Now one way to see
a problem is to say
00:28:48.520 --> 00:28:52.020
we can get the answer of all
zeroes by putting constants.
00:28:52.020 --> 00:28:55.480
All that's saying in
words the differences
00:28:55.480 --> 00:28:58.720
of a constant factor
are all zeroes, right?
00:28:58.720 --> 00:29:00.610
That's all that happened.
00:29:00.610 --> 00:29:06.410
Another way to see a problem if
I had this system of equations,
00:29:06.410 --> 00:29:08.470
how would you see that
there's a problem,
00:29:08.470 --> 00:29:10.960
and how would you see that
there is sometimes an answer
00:29:10.960 --> 00:29:13.270
and even decide when?
00:29:13.270 --> 00:29:17.360
I don't know if you
can take a quick look.
00:29:17.360 --> 00:29:20.990
If I had three equations,
x_1-x_3 is b_1,
00:29:20.990 --> 00:29:23.150
this equals b_2,
this equals b_3.
00:29:27.330 --> 00:29:32.010
Do you see something that I
can do to the left sides that's
00:29:32.010 --> 00:29:36.070
important somehow?
00:29:36.070 --> 00:29:39.510
Suppose I add those
left-hand sides.
00:29:39.510 --> 00:29:41.040
What do I get?
00:29:41.040 --> 00:29:42.510
And I'm allowed
to do that, right?
00:29:42.510 --> 00:29:45.510
If I've got three equations
I'm allowed to add them,
00:29:45.510 --> 00:29:50.970
and I would get zero, if I
add, I get zero equals --
00:29:50.970 --> 00:29:53.370
I have to add the right-sides
of course -- b_1+b_2+b_3.
00:29:57.810 --> 00:30:00.890
I hesitate to say a fourth
equation because it's not
00:30:00.890 --> 00:30:02.990
independent of those
three, but it's
00:30:02.990 --> 00:30:04.930
a consequence of those three.
00:30:04.930 --> 00:30:11.470
So actually this is telling
me when I could get an answer
00:30:11.470 --> 00:30:14.390
and when I couldn't.
00:30:14.390 --> 00:30:16.500
If I get zero on
the left side I have
00:30:16.500 --> 00:30:19.540
to have zero on the
right side or I'm lost.
00:30:19.540 --> 00:30:23.640
So I could actually solve
this when b_1+b_2+b_3=0.
00:30:30.920 --> 00:30:33.530
So I've taken a step there.
00:30:33.530 --> 00:30:37.210
I've said that okay,
we're in trouble often,
00:30:37.210 --> 00:30:42.300
but in case the right-side
adds up to zero then we're not.
00:30:42.300 --> 00:30:47.670
And if you'll allow me to
jump to a mechanical meaning
00:30:47.670 --> 00:30:53.940
of this, if these were
springs or something, masses,
00:30:53.940 --> 00:30:58.210
and these were forces on
them -- so I'm solving
00:30:58.210 --> 00:31:02.150
for displacements of masses
that we'll see very soon,
00:31:02.150 --> 00:31:07.790
and these are forces -- what
that equation is saying is,
00:31:07.790 --> 00:31:12.200
because they're sorta cyclical,
it's somehow saying that
00:31:12.200 --> 00:31:17.230
if the forces add up to zero,
if the resulting force is zero,
00:31:17.230 --> 00:31:19.160
then you're okay.
00:31:19.160 --> 00:31:23.060
The springs and masses don't
like take off, or start
00:31:23.060 --> 00:31:25.750
spinning or whatever.
00:31:25.750 --> 00:31:30.030
So there's a physical
meaning for that condition
00:31:30.030 --> 00:31:35.730
that it's okay provided,
if, the b's add up to zero.
00:31:35.730 --> 00:31:38.850
But of course, if the b's don't
add up to zero we're lost.
00:31:38.850 --> 00:31:40.790
Right yeah.
00:31:40.790 --> 00:31:42.070
Okay.
00:31:42.070 --> 00:31:52.200
So Cx=b could be solved
sometimes, but not always.
00:31:52.200 --> 00:31:55.520
The difficulty with C is
showing up several ways.
00:31:55.520 --> 00:32:00.320
It's showing up in C times
a vector x giving zero.
00:32:00.320 --> 00:32:02.830
That's bad news.
00:32:02.830 --> 00:32:05.430
Because no C inverse
can bring you back.
00:32:05.430 --> 00:32:07.650
I mean it's like you
can't come back from zero.
00:32:07.650 --> 00:32:11.600
Once you get to zero, C inverse
can never bring you back
00:32:11.600 --> 00:32:14.010
to x, right?
00:32:14.010 --> 00:32:21.220
A took x into b up there, and
then A inverse brought back x.
00:32:21.220 --> 00:32:23.150
But here there's no
way to bring back
00:32:23.150 --> 00:32:25.620
that x because I can't
multiply zero by anything
00:32:25.620 --> 00:32:27.380
and get back to x.
00:32:27.380 --> 00:32:30.150
So that's why I see
it's got troubles here.
00:32:30.150 --> 00:32:33.410
Here I see it's got troubles
because if I add the left sides
00:32:33.410 --> 00:32:35.210
I get zero.
00:32:35.210 --> 00:32:37.590
And therefore the right
sides must add to zero.
00:32:37.590 --> 00:32:41.220
So you've got
trouble several ways.
00:32:41.220 --> 00:32:45.290
Ah, let's see another way,
let's see geometrically
00:32:45.290 --> 00:32:46.890
why were in trouble.
00:32:46.890 --> 00:32:53.930
Okay, so let me draw a picture
to go with that picture.
00:32:53.930 --> 00:32:56.880
So there's
three-dimensional space.
00:32:56.880 --> 00:33:00.940
I didn't change u,
I didn't change v,
00:33:00.940 --> 00:33:06.450
but I changed w to minus
one, what does that mean?
00:33:06.450 --> 00:33:09.840
Minus one sort of going
this way maybe, zero,
00:33:09.840 --> 00:33:13.670
one is the z direction,
somehow I changed it to there.
00:33:13.670 --> 00:33:17.630
So this is w star
maybe, a different w.
00:33:17.630 --> 00:33:23.330
This is the w that
gave me problems.
00:33:23.330 --> 00:33:26.160
What's the problem?
00:33:26.160 --> 00:33:37.170
How does the picture
show the problem?
00:33:37.170 --> 00:33:41.030
What's the problem with those
three vectors, those three
00:33:41.030 --> 00:33:45.730
columns of C?
00:33:45.730 --> 00:33:46.230
Yeah?
00:33:46.230 --> 00:33:47.380
AUDIENCE: [UNINTELLIGIBLE]
00:33:47.380 --> 00:33:49.580
PROFESSOR STRANG: There
in the same plane.
00:33:49.580 --> 00:33:53.560
There in the same plane.
w* gave us nothing new.
00:33:53.560 --> 00:33:56.690
We had the combinations
of u and v made a plane,
00:33:56.690 --> 00:34:00.280
and w* happened to
fall in that plane.
00:34:00.280 --> 00:34:06.360
So this is a plane
here somehow, and goes
00:34:06.360 --> 00:34:09.340
through the origin of course.
00:34:09.340 --> 00:34:10.310
What is that plane?
00:34:10.310 --> 00:34:17.830
This is all combinations,
all combinations of u, v,
00:34:17.830 --> 00:34:21.370
and the third guy, w*.
00:34:21.370 --> 00:34:22.040
Right.
00:34:22.040 --> 00:34:23.950
It's a plane, and
I drew a triangle,
00:34:23.950 --> 00:34:28.200
but of course, I should draw
the plane goes out to infinity.
00:34:28.200 --> 00:34:32.060
But the point is there
are lots of b's, lots
00:34:32.060 --> 00:34:36.590
of right-hand sides
not on that plane.
00:34:36.590 --> 00:34:37.190
Okay.
00:34:37.190 --> 00:34:43.830
Now if I drew all combinations
of u, v, w, the original w,
00:34:43.830 --> 00:34:45.180
what have I got?
00:34:45.180 --> 00:34:48.750
So let me bring that
picture back for a moment.
00:34:48.750 --> 00:34:51.020
If I took all
combinations of those
00:34:51.020 --> 00:34:55.560
does w lie in the
plane of u and v?
00:34:55.560 --> 00:34:56.580
No, right?
00:34:56.580 --> 00:34:58.760
I would call it independent.
00:34:58.760 --> 00:35:00.900
These three vectors
are independent.
00:35:00.900 --> 00:35:05.370
These three, u, v, and w*
I would call dependent.
00:35:05.370 --> 00:35:09.670
Because the third guy was a
combination of the first two.
00:35:09.670 --> 00:35:13.310
Okay, so tell me
what do I get now?
00:35:13.310 --> 00:35:16.610
So now you're really
up to 3-D. What
00:35:16.610 --> 00:35:20.580
do you get if you take all
combinations of u, v, and w?
00:35:20.580 --> 00:35:23.640
AUDIENCE: [INAUDIBLE].
00:35:23.640 --> 00:35:25.500
PROFESSOR STRANG: Say it again.
00:35:25.500 --> 00:35:29.130
The whole space.
00:35:29.130 --> 00:35:31.495
If taking all
combinations of u, v, w
00:35:31.495 --> 00:35:33.390
will give you the whole space.
00:35:33.390 --> 00:35:34.550
Why is that?
00:35:34.550 --> 00:35:38.620
Well we just showed--
when it was A we
00:35:38.620 --> 00:35:43.500
showed that we
could get every b.
00:35:43.500 --> 00:35:48.330
We wanted the combination
that gave b and we found it.
00:35:48.330 --> 00:35:53.990
So in the beginning when we
were working with u, v, w,
00:35:53.990 --> 00:36:01.330
we found -- and this
was short hand here --
00:36:01.330 --> 00:36:04.200
this said find a
combination to give b,
00:36:04.200 --> 00:36:06.830
and this says that
combination will work.
00:36:06.830 --> 00:36:09.310
And we wrote out what x was.
00:36:09.310 --> 00:36:13.170
Now what's the
difference-- Okay, here.
00:36:13.170 --> 00:36:21.080
So those were dependent,
sorry, those were independent.
00:36:21.080 --> 00:36:24.580
I would even call those
three vectors a basis
00:36:24.580 --> 00:36:25.860
for three-dimensional space.
00:36:25.860 --> 00:36:28.510
That word "basis" is a big deal.
00:36:28.510 --> 00:36:31.065
So a basis for
five-dimensional space
00:36:31.065 --> 00:36:35.610
is five vectors that
are independent.
00:36:35.610 --> 00:36:37.190
That's one way to say it.
00:36:37.190 --> 00:36:39.690
The second way to say it
would be their combinations
00:36:39.690 --> 00:36:42.390
give the whole
five-dimensional space.
00:36:42.390 --> 00:36:45.590
A third way to say it-- See if
you can finish this sentence.
00:36:45.590 --> 00:36:48.140
This is for the
independent, the good guys.
00:36:48.140 --> 00:36:54.000
If I put those five vectors
into a five by five matrix,
00:36:54.000 --> 00:37:01.530
that matrix will
be... invertible.
00:37:01.530 --> 00:37:04.060
That matrix will be invertible.
00:37:04.060 --> 00:37:06.650
So an invertible matrix
is one with a basis
00:37:06.650 --> 00:37:09.240
sitting in its columns.
00:37:09.240 --> 00:37:12.810
It's a transform that
has an inverse transform.
00:37:12.810 --> 00:37:16.300
This matrix is not invertible,
those three vectors
00:37:16.300 --> 00:37:17.970
are not a basis.
00:37:17.970 --> 00:37:21.980
Their combinations
are only in a plane.
00:37:21.980 --> 00:37:24.390
By the way, a plane
as a subspace.
00:37:24.390 --> 00:37:27.490
A plane would be a
typical subspace.
00:37:27.490 --> 00:37:29.030
It's like fill it out.
00:37:29.030 --> 00:37:31.900
You took all the combinations,
you did your job,
00:37:31.900 --> 00:37:37.460
but in that case the whole space
would count as a subspace too.
00:37:37.460 --> 00:37:39.540
That's the way
you get subspaces,
00:37:39.540 --> 00:37:42.010
by taking all combinations.
00:37:42.010 --> 00:37:46.180
Okay, now I'm even going to
push you one more step and then
00:37:46.180 --> 00:37:50.900
this example is complete.
00:37:50.900 --> 00:37:56.250
Can you tell me what
vectors do you get?
00:37:56.250 --> 00:37:58.190
All combinations of u, v, w.
00:37:58.190 --> 00:37:59.710
Let me try to write something.
00:37:59.710 --> 00:38:08.190
This gives only a plane.
00:38:08.190 --> 00:38:10.470
Because we've got two
independent vectors but not
00:38:10.470 --> 00:38:12.180
the third.
00:38:12.180 --> 00:38:15.370
Okay.
00:38:15.370 --> 00:38:17.570
I don't know if I
should even ask.
00:38:17.570 --> 00:38:20.960
Do we know an equation
for that plane?
00:38:20.960 --> 00:38:25.880
Well I think we do if we
think about it correctly.
00:38:25.880 --> 00:38:33.640
All combinations of u, v, w* is
the same as saying all vectors
00:38:33.640 --> 00:38:41.610
C times x, right?
00:38:41.610 --> 00:38:48.780
Do you agree that those two
are exactly the same thing?
00:38:48.780 --> 00:38:51.940
This is the key,
because we're moving up
00:38:51.940 --> 00:38:56.670
to vectors, combinations,
and now comes subspaces.
00:38:56.670 --> 00:38:59.590
If I take all
combinations of u, v, w*,
00:38:59.590 --> 00:39:02.610
I say that that's the same
as all vectors C times x,
00:39:02.610 --> 00:39:07.520
why's that?
00:39:07.520 --> 00:39:12.670
It's what I said in the very
first sentence at 4 o'clock.
00:39:12.670 --> 00:39:17.210
The combinations of u, v,
w*, how do I produce them?
00:39:17.210 --> 00:39:21.260
I create the matrix
with those columns.
00:39:21.260 --> 00:39:28.150
I multiply them by x's, and
I get all the combinations.
00:39:28.150 --> 00:39:31.510
And this is just C times x.
00:39:31.510 --> 00:39:35.870
So what I've said there
is just another way
00:39:35.870 --> 00:39:38.670
of saying how does matrix
multiplication work.
00:39:38.670 --> 00:39:46.440
Put the guys in its columns
and multiply by a vector.
00:39:46.440 --> 00:39:49.190
So we're getting all
vectors C times x,
00:39:49.190 --> 00:39:55.870
and now I was going to stretch
it that little bit further.
00:39:55.870 --> 00:39:58.120
Can we describe
what vectors we get?
00:39:58.120 --> 00:40:02.220
So that's my question.
00:40:02.220 --> 00:40:09.550
What b's -- so this is b =
[b 1, b 2, b 3] -- do we get?
00:40:09.550 --> 00:40:14.912
We don't get them all.
00:40:14.912 --> 00:40:16.120
Right, we don't get them all.
00:40:16.120 --> 00:40:21.050
That's the trouble with C.
We only get a plane of them.
00:40:21.050 --> 00:40:25.440
And now can you
tell me which b's
00:40:25.440 --> 00:40:33.480
we do get when we look at all
combinations of these three
00:40:33.480 --> 00:40:36.910
dependent vectors.
00:40:36.910 --> 00:40:39.850
Well we've done a lot today.
00:40:39.850 --> 00:40:42.850
Let me just tell you the
answer because it's here.
00:40:42.850 --> 00:40:45.560
The b's have to add to zero.
00:40:45.560 --> 00:40:49.400
That's the equation that
the b's have to satisfy.
00:40:49.400 --> 00:40:54.820
Because when we wrote out Cx
we noticed that the components
00:40:54.820 --> 00:40:58.550
always added to zero.
00:40:58.550 --> 00:41:00.270
Which b's do we get?
00:41:00.270 --> 00:41:05.260
We get the ones where the
components add to zero.
00:41:05.260 --> 00:41:10.170
In other words that's the
equation of the plane,
00:41:10.170 --> 00:41:11.000
you could say.
00:41:11.000 --> 00:41:11.500
Yeah.
00:41:11.500 --> 00:41:13.780
Actually that's a good
way to look at it.
00:41:13.780 --> 00:41:19.720
All these vectors
are on the plane.
00:41:19.720 --> 00:41:23.520
Do the components
of u add to zero?
00:41:23.520 --> 00:41:25.190
Look at u.
00:41:25.190 --> 00:41:26.750
Yes.
00:41:26.750 --> 00:41:30.930
Do the components
of v add to zero?
00:41:30.930 --> 00:41:31.510
Yes.
00:41:31.510 --> 00:41:32.580
Add them up.
00:41:32.580 --> 00:41:37.010
Do the components of w*, now
that you've fixed it correctly,
00:41:37.010 --> 00:41:38.001
do they add to zero?
00:41:38.001 --> 00:41:38.500
Yes.
00:41:38.500 --> 00:41:40.470
So all the combinations
will add to zero.
00:41:40.470 --> 00:41:42.800
That's the plane.
00:41:42.800 --> 00:41:44.370
That's the plane.
00:41:44.370 --> 00:41:47.880
You see there are so many
different ways to see,
00:41:47.880 --> 00:41:50.610
and none of this is
difficult, but it's
00:41:50.610 --> 00:41:55.100
coming fast because we're
seeing the same thing
00:41:55.100 --> 00:41:56.310
in different languages.
00:41:56.310 --> 00:41:59.620
We're seeing it geometrically
in a picture of a plane.
00:41:59.620 --> 00:42:02.190
We're seeing it as a
combination of vectors.
00:42:02.190 --> 00:42:05.310
We're seeing it as a
multiplication by a matrix.
00:42:05.310 --> 00:42:11.280
And we saw it sort
of here by operation,
00:42:11.280 --> 00:42:16.140
operating and simplifying,
and getting the key fact out
00:42:16.140 --> 00:42:21.000
of the equations.
00:42:21.000 --> 00:42:22.880
Well, okay.
00:42:22.880 --> 00:42:28.580
I wanted to give you this
example, the two examples,
00:42:28.580 --> 00:42:31.380
because they bring out
so many of the key ideas.
00:42:31.380 --> 00:42:34.990
The key idea of a subspace.
00:42:34.990 --> 00:42:38.670
Shall I just say a little
about what that word means?
00:42:38.670 --> 00:42:41.430
A subspace.
00:42:41.430 --> 00:42:43.560
What's a subspace?
00:42:43.560 --> 00:42:48.360
Well, what's a vector
space first of all?
00:42:48.360 --> 00:42:51.570
A vector space is
a bunch of vectors.
00:42:51.570 --> 00:42:54.770
And the rule is you
have to be able to take
00:42:54.770 --> 00:42:56.130
their combinations.
00:42:56.130 --> 00:42:57.760
That what linear algebra does.
00:42:57.760 --> 00:42:59.370
Takes combinations.
00:42:59.370 --> 00:43:05.810
So a vector space is one where
you take all combinations.
00:43:05.810 --> 00:43:09.690
So if I only took just this
triangle that would not
00:43:09.690 --> 00:43:14.380
be a subspace because one
combination would be 2u
00:43:14.380 --> 00:43:16.190
and it would be out
of the triangle.
00:43:16.190 --> 00:43:22.210
So a subspace, just
think of it as a plane,
00:43:22.210 --> 00:43:25.370
but then of course it could
be in higher dimensions.
00:43:25.370 --> 00:43:28.010
You know it could be a
7-dimensional subspace
00:43:28.010 --> 00:43:30.820
inside a 15-dimensional space.
00:43:30.820 --> 00:43:37.780
And I don't know if you're good
at visualizing that, I'm not.
00:43:37.780 --> 00:43:38.730
Never mind.
00:43:38.730 --> 00:43:41.730
You you've got seven
vectors, you think okay,
00:43:41.730 --> 00:43:44.800
their combinations give us
seven-dimensional subspace.
00:43:44.800 --> 00:43:47.250
Each factor has 15 components.
00:43:47.250 --> 00:43:48.300
No problem.
00:43:48.300 --> 00:43:50.110
I mean no problem
for MATLAB certainly.
00:43:50.110 --> 00:43:53.080
It's got what, a matrix
with a 105 entries.
00:43:53.080 --> 00:43:55.770
It deals with that instantly.
00:43:55.770 --> 00:44:02.480
Okay, so a subspace is like a
vector space inside a bigger
00:44:02.480 --> 00:44:03.200
one.
00:44:03.200 --> 00:44:06.460
That's why the prefix
"sub-" is there.
00:44:06.460 --> 00:44:07.160
Right?
00:44:07.160 --> 00:44:11.720
And mathematics always counts
the biggest possibility too,
00:44:11.720 --> 00:44:13.640
which would be the whole space.
00:44:13.640 --> 00:44:15.680
And what's the smallest?
00:44:15.680 --> 00:44:19.160
So what's the smallest
subspace of R^3?
00:44:19.160 --> 00:44:21.880
So I have 3-dimensional
space-- you can tell me all
00:44:21.880 --> 00:44:23.880
the subspaces of R^3.
00:44:23.880 --> 00:44:25.980
So there is one, a plane.
00:44:25.980 --> 00:44:28.180
Yeah, tell me all
the subspaces of R^3.
00:44:28.180 --> 00:44:31.450
And then you'll have
that word kind of down.
00:44:31.450 --> 00:44:33.050
AUDIENCE: [UNINTELLIGIBLE]
00:44:33.050 --> 00:44:34.660
PROFESSOR STRANG: A line.
00:44:34.660 --> 00:44:37.420
So planes and lines,
those you could say,
00:44:37.420 --> 00:44:39.490
the real, the proper subspaces.
00:44:39.490 --> 00:44:41.330
The best, the right ones.
00:44:41.330 --> 00:44:45.230
But there are a couple more
possibilities which are?
00:44:45.230 --> 00:44:46.337
AUDIENCE: [UNINTELLIGIBLE]
00:44:46.337 --> 00:44:47.420
PROFESSOR STRANG: A point.
00:44:47.420 --> 00:44:49.140
Which point?
00:44:49.140 --> 00:44:50.010
The origin.
00:44:50.010 --> 00:44:50.980
Only the origin.
00:44:50.980 --> 00:44:56.020
Because if you tried to say that
point was a subspace, no way.
00:44:56.020 --> 00:44:56.910
Why not?
00:44:56.910 --> 00:44:58.520
Because I wouldn't
be able to multiply
00:44:58.520 --> 00:45:03.870
that vector by five and I
would be away from the point.
00:45:03.870 --> 00:45:08.790
But the zero subspace,
the really small subspace
00:45:08.790 --> 00:45:12.670
that just has the zero vector,
it's got one vector in it.
00:45:12.670 --> 00:45:13.330
Not empty.
00:45:13.330 --> 00:45:15.520
It's got that one
point but that's all.
00:45:15.520 --> 00:45:20.090
Okay, so planes,
lines, the origin,
00:45:20.090 --> 00:45:25.030
and then the other possibility
for a subspaces is?
00:45:25.030 --> 00:45:26.000
The whole space.
00:45:26.000 --> 00:45:26.520
Right.
00:45:26.520 --> 00:45:27.080
Right.
00:45:27.080 --> 00:45:30.390
So the dimensions could be
three for the whole space,
00:45:30.390 --> 00:45:35.420
two for a plane, one for
a line, zero for a point.
00:45:35.420 --> 00:45:41.300
It just kicks together.
00:45:41.300 --> 00:45:42.300
How are we for time?
00:45:42.300 --> 00:45:48.120
Maybe I went more than half, but
now is a chance to just ask me,
00:45:48.120 --> 00:45:53.160
if you want to, like
anything about the course.
00:45:53.160 --> 00:45:54.540
Is at all linear algebra?
00:45:54.540 --> 00:45:55.380
No.
00:45:55.380 --> 00:46:02.520
But I think I can't do
anything more helpful to you
00:46:02.520 --> 00:46:06.230
then to for you
to begin to see--
00:46:06.230 --> 00:46:09.960
when you look at a matrix,
begin to see what is it doing.
00:46:09.960 --> 00:46:11.640
What is it about.
00:46:11.640 --> 00:46:14.910
Right, and of course
matrices can be rectangular.
00:46:14.910 --> 00:46:17.970
So I'll give you a
hint about what's
00:46:17.970 --> 00:46:20.780
coming in the course itself.
00:46:20.780 --> 00:46:26.340
We'll have rectangular
matrices A, okay.
00:46:26.340 --> 00:46:28.310
They're not invertible.
00:46:28.310 --> 00:46:30.760
They're taking
seven-dimensional space
00:46:30.760 --> 00:46:33.720
to three-dimensional
space or something.
00:46:33.720 --> 00:46:35.750
You can't invert that.
00:46:35.750 --> 00:46:40.070
What comes up every time-- I
sort of got the idea finally.
00:46:40.070 --> 00:46:45.510
Every time I see a rectangular
matrix, maybe seven by three,
00:46:45.510 --> 00:46:48.890
that would be seven
rows by three columns.
00:46:48.890 --> 00:46:53.170
Then what comes up with
a rectangular matrix
00:46:53.170 --> 00:46:57.820
A is sooner or later A
transpose sticks its nose in
00:46:57.820 --> 00:47:07.970
and multiplies that A. And we
couldn't do it for our A here.
00:47:07.970 --> 00:47:10.730
Actually if I did it for
that original matrix A
00:47:10.730 --> 00:47:14.770
I would get something
you'd recognize.
00:47:14.770 --> 00:47:18.790
What I want to say is
that the course focuses
00:47:18.790 --> 00:47:23.860
on A transpose A. I'll just
say now that that matrix always
00:47:23.860 --> 00:47:27.390
comes out square, because this
would be three times seven
00:47:27.390 --> 00:47:31.380
times seven times three, so
this would be three by three,
00:47:31.380 --> 00:47:34.300
and it always comes
out symmetric.
00:47:34.300 --> 00:47:36.300
That's the nice thing.
00:47:36.300 --> 00:47:37.190
And even more.
00:47:37.190 --> 00:47:39.760
We'll see more.
00:47:39.760 --> 00:47:41.590
That's like a hint.
00:47:41.590 --> 00:47:48.320
Watch for A transpose
A. And watch for it
00:47:48.320 --> 00:47:52.740
in applications of all kinds.
00:47:52.740 --> 00:47:56.740
In networks an A
will be associated
00:47:56.740 --> 00:47:59.270
with Kirchhoff's voltage
law, and A transpose
00:47:59.270 --> 00:48:00.530
with Kirchhoff's current law.
00:48:00.530 --> 00:48:04.720
They just teamed up together.
00:48:04.720 --> 00:48:06.780
We'll see more.
00:48:06.780 --> 00:48:10.230
Alright now let me give you
a chance to ask any question.
00:48:10.230 --> 00:48:14.230
Whatever.
00:48:14.230 --> 00:48:15.370
Homework.
00:48:15.370 --> 00:48:17.460
Did I mention homework?
00:48:17.460 --> 00:48:21.450
You may have said
that's a crazy homework
00:48:21.450 --> 00:48:24.730
to say three problems 1.1.
00:48:24.730 --> 00:48:30.200
I've never done this
before so essentially you
00:48:30.200 --> 00:48:34.380
can get away with
anything this week,
00:48:34.380 --> 00:48:37.780
and indefinitely actually.
00:48:37.780 --> 00:48:42.300
How many are-- Is this the
first day of MIT classes?
00:48:42.300 --> 00:48:43.000
Oh wow.
00:48:43.000 --> 00:48:43.840
Okay.
00:48:43.840 --> 00:48:46.380
Well, welcome to MIT.
00:48:46.380 --> 00:48:49.930
I hope you like it.
00:48:49.930 --> 00:48:56.050
It's not so high
pressure or whatever
00:48:56.050 --> 00:48:57.560
is associated with MIT.
00:48:57.560 --> 00:49:01.450
It's kind of tolerant.
00:49:01.450 --> 00:49:04.160
If my advisees ask for
something I always say yes.
00:49:04.160 --> 00:49:05.610
It's easier that way.
00:49:05.610 --> 00:49:11.030
AUDIENCE: [LAUGHTER].
00:49:11.030 --> 00:49:15.400
PROFESSOR STRANG: And
let me just again--
00:49:15.400 --> 00:49:19.000
and I'll say it
often and in private.
00:49:19.000 --> 00:49:21.170
This is like a grown-up course.
00:49:21.170 --> 00:49:23.430
I'm figuring you're
here to learn,
00:49:23.430 --> 00:49:26.090
so it's not my job to force it.
00:49:26.090 --> 00:49:31.570
My job is to help it, and
hope this is some help.