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PROFESSOR STRANG: This is the
one and only review, you could
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00:00:27 --> 00:00:30
say, of linear algebra.
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I just think linear algebra
is very important.
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00:00:33 --> 00:00:35
You may have got that idea.
13
00:00:35 --> 00:00:40
And my website even has
a little essay called
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Too Much Calculus.
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00:00:43 --> 00:00:49
Because I think it's crazy for
all the U.S. universities do
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00:00:49 --> 00:00:52
this pretty much, you get
semester after semester in
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00:00:52 --> 00:00:56
differential calculus, integral
calculus, ultimately
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00:00:56 --> 00:00:57
differential equations.
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00:00:57 --> 00:01:00
You run out of steam before
the good stuff, before
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00:01:00 --> 00:01:03
you run out of time.
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And anybody who computes, who's
living in the real world
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is using linear algebra.
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You're taking a differential
equation, you're taking your
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model, making it discrete and
computing with matrices.
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The world's digital
now, not analog.
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00:01:27 --> 00:01:33
I hope it's OK to start the
course with linear algebra.
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00:01:33 --> 00:01:39
But many engineering curricula
don't fully recognize that and
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00:01:39 --> 00:01:43
so if you haven't had an
official course, linear
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00:01:43 --> 00:01:46
algebra, stay with 18.085.
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00:01:46 --> 00:01:48
This is a good way to learn it.
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00:01:48 --> 00:01:51
You're sort of learning
what's important.
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00:01:51 --> 00:01:58
So my review would be-- and
then this, future Wednesdays
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will be in our regular room for
homework, review, questions of
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00:02:02 --> 00:02:06
all kinds, and today
questions too.
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00:02:06 --> 00:02:11
Shall I just fire away for the
first half of the time to give
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you a sense of how I see
the subject, or at least
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00:02:16 --> 00:02:17
within that limited time.
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00:02:17 --> 00:02:22
And then questions
are totally welcome.
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00:02:22 --> 00:02:24
Always welcome, actually.
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00:02:24 --> 00:02:24
Right?
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00:02:24 --> 00:02:26
So I'll just start right up.
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00:02:26 --> 00:02:32
So essentially linear algebra
progresses starting with
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vectors to matrices and
then finally to subspaces.
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00:02:42 --> 00:02:45
So that's, like,
the abstraction.
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00:02:45 --> 00:02:49
You could say abstraction,
but it's not difficult,
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00:02:49 --> 00:02:50
that you want to see.
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00:02:50 --> 00:02:53
Until you see the idea of
a subspace, you haven't
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00:02:53 --> 00:02:56
really got linear algebra.
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00:02:56 --> 00:02:58
Okay, so I'll start
at the beginning.
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What do you do with vectors?
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Answer; you take their
linear combinations.
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That's what you can
do with a vector.
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You can multiply it by a number
and you can add or subtract.
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So that's the key operation.
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Suppose I have
vectors u, v and w.
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Let me take three of them.
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So I can take their
combinations.
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So some combination will be,
say some number times u plus
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some number times v plus
some number times w.
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So these numbers are
called scalers.
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So these would be
called scalers.
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And the whole thing is
a linear combination.
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Let me abbreviate those
words, linear combination.
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And you get some answer, say b.
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But let's put it down, make
this whole discussion specific.
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Yeah, I started a
little early, I think.
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I'm going to take three
vectors; u, v and w and
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take their combinations.
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They're carefully chosen.
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My u is going to be .
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And I'll take vectors
in three dimensions.
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So that means their
combinations will be in
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three dimensions, R^3,
three-dimensional space.
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So that'll be u and then v,
let's take zero, I think,
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one and minus one.
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Okay.
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Suppose I stopped there and
took their linear combinations.
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It's very helpful to
see a picture in
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three-dimensional space.
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I mean the great thing about
linear algebra, it moves into
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n-dimensional space,
10-dimensional,
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100-dimensional, where we can't
visualize, but yet, our
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00:04:59 --> 00:05:02
instinct is right if
we just follow.
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00:05:02 --> 00:05:07
So what's your instinct if I
took those two vectors, and
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00:05:07 --> 00:05:10
notice they're not on the same
line, one isn't a multiple of
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00:05:10 --> 00:05:13
the other, they go in
different directions.
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00:05:13 --> 00:05:17
If I took their combinations,
say x_1*u+x_2*v.
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00:05:17 --> 00:05:20
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Oh now, let me push, this
is a serious question.
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If I took all their
combinations.
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00:05:26 --> 00:05:28
So let me try to
draw a little bit.
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Okay.
93
00:05:29 --> 00:05:33
I'm in three-dimensional
space and u goes somewhere,
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00:05:33 --> 00:05:39
maybe there and v goes
somewhere, maybe here.
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00:05:39 --> 00:05:42
Now suppose I take all the
combinations, so I could
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00:05:42 --> 00:05:45
multiply that first guy
by any number, that
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would fill the line.
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00:05:46 --> 00:05:49
I can multiply that
second guy, v.
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00:05:49 --> 00:05:51
So this was u and this was v.
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00:05:51 --> 00:05:54
I can multiply that by
any number x_2, that
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00:05:54 --> 00:05:56
would fill its line.
102
00:05:56 --> 00:05:59
Each of those lines I would
later call a one-dimensional
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00:05:59 --> 00:06:01
subspace, just a line.
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00:06:01 --> 00:06:06
But now, what happens if I take
all combinations of the two?
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00:06:06 --> 00:06:08
What do you think?
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00:06:08 --> 00:06:09
You got a plane.
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00:06:09 --> 00:06:11
Get a plane.
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00:06:11 --> 00:06:14
If I take anything on this line
and anything on this line and
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00:06:14 --> 00:06:19
add them up you can see that
I'm not going to fill 3-D.
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00:06:19 --> 00:06:23
But I'm going to fill a
plane and that maybe
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00:06:23 --> 00:06:25
takes a little thinking.
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00:06:25 --> 00:06:29
It just, then it becomes sort
of, you see that that's
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what it has to be.
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00:06:31 --> 00:06:34
Ok, now I'm going to
have a third vector.
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00:06:34 --> 00:06:38
Ok, my third vector
will be .
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00:06:38 --> 00:06:43
Ok, so that is zero
in the x, zero in the y and
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00:06:43 --> 00:06:44
one in the z direction.
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So there's W.
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00:06:48 --> 00:06:51
Now I want to take
their combinations.
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00:06:51 --> 00:06:54
So let me do that
very specifically.
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00:06:54 --> 00:06:57
How do I take combinations?
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00:06:57 --> 00:06:58
This is important.
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00:06:58 --> 00:07:01
Seems it's very simple,
but important.
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00:07:01 --> 00:07:07
I like to think of taking the
combinations of some vectors,
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00:07:07 --> 00:07:11
I'm always putting vectors
into the columns of a matrix.
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So now I'm going to move
to step two; matrix.
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I'm going to move to step two
and maybe I'll put it-- well
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not, I better put it here.
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Ok, step two is the matrix has
those vectors in its columns.
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00:07:26 --> 00:07:31
So in this case, it's
three by three.
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00:07:31 --> 00:07:38
That's my matrix and I'm
going to call it A.
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00:07:38 --> 00:07:44
How do I take
combinations of vectors?
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00:07:44 --> 00:07:49
I should have maybe done it in
detail here, but I'll just
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00:07:49 --> 00:07:52
do it with a matrix here.
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00:07:52 --> 00:07:54
Watch this now.
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If I multiply A by the vector
of x's, what that does, so this
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00:08:05 --> 00:08:08
is now A times x, so very
important, a matrix
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00:08:08 --> 00:08:10
times a vector.
139
00:08:10 --> 00:08:12
What does it do?
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00:08:12 --> 00:08:15
The output is just what I want.
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00:08:15 --> 00:08:17
This is the output.
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00:08:17 --> 00:08:22
It takes x_1 times the first
column plus x_2 times
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00:08:22 --> 00:08:24
the second plus x_3
times the third.
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00:08:24 --> 00:08:26
That's the way matrix
multiplication
145
00:08:26 --> 00:08:28
works; by columns.
146
00:08:28 --> 00:08:30
And you don't always see that.
147
00:08:30 --> 00:08:32
Because what do you see?
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00:08:32 --> 00:08:34
You probably know how to
multiply that matrix
149
00:08:34 --> 00:08:35
by that vector.
150
00:08:35 --> 00:08:38
Let me ask you to do it.
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What do you get?
152
00:08:40 --> 00:08:43
So everybody does it a
component at a time.
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00:08:43 --> 00:08:48
So what's the first component
of the answer? x_1, yeah.
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00:08:48 --> 00:08:49
How do you get that?
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00:08:49 --> 00:08:52
It's row times the vector.
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00:08:52 --> 00:08:56
And when I say "times", I
really mean that dot product.
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00:08:56 --> 00:08:59
This plus this
plus this is x_1.
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00:09:00 --> 00:09:04
And what about the second row?
159
00:09:05 --> 00:09:05
-x_1+x_2.
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00:09:06 --> 00:09:08
Or I'll just say x_2-x_1.
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00:09:09 --> 00:09:12
And the third guy, the
third component would
162
00:09:12 --> 00:09:22
be x_3-x_2, right?
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00:09:22 --> 00:09:26
So right away I'm going to say,
I'm going to call this matrix
164
00:09:26 --> 00:09:28
A a difference matrix.
165
00:09:28 --> 00:09:31
It always helps to
give names to things.
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00:09:31 --> 00:09:35
So this A is a difference
matrix because it takes
167
00:09:35 --> 00:09:37
differences of the x's.
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00:09:37 --> 00:09:41
And I would even say a first
difference matrix because it's
169
00:09:41 --> 00:09:44
just the straightforward
difference and we'll see second
170
00:09:44 --> 00:09:48
differences in class Friday.
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00:09:48 --> 00:09:50
So that's what A does.
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00:09:50 --> 00:09:54
But you remember my first point
was that when a matrix
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00:09:54 --> 00:09:58
multiplies a vector, the result
is a combination
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00:09:58 --> 00:10:01
of the columns.
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00:10:01 --> 00:10:04
And that's not always, because
see, I'm looking at the
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00:10:04 --> 00:10:07
picture not just by numbers.
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00:10:07 --> 00:10:10
You know, with numbers I'm
just doing this stuff.
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00:10:10 --> 00:10:12
But now I'm stepping
back a little bit and
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00:10:12 --> 00:10:14
saying I'm combining.
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00:10:14 --> 00:10:16
It's this vector times x_1.
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00:10:18 --> 00:10:22
That vector times x_1 plus this
vector times x_2 plus that one
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00:10:22 --> 00:10:25
times x_3 added together
gives me this.
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00:10:25 --> 00:10:28
Saying nothing
complicated here.
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00:10:28 --> 00:10:36
It's just look at it
by vectors, also.
185
00:10:36 --> 00:10:39
It's a little
interesting, already.
186
00:10:39 --> 00:10:45
Here we multiplied these
vectors by numbers.
187
00:10:45 --> 00:10:47
x_1, x_2, x_3.
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00:10:47 --> 00:10:49
That was our thinking here.
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00:10:49 --> 00:10:50
Now our thinking here
is a little, we've
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00:10:50 --> 00:10:52
switched slightly.
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00:10:52 --> 00:10:57
Now I'm multiplying the matrix
times the numbers in x.
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00:10:57 --> 00:11:01
Just a slight switch, multiply
the matrix times the number.
193
00:11:01 --> 00:11:03
And I get some answer, b.
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00:11:03 --> 00:11:07
Which is this, this is b.
195
00:11:07 --> 00:11:12
And of course, I can do a
specific example like, suppose
196
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I take, well, I could take
the squares to be in x.
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00:11:19 --> 00:11:27
So suppose I take A times the
first three squares, .
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What answer would I get?
199
00:11:30 --> 00:11:35
Just to keep it clear that
we're very specific here.
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00:11:35 --> 00:11:38
So what would be the output b?
201
00:11:38 --> 00:11:42
I think of this as the input,
the , the x's.
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00:11:42 --> 00:11:48
Now the machine is a multiply
by A and here's the output.
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00:11:48 --> 00:11:49
And what would be the output?
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00:11:49 --> 00:11:51
What numbers am I
going to get there?
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00:11:51 --> 00:11:52
Yeah?
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00:11:52 --> 00:11:57
One, three, something?
.
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00:11:57 --> 00:12:03
Which is actually a little neat
that you find the differences
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00:12:03 --> 00:12:06
of the squares are
the odd numbers.
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00:12:06 --> 00:12:11
That appealed to me
in school somehow.
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That was already a
bad sign, right?
211
00:12:13 --> 00:12:18
This dumb kid notices that you
take differences of squares and
212
00:12:18 --> 00:12:21
get odd numbers, whatever.
213
00:12:21 --> 00:12:25
So now is a big step.
214
00:12:25 --> 00:12:29
This was the forward
direction, right?
215
00:12:29 --> 00:12:31
Input and there's the output.
216
00:12:31 --> 00:12:38
But now the real reality,
that's easy and important, but
217
00:12:38 --> 00:12:48
the more deep problem is, what
if I give you b and ask for x?
218
00:12:48 --> 00:12:52
So again, we're switching
the direction here.
219
00:12:52 --> 00:12:55
We're solving an equation
now, or three equations
220
00:12:55 --> 00:12:57
and three unknowns, Ax=b.
221
00:12:58 --> 00:13:02
So if I give you this
b, can you get x?
222
00:13:02 --> 00:13:06
How do you solve
three equations?
223
00:13:06 --> 00:13:07
We're looking backwards.
224
00:13:07 --> 00:13:15
Now that won't be too hard for
this particular matrix that I
225
00:13:15 --> 00:13:23
chose because its triangular
will be able to go backwards.
226
00:13:23 --> 00:13:25
So let me do that.
227
00:13:25 --> 00:13:28
Let me take b to be.
228
00:13:28 --> 00:13:31
It's a vector, it's
got three components.
229
00:13:31 --> 00:13:36
And now I'm going to go
backwards to find x.
230
00:13:36 --> 00:13:39
Or we will.
231
00:13:39 --> 00:13:41
So do you see the
three equations?
232
00:13:41 --> 00:13:46
Here they are; x_1 is b_1, this
is b_2, that difference is b_3.
233
00:13:47 --> 00:13:49
Those are my three equations.
234
00:13:49 --> 00:13:53
Three unknown x's, three
known right-hand sides.
235
00:13:53 --> 00:13:58
Or I think of it as A times
x, as a matrix times
236
00:13:58 --> 00:14:00
x giving a vector b.
237
00:14:00 --> 00:14:02
What's the answer?
238
00:14:02 --> 00:14:05
As I said, we're going
to be able to do this.
239
00:14:05 --> 00:14:09
We're going to be able to solve
this system easily because
240
00:14:09 --> 00:14:11
it's already triangular.
241
00:14:11 --> 00:14:16
And it's actually lower
triangular so that means
242
00:14:16 --> 00:14:17
we'll start from the top.
243
00:14:17 --> 00:14:24
So the answers, the
solution will be what?
244
00:14:24 --> 00:14:30
Let's make room for it. x_1,
x_2, and x_3 I want to find.
245
00:14:30 --> 00:14:33
And what's the answer?
246
00:14:33 --> 00:14:37
Can we just go from
top to bottom now?
247
00:14:37 --> 00:14:42
What's x_1? b_1, great.
248
00:14:42 --> 00:14:42
What's x_2?
249
00:14:44 --> 00:14:45
So x_2-x_1.
250
00:14:47 --> 00:14:48
These are my equations.
251
00:14:48 --> 00:14:48
So what's x_2-x_1?
252
00:14:50 --> 00:14:59
Well, is b_2, so what is
x_2? b_1+b_2, right?
253
00:14:59 --> 00:15:02
And what's x_3?
254
00:15:03 --> 00:15:05
What do we need there for x_3?
255
00:15:05 --> 00:15:08
So I'm looking at
the third equation.
256
00:15:08 --> 00:15:11
That'll determine x_3.
257
00:15:11 --> 00:15:14
When I see it this way,
I see those ones and I
258
00:15:14 --> 00:15:16
see it multiplying X 3.
259
00:15:16 --> 00:15:20
And what do I get?
260
00:15:20 --> 00:15:25
Yeah, so x_3 minus this guy
is b_3, so I have to add
261
00:15:25 --> 00:15:28
in another b_3, right?
262
00:15:28 --> 00:15:31
I'm doing sort of
substitution down as I go.
263
00:15:31 --> 00:15:37
Once I learned that x_1 was b_1
I used it there to find x_2.
264
00:15:38 --> 00:15:40
And now I'll use
x_2 to find x_3.
265
00:15:40 --> 00:15:47
And what do I get again?
x_3 is, I'll put
266
00:15:47 --> 00:15:48
the x_2 over there.
267
00:15:48 --> 00:15:50
I think you've got
it. b_1+b_2+b_3.
268
00:15:55 --> 00:15:59
So that's the solution.
269
00:15:59 --> 00:16:02
Not difficult because the
matrix was triangular.
270
00:16:02 --> 00:16:06
But let's think about
that solution.
271
00:16:06 --> 00:16:13
That solution is a
matrix times b.
272
00:16:13 --> 00:16:18
When you look at that, so
this is like a good early
273
00:16:18 --> 00:16:19
step in linear algebra.
274
00:16:19 --> 00:16:26
When I look at that I see
a matrix multiplying b.
275
00:16:26 --> 00:16:29
You take that step up
to seeing a matrix.
276
00:16:29 --> 00:16:30
And you can just read it off.
277
00:16:30 --> 00:16:34
So let me say, what's the
matrix there that's multiplying
278
00:16:34 --> 00:16:41
b to give that answer?
279
00:16:41 --> 00:16:46
Remember the columns of this
matrix-- well, I don't know
280
00:16:46 --> 00:16:47
how you want to read it off.
281
00:16:47 --> 00:16:52
But one way is the think the
columns of that matrix are
282
00:16:52 --> 00:16:57
multiplying b_1, b_2,
and b_3 to give this.
283
00:16:57 --> 00:17:01
So what's the first
column of the matrix?
284
00:17:01 --> 00:17:05
It's whatever I'm reading
off, the coefficients really
285
00:17:05 --> 00:17:09
of b_1 here; .
286
00:17:09 --> 00:17:13
And what's the second column
of the matrix? .
287
00:17:13 --> 00:17:13
Good.
288
00:17:13 --> 00:17:16
Zero b_2's, one, one.
289
00:17:16 --> 00:17:19
And the third is? .
290
00:17:19 --> 00:17:23
Good.
291
00:17:23 --> 00:17:27
Now, so lots of things
to comment here.
292
00:17:27 --> 00:17:30
Let me write up again
here, this is x.
293
00:17:30 --> 00:17:34
That was the answer.
294
00:17:34 --> 00:17:38
It's a matrix times b.
295
00:17:38 --> 00:17:42
And what's the name
of that matrix?
296
00:17:42 --> 00:17:44
It's the inverse matrix.
297
00:17:44 --> 00:17:49
If Ax gives b, the inverse
matrix does it the other way
298
00:17:49 --> 00:17:50
around, x is A inverse b.
299
00:17:50 --> 00:17:53
Let me just put that over here.
300
00:17:53 --> 00:18:01
If Ax is b, then x
should be A inverse b.
301
00:18:01 --> 00:18:04
So we had inverse, I wrote down
inverse this morning but
302
00:18:04 --> 00:18:10
without saying the point, but
so you see how that comes?
303
00:18:10 --> 00:18:12
I mean, if I want to go
formally, I multiply both
304
00:18:12 --> 00:18:15
sides by A inverse.
305
00:18:15 --> 00:18:18
If there is an A inverse.
306
00:18:18 --> 00:18:20
That's a critical
thing as we saw.
307
00:18:20 --> 00:18:21
Is the matrix invertible?
308
00:18:21 --> 00:18:24
The answer here is, yes,
there is an inverse.
309
00:18:24 --> 00:18:26
And what does that really mean?
310
00:18:26 --> 00:18:32
The inverse is the thing that
takes us from b back to x.
311
00:18:32 --> 00:18:36
Think of A as kind of a,
multiplying by A is kind of a
312
00:18:36 --> 00:18:40
mapping, mathematicians use
the word, or transform.
313
00:18:40 --> 00:18:42
Transform would be good.
314
00:18:42 --> 00:18:45
Transform from x to b.
315
00:18:45 --> 00:18:49
And this is the
inverse transform.
316
00:18:49 --> 00:18:53
So it doesn't happen to be the
discrete Fourier transform or a
317
00:18:53 --> 00:18:56
wavelet transform, it's
a-- well, actually we
318
00:18:56 --> 00:18:57
could give it a name.
319
00:18:57 --> 00:19:00
This is kind of a difference
transform, right?
320
00:19:00 --> 00:19:03
That's what A did,
took differences.
321
00:19:03 --> 00:19:06
So what does A inverse do?
322
00:19:06 --> 00:19:09
It takes sums.
323
00:19:09 --> 00:19:10
It takes sums.
324
00:19:10 --> 00:19:15
That's why you see one, one and
one, one, one along the rows
325
00:19:15 --> 00:19:18
because it's just adding, and
you see it here in
326
00:19:18 --> 00:19:20
fully display.
327
00:19:20 --> 00:19:21
It's a sum matrix.
328
00:19:21 --> 00:19:25
I might as well
call it S for sum.
329
00:19:25 --> 00:19:28
So that matrix, that sum
matrix is the inverse of
330
00:19:28 --> 00:19:33
the different matrix.
331
00:19:33 --> 00:19:40
And maybe, since I hit on
calculus earlier, you could say
332
00:19:40 --> 00:19:45
that calculus is all about
one thing and it's inverse.
333
00:19:45 --> 00:19:50
The derivative is
A, and what's S?
334
00:19:50 --> 00:19:51
In calculus.
335
00:19:51 --> 00:19:53
The integral.
336
00:19:53 --> 00:19:58
The whole subject is about one
operation, now admittedly it's
337
00:19:58 --> 00:20:03
not a matrix, it operates on
functions instead of just
338
00:20:03 --> 00:20:06
little vectors, but
that's the main point.
339
00:20:06 --> 00:20:10
The fundamental theorem of
calculus is telling us that
340
00:20:10 --> 00:20:14
integration's the inverse
of differentiation.
341
00:20:14 --> 00:20:20
So this is good and if I put in
B equal one, three, five for
342
00:20:20 --> 00:20:25
example just to put in some
numbers, if I put in b equal
343
00:20:25 --> 00:20:33
, what would the x
that comes out be? .
344
00:20:33 --> 00:20:33
Right?
345
00:20:33 --> 00:20:35
Because it takes us back.
346
00:20:35 --> 00:20:38
Here, previously we started,
we took differences of
347
00:20:38 --> 00:20:40
, got .
348
00:20:40 --> 00:20:46
Now if we take sums of <1,
3, 5>, we get .
349
00:20:46 --> 00:20:51
Now we have a system
of linear equations.
350
00:20:51 --> 00:20:55
Now I want to step back
and see what was good
351
00:20:55 --> 00:20:56
about this matrix.
352
00:20:56 --> 00:20:59
Somehow it has an inverse.
353
00:20:59 --> 00:21:03
Ax=b has a solution,
in other words.
354
00:21:03 --> 00:21:05
And it has only one
solution, right?
355
00:21:05 --> 00:21:07
Because we worked it out.
356
00:21:07 --> 00:21:08
We had no choice.
357
00:21:08 --> 00:21:10
That was it.
358
00:21:10 --> 00:21:12
So there's just one solution.
359
00:21:12 --> 00:21:14
There's always one and
only one solution.
360
00:21:14 --> 00:21:18
It's like a perfect transform
from the x's to the
361
00:21:18 --> 00:21:20
b's and back again.
362
00:21:20 --> 00:21:24
Yeah so that's what an
invertible matrix is.
363
00:21:24 --> 00:21:29
It's a perfect map from one
set of x's to the x's and
364
00:21:29 --> 00:21:33
you can get back again.
365
00:21:33 --> 00:21:36
Questions always.
366
00:21:36 --> 00:21:39
Now I think I'm ready
for another example.
367
00:21:39 --> 00:21:41
There are only two examples.
368
00:21:41 --> 00:21:47
And actually these two examples
are on the 18.06 web page.
369
00:21:47 --> 00:21:53
If some people asked after
class how to get sort of a
370
00:21:53 --> 00:22:02
review of linear algebra, well
the 18.06 website would be
371
00:22:02 --> 00:22:06
a definitely a possibility.
372
00:22:06 --> 00:22:12
Well, I'll put down the open
courseware website; mit.edu and
373
00:22:12 --> 00:22:15
then you would look at the
linear algebra course
374
00:22:15 --> 00:22:20
or the math one.
375
00:22:20 --> 00:22:26
What is it? web.math.edu,
is that it?
376
00:22:26 --> 00:22:34
No, maybe that's an
MIT-- so is it math?
377
00:22:34 --> 00:22:39
I can't live without
edu at the end, right?
378
00:22:39 --> 00:22:41
Is it just edu?
379
00:22:41 --> 00:22:49
Whatever!
380
00:22:49 --> 00:22:54
So that website has, well, all
the old exams you could ever
381
00:22:54 --> 00:22:55
want if you wanted any.
382
00:22:55 --> 00:23:04
And it has this example
and you click on Starting
383
00:23:04 --> 00:23:06
With Two Matrices.
384
00:23:06 --> 00:23:09
And this is one of them.
385
00:23:09 --> 00:23:11
Ok, ready for the other.
386
00:23:11 --> 00:23:14
So here comes the second
matrix, second example
387
00:23:14 --> 00:23:16
that you can contrast.
388
00:23:16 --> 00:23:20
Second example is going
to have the same u.
389
00:23:20 --> 00:23:25
Let me put, our matrix I'm
going to call it, what
390
00:23:25 --> 00:23:26
am I going to call it?
391
00:23:26 --> 00:23:30
Maybe C.
392
00:23:30 --> 00:23:33
So it'll have the same u.
393
00:23:33 --> 00:23:36
And the same v.
394
00:23:36 --> 00:23:39
But I'm going to change w.
395
00:23:39 --> 00:23:41
And that's going to make
all the difference.
396
00:23:41 --> 00:23:48
My w, I'm going to
make that into w.
397
00:23:48 --> 00:23:52
So now I have three vectors.
398
00:23:52 --> 00:23:55
I can take their combinations.
399
00:23:55 --> 00:23:57
I can look at the
equation Cx=b
400
00:23:57 --> 00:23:57
.
401
00:23:58 --> 00:24:00
I can try to solve it.
402
00:24:00 --> 00:24:06
All the normal stuff with
those combinations of
403
00:24:06 --> 00:24:09
those three vectors.
404
00:24:09 --> 00:24:12
And we'll see a difference.
405
00:24:12 --> 00:24:17
So now, what happens if I do,
could I even like do just
406
00:24:17 --> 00:24:24
a little erase to
deal with C now?
407
00:24:24 --> 00:24:31
How does C differ if I change
this multiplication from A
408
00:24:31 --> 00:24:34
to C to this new matrix.
409
00:24:34 --> 00:24:38
Then what we've done is to put
in a minus one there, right?
410
00:24:38 --> 00:24:41
That's the only change we made.
411
00:24:41 --> 00:24:49
And what's the change in Cx?
412
00:24:49 --> 00:24:52
I've changed the first row, so
I'm going to change the first
413
00:24:52 --> 00:24:56
row of the answer
to what? x_1-x_3.
414
00:24:56 --> 00:25:04
415
00:25:04 --> 00:25:06
You could say again, as I said
this morning, you've sort of
416
00:25:06 --> 00:25:09
changed the boundary
condition maybe.
417
00:25:09 --> 00:25:14
You've made this difference
equation somehow circular.
418
00:25:14 --> 00:25:23
That's why I'm using
that letter C.
419
00:25:23 --> 00:25:25
Is it different?
420
00:25:25 --> 00:25:26
Ah, yes!
421
00:25:26 --> 00:25:29
I didn't get it right here.
422
00:25:29 --> 00:25:33
Thank you, thank you very much.
423
00:25:33 --> 00:25:35
Absolutely.
424
00:25:35 --> 00:25:37
I mean that would have been
another matrix that we could
425
00:25:37 --> 00:25:39
think about but it wouldn't
have made the point I wanted,
426
00:25:39 --> 00:25:42
so thanks, that's
absolutely great.
427
00:25:42 --> 00:25:48
So now it's correct here and
this is correct and I can
428
00:25:48 --> 00:25:53
look at equations but
can I solve them?
429
00:25:53 --> 00:25:56
Can I solve them?
430
00:25:56 --> 00:26:00
And you're guessing already,
no we can't do it.
431
00:26:00 --> 00:26:02
Right?
432
00:26:02 --> 00:26:08
So now let me maybe go to a
board, work below, because I'd
433
00:26:08 --> 00:26:12
hate to erase, that was so
great, that being able to solve
434
00:26:12 --> 00:26:17
it in a nice clear solution
and some matrix coming in.
435
00:26:17 --> 00:26:19
But now, how about this one?
436
00:26:19 --> 00:26:24
OK.
437
00:26:24 --> 00:26:27
One comment I should
have made here.
438
00:26:27 --> 00:26:30
Suppose the b's were zero.
439
00:26:30 --> 00:26:33
Suppose I was looking at
originally at A times x
440
00:26:33 --> 00:26:38
equal all zeroes, What's x?
441
00:26:38 --> 00:26:43
If all the b's were zero in
this, this was the one that
442
00:26:43 --> 00:26:45
dealt with the matrix A.
443
00:26:45 --> 00:26:50
If all the b's are zero
then the x's are zero.
444
00:26:50 --> 00:26:55
The only way to get zero
right-hand sides, b's,
445
00:26:55 --> 00:26:58
was to have zero x's.
446
00:26:58 --> 00:27:01
Right?
447
00:27:01 --> 00:27:06
If you wanted to get zero
out, you had to put zero in.
448
00:27:06 --> 00:27:09
Well, you can always put zero
in and get zero out, but here
449
00:27:09 --> 00:27:12
you can put other vectors
in and get zero out.
450
00:27:12 --> 00:27:17
So I want to say there's a
solution with zeroes out,
451
00:27:17 --> 00:27:22
coming out of C, but some
non zeroes going in.
452
00:27:22 --> 00:27:27
And of course we know from this
morning that that's a signal
453
00:27:27 --> 00:27:30
that it's a different sort of
matrix, there won't be an
454
00:27:30 --> 00:27:36
inverse, we've got questions.
455
00:27:36 --> 00:27:39
Tell me all the solutions.
456
00:27:39 --> 00:27:42
All the solutions, so actually
not just one, well you could
457
00:27:42 --> 00:27:44
tell me one, tell me one first.
458
00:27:44 --> 00:27:45
AUDIENCE: [UNINELLIGIBLE]
459
00:27:45 --> 00:27:46
PROFESSOR STRANG: .
460
00:27:46 --> 00:27:47
OK.
461
00:27:47 --> 00:27:48
Now tell me all.
462
00:27:48 --> 00:27:48
AUDIENCE:
463
00:27:48 --> 00:27:50
C, C, C.
464
00:27:50 --> 00:27:51
PROFESSOR STRANG: C, C>.
465
00:27:51 --> 00:27:52
Yeah.
466
00:27:52 --> 00:27:55
That whole line
through .
467
00:27:55 --> 00:27:57
And that would be normal.
468
00:27:57 --> 00:28:01
So this is a line of solutions.
469
00:28:01 --> 00:28:01
Right.
470
00:28:01 --> 00:28:03
A line of a solutions.
471
00:28:03 --> 00:28:06
I think of as
in some solution space,
472
00:28:06 --> 00:28:08
and then all multiples.
473
00:28:08 --> 00:28:10
That whole line.
474
00:28:10 --> 00:28:13
Later I would say
it's a subspace.
475
00:28:13 --> 00:28:18
When I say what that word
subspace means it's just this--
476
00:28:18 --> 00:28:23
linear algebra's done its
job beyond just .
477
00:28:23 --> 00:28:25
OK.
478
00:28:25 --> 00:28:33
So, again, it's this fact
of-- if we only know
479
00:28:33 --> 00:28:38
the differences-- Yeah.
480
00:28:38 --> 00:28:42
You can see different ways
that this has got problems.
481
00:28:42 --> 00:28:43
So that's C times x.
482
00:28:43 --> 00:28:49
Now one way to see a problem
is to say we can get the
483
00:28:49 --> 00:28:52
answer of all zeroes
by putting constants.
484
00:28:52 --> 00:28:56
All that's saying in words the
differences of a constant
485
00:28:56 --> 00:28:58
factor are all zeroes, right?
486
00:28:58 --> 00:29:00
That's all that happened.
487
00:29:00 --> 00:29:06
Another way to see a problem if
I had this system of equations,
488
00:29:06 --> 00:29:09
how would you see that there's
a problem, and how would you
489
00:29:09 --> 00:29:13
see that there is sometimes an
answer and even decide when?
490
00:29:13 --> 00:29:17
I don't know if you can
take a quick look.
491
00:29:17 --> 00:29:21
If I had three equations,
x_1-x_3 is b_1, this equals
492
00:29:21 --> 00:29:23
b_2, this equals b_3.
493
00:29:23 --> 00:29:27
494
00:29:27 --> 00:29:31
Do you see something that I
can do to the left sides
495
00:29:31 --> 00:29:36
that's important somehow?
496
00:29:36 --> 00:29:39
Suppose I add those
left-hand sides.
497
00:29:39 --> 00:29:41
What do I get?
498
00:29:41 --> 00:29:42
And I'm allowed to
do that, right?
499
00:29:42 --> 00:29:46
If I've got three equations I'm
allowed to add them, and I
500
00:29:46 --> 00:29:51
would get zero, if I add, I get
zero equals-- I have to
501
00:29:51 --> 00:29:53
add the right-sides of
course-- b_1+b_2+b_3.
502
00:29:57 --> 00:30:00
I hesitate to say a fourth
equation because it's not
503
00:30:00 --> 00:30:03
independent of those three,
but it's a consequence
504
00:30:03 --> 00:30:04
of those three.
505
00:30:04 --> 00:30:11
So actually this is telling me
when I could get an answer
506
00:30:11 --> 00:30:14
and when I couldn't.
507
00:30:14 --> 00:30:17
If I get zero on the left side
I have to have zero on the
508
00:30:17 --> 00:30:19
right side or I'm lost.
509
00:30:19 --> 00:30:23
So I could actually solve
this when b_1+b_2+b_3=0.
510
00:30:30 --> 00:30:33
So I've taken a step there.
511
00:30:33 --> 00:30:38
I've said that okay, we're in
trouble often, but in case
512
00:30:38 --> 00:30:42
the right-side adds up
to zero them or not.
513
00:30:42 --> 00:30:47
And if you'll allow me to jump
to a mechanical meaning of
514
00:30:47 --> 00:30:54
this, if these were springs or
something, masses, and these
515
00:30:54 --> 00:30:59
were forces on them-- so I'm
solving for displacements of
516
00:30:59 --> 00:31:05
masses that we'll see very
soon, and these are forces--
517
00:31:05 --> 00:31:09
what that equation is saying
is-- because they're sorta
518
00:31:09 --> 00:31:13
cyclical-- it's somehow saying
that if the forces add up to
519
00:31:13 --> 00:31:19
zero, if the resulting force
is zero, then you're OK.
520
00:31:19 --> 00:31:23
The springs and masses don't
like take off, or start
521
00:31:23 --> 00:31:25
spinning or whatever.
522
00:31:25 --> 00:31:30
So there's a physical meaning
for that condition that
523
00:31:30 --> 00:31:35
it's OK provided if the
b's add up to zero.
524
00:31:35 --> 00:31:38
But of course, if the b's don't
add up to zero we're lost.
525
00:31:38 --> 00:31:40
Right yeah.
526
00:31:40 --> 00:31:42
OK.
527
00:31:42 --> 00:31:52
So Cx=b could be solved
sometimes, but not always.
528
00:31:52 --> 00:31:55
The difficulty with C is
showing up several ways.
529
00:31:55 --> 00:32:00
It's showing up in a C times
a vector x giving zero.
530
00:32:00 --> 00:32:02
That's bad news.
531
00:32:02 --> 00:32:05
Because no C inverse
can bring you back.
532
00:32:05 --> 00:32:07
I mean it's like you can't
come back from zero.
533
00:32:07 --> 00:32:11
Once you get to zero, C
inverse can never bring
534
00:32:11 --> 00:32:14
you back to x, right?
535
00:32:14 --> 00:32:21
A took x into b up there, and
then A inverse brought back x.
536
00:32:21 --> 00:32:23
But here there's no way to
bring back that x because
537
00:32:23 --> 00:32:27
I can't multiply zero by
anything and get back to x.
538
00:32:27 --> 00:32:30
So that's why I see it's
got troubles here.
539
00:32:30 --> 00:32:32
Here I see it's got troubles
because if I add the
540
00:32:32 --> 00:32:35
left-sides I get zero.
541
00:32:35 --> 00:32:37
And therefore the right-sides
must add to zero.
542
00:32:37 --> 00:32:41
So you've got trouble
several ways.
543
00:32:41 --> 00:32:45
Ah, let's see another way,
let's see geometrically
544
00:32:45 --> 00:32:46
why were in trouble.
545
00:32:46 --> 00:32:53
OK, so let me draw a picture
to go with that picture.
546
00:32:53 --> 00:32:56
So there's
three-dimensional space.
547
00:32:56 --> 00:33:01
I didn't change u, I didn't
change v, but I changed
548
00:33:01 --> 00:33:05
w to minus one.
549
00:33:05 --> 00:33:05
What does that mean?
550
00:33:05 --> 00:33:10
Minus one sort of going this
way maybe, zero, one is the z
551
00:33:10 --> 00:33:13
direction, somehow I
change it to there.
552
00:33:13 --> 00:33:17
So this is w* star
maybe, a different w.
553
00:33:17 --> 00:33:23
This is the w that
gave me problems.
554
00:33:23 --> 00:33:26
What's the problem?
555
00:33:26 --> 00:33:37
How does the picture
show the problem?
556
00:33:37 --> 00:33:40
What's the problem with
those three vectors,
557
00:33:40 --> 00:33:45
those three columns of C?
558
00:33:45 --> 00:33:46
Yeah?
559
00:33:46 --> 00:33:47
AUDIENCE: [UNINTELLIGIBLE]
560
00:33:47 --> 00:33:49
PROFESSOR STRANG: There
in the same plane.
561
00:33:49 --> 00:33:53
There in the same plane.
w* gave us nothing new.
562
00:33:53 --> 00:33:58
We had a combinations of u and
v made a plane, and w* happened
563
00:33:58 --> 00:34:00
to fall in that plane.
564
00:34:00 --> 00:34:06
So this is a plane here
somehow, and goes through
565
00:34:06 --> 00:34:09
the origin of course.
566
00:34:09 --> 00:34:10
What is that plane?
567
00:34:10 --> 00:34:17
This is all combinations,
all combinations of u, v,
568
00:34:17 --> 00:34:19
and the third guy, w*.
569
00:34:21 --> 00:34:22
Right.
570
00:34:22 --> 00:34:25
It's a plane, and I drew a
triangle, but of course, I
571
00:34:25 --> 00:34:28
should draw the plane
goes out to infinity.
572
00:34:28 --> 00:34:32
But the point is there are lots
of b's, lots of right-hand
573
00:34:32 --> 00:34:36
sides not on that plane.
574
00:34:36 --> 00:34:37
OK.
575
00:34:37 --> 00:34:42
Now if I drew all combinations
of u, v, w, the original
576
00:34:42 --> 00:34:45
w, what have I got?
577
00:34:45 --> 00:34:48
So let me bring that
picture back for a moment.
578
00:34:48 --> 00:34:52
If I took all combinations
of those does w lie in
579
00:34:52 --> 00:34:55
the plane of u and v?
580
00:34:55 --> 00:34:56
No, right?
581
00:34:56 --> 00:34:58
I would call it independent.
582
00:34:58 --> 00:35:00
These three vectors
are independent.
583
00:35:00 --> 00:35:05
These three, u, v, and w*
I would call dependent.
584
00:35:05 --> 00:35:09
Because the third guy was a
combination of the first two.
585
00:35:09 --> 00:35:13
OK, so tell me what
do I get now?
586
00:35:13 --> 00:35:16
So now you're really up to 3-D.
587
00:35:16 --> 00:35:20
What do you get if you take all
combinations of u, v, and w?
588
00:35:20 --> 00:35:23
AUDIENCE: [INAUDIBLE].
589
00:35:23 --> 00:35:25
PROFESSOR STRANG: Say it again.
590
00:35:25 --> 00:35:29
The whole space.
591
00:35:29 --> 00:35:31
If taking all combinations
of u, v, w will give
592
00:35:31 --> 00:35:33
you the whole space.
593
00:35:33 --> 00:35:34
Why is that?
594
00:35:34 --> 00:35:40
Well we just showed-- when
it was A we showed that
595
00:35:40 --> 00:35:43
we could get every b.
596
00:35:43 --> 00:35:48
We wanted the combination
that gave b and we found it.
597
00:35:48 --> 00:35:54
So in the beginning when we
were working with u, v, w, we
598
00:35:54 --> 00:36:02
found-- and this was short hand
here-- this said find a
599
00:36:02 --> 00:36:05
combination to give b, and this
says that combination
600
00:36:05 --> 00:36:06
will work.
601
00:36:06 --> 00:36:09
And we wrote out what x was.
602
00:36:09 --> 00:36:13
Now what's the
difference-- OK-- here.
603
00:36:13 --> 00:36:21
So those were dependent, sorry,
those were independent.
604
00:36:21 --> 00:36:24
I would even called those
three vectors a basis for
605
00:36:24 --> 00:36:25
three-dimensional space.
606
00:36:25 --> 00:36:28
That word basis is a big deal.
607
00:36:28 --> 00:36:33
So a basis for five-dimensional
space is five vectors
608
00:36:33 --> 00:36:35
that are independent.
609
00:36:35 --> 00:36:37
That's one way to say it.
610
00:36:37 --> 00:36:39
The second way to say it would
be there combinations give the
611
00:36:39 --> 00:36:42
whole 5-dimensional space.
612
00:36:42 --> 00:36:45
A third way to say it-- see if
you can finish this sentence--
613
00:36:45 --> 00:36:49
this is for the independent,
the good guys-- if I put those
614
00:36:49 --> 00:36:55
five vectors into a five by
five matrix, that matrix
615
00:36:55 --> 00:37:01
will be-- invertible.
616
00:37:01 --> 00:37:04
That matrix will be invertible.
617
00:37:04 --> 00:37:06
So an invertible matrix is
one with a basis sitting
618
00:37:06 --> 00:37:09
in it's columns.
619
00:37:09 --> 00:37:12
It's a transform that has
an inverse transform.
620
00:37:12 --> 00:37:16
This matrix is not invertible,
those three vectors
621
00:37:16 --> 00:37:17
are not a basis.
622
00:37:17 --> 00:37:21
Their combinations
are only in a plane.
623
00:37:21 --> 00:37:24
By the way, a plane
as a subspace.
624
00:37:24 --> 00:37:27
A plane would be a
typical subspace.
625
00:37:27 --> 00:37:29
It's like fill it out.
626
00:37:29 --> 00:37:33
You took all the combinations,
you did your job, but in that
627
00:37:33 --> 00:37:37
case the whole space would
count as a subspace too.
628
00:37:37 --> 00:37:40
That's the way you get
subspaces, by taking
629
00:37:40 --> 00:37:42
all combinations.
630
00:37:42 --> 00:37:46
OK, now I'm even going to push
you one more step and then
631
00:37:46 --> 00:37:50
this example is complete.
632
00:37:50 --> 00:37:56
Can you tell me what
vectors do you get?
633
00:37:56 --> 00:37:58
All combinations of u, v, w.
634
00:37:58 --> 00:37:59
Let me try to write something.
635
00:37:59 --> 00:38:08
This gives only a plane.
636
00:38:08 --> 00:38:10
Because we've got two
independent vectors
637
00:38:10 --> 00:38:12
but not the third.
638
00:38:12 --> 00:38:15
OK.
639
00:38:15 --> 00:38:17
I don't know if I
should even ask.
640
00:38:17 --> 00:38:20
Do we know an equation
for that plane?
641
00:38:20 --> 00:38:25
Well I think we do if we
think about it correctly.
642
00:38:25 --> 00:38:30
All combinations of u, v, w*
is the same as saying all
643
00:38:30 --> 00:38:41
vectors C times x, right?
644
00:38:41 --> 00:38:48
Do you agree that those two
are exactly the same thing?
645
00:38:48 --> 00:38:52
This is the key, because
we're moving up to
646
00:38:52 --> 00:38:56
vectors, combinations,
and now comes subspaces.
647
00:38:56 --> 00:39:00
If I take all combinations of
u, v, w*, I say that that's
648
00:39:00 --> 00:39:07
the same as all vectors
C times x, why's that?
649
00:39:07 --> 00:39:12
It's what I said in the very
first sentence at 4 o'clock.
650
00:39:12 --> 00:39:17
The combinations of u, v,
w*, how do I produce them?
651
00:39:17 --> 00:39:21
I create the matrix
with those columns.
652
00:39:21 --> 00:39:28
I multiply them by x's, and
I get all the combinations.
653
00:39:28 --> 00:39:31
And this is just C times x.
654
00:39:31 --> 00:39:36
So what I've said there is just
another way of saying how does
655
00:39:36 --> 00:39:38
matrix multiplication work.
656
00:39:38 --> 00:39:46
Put the guys in it's columns
and multiply by a vector.
657
00:39:46 --> 00:39:49
So we're getting all vectors
C times x, and now I was
658
00:39:49 --> 00:39:55
going to stretch it that
little bit further.
659
00:39:55 --> 00:39:58
Can we describe what
vectors we get?
660
00:39:58 --> 00:40:02
So that's my question.
661
00:40:02 --> 00:40:09
What b's, so this is b equal
b_1, b_2, b_3 do we get?
662
00:40:09 --> 00:40:14
We don't get them all.
663
00:40:14 --> 00:40:16
Right, we don't get them all.
664
00:40:16 --> 00:40:18
That's the trouble with C.
665
00:40:18 --> 00:40:21
We only get a plane of them.
666
00:40:21 --> 00:40:30
And now can you tell me which
b's we do get when we look at
667
00:40:30 --> 00:40:36
all combinations of these
three dependent vectors.
668
00:40:36 --> 00:40:39
Well we've done a lot today.
669
00:40:39 --> 00:40:42
Let me just tell you the
answer because it's here.
670
00:40:42 --> 00:40:45
The b's have to add to zero.
671
00:40:45 --> 00:40:49
That's the equation that
the b's have to satisfy.
672
00:40:49 --> 00:40:54
Because when we wrote out Cx
we notice that the components
673
00:40:54 --> 00:40:58
always added to zero.
674
00:40:58 --> 00:41:00
Which b's do we get?
675
00:41:00 --> 00:41:05
We get the ones where the
components add to zero.
676
00:41:05 --> 00:41:09
In other words that's
the equation of the
677
00:41:09 --> 00:41:11
plane you could say.
678
00:41:11 --> 00:41:11
Yeah.
679
00:41:11 --> 00:41:13
Actually that's a good
way to look at it.
680
00:41:13 --> 00:41:19
All these vectors
are on the plane.
681
00:41:19 --> 00:41:25
Do the components of u
add to zero? look at u.
682
00:41:25 --> 00:41:26
Yes.
683
00:41:26 --> 00:41:30
Do the components
of v add to zero?
684
00:41:30 --> 00:41:31
Yes.
685
00:41:31 --> 00:41:32
Add them up.
686
00:41:32 --> 00:41:37
Does the components of w*, now
that you've fix it correctly,
687
00:41:37 --> 00:41:38
do they add to zero?
688
00:41:38 --> 00:41:38
Yes.
689
00:41:38 --> 00:41:40
So all the combinations
will add to zero.
690
00:41:40 --> 00:41:42
That's the plane.
691
00:41:42 --> 00:41:44
That's the plane.
692
00:41:44 --> 00:41:48
You see there are so many
different ways to C, and none
693
00:41:48 --> 00:41:52
of this is difficult, but it's
coming fast because we're
694
00:41:52 --> 00:41:56
seeing the same thing in
different languages.
695
00:41:56 --> 00:41:59
We're seeing it geometrically
in a picture of a plane.
696
00:41:59 --> 00:42:02
We're seeing it as a
combination of vectors.
697
00:42:02 --> 00:42:05
We're seeing it as a
multiplication by a matrix.
698
00:42:05 --> 00:42:11
And we saw it sort of here by
operation, operating and
699
00:42:11 --> 00:42:21
simplifying, and getting the
key fact out of the equations.
700
00:42:21 --> 00:42:22
Well OK.
701
00:42:22 --> 00:42:28
I wanted to give you this
example, the two examples,
702
00:42:28 --> 00:42:31
because they bring out so
many of the key ideas.
703
00:42:31 --> 00:42:34
The key idea of a subspace.
704
00:42:34 --> 00:42:38
Shall I just say a little
about what that word means?
705
00:42:38 --> 00:42:41
A subspace.
706
00:42:41 --> 00:42:43
What's a subspace?
707
00:42:43 --> 00:42:48
Well, what's a vector
space first of all?
708
00:42:48 --> 00:42:51
A vector space is a
bunch of vectors.
709
00:42:51 --> 00:42:54
And the rule is you have
to be able to take
710
00:42:54 --> 00:42:56
their combinations.
711
00:42:56 --> 00:42:57
That what linear algebra does.
712
00:42:57 --> 00:42:59
Takes combinations.
713
00:42:59 --> 00:43:05
So a vector space is one where
you take all combinations.
714
00:43:05 --> 00:43:09
So if I only took just this
triangle that would not be
715
00:43:09 --> 00:43:14
a subspace because one
combination would be 2u and it
716
00:43:14 --> 00:43:16
would be out of the triangle.
717
00:43:16 --> 00:43:22
So a subspace, just think of
it as a plane, but then of
718
00:43:22 --> 00:43:25
course it could be in
higher dimensions.
719
00:43:25 --> 00:43:28
You know it could be a
7-dimensional subspace inside
720
00:43:28 --> 00:43:30
a 15-dimensional space.
721
00:43:30 --> 00:43:37
And I don't know if you're good
at visualizing that, I'm not.
722
00:43:37 --> 00:43:38
Never mind.
723
00:43:38 --> 00:43:41
You you've got seven vectors,
you think OK, their
724
00:43:41 --> 00:43:44
combinations give us
seven-dimensional subspace.
725
00:43:44 --> 00:43:47
Each factor has 15 components.
726
00:43:47 --> 00:43:48
No problem.
727
00:43:48 --> 00:43:50
I mean no problem for
MATLAB certainly.
728
00:43:50 --> 00:43:53
It's got what, a matrix
with a 105 entries.
729
00:43:53 --> 00:43:55
It deals with that instantly.
730
00:43:55 --> 00:44:01
OK, so a subspace is
like a vector space
731
00:44:01 --> 00:44:03
inside a bigger one.
732
00:44:03 --> 00:44:06
That's why the prefix
sub is there.
733
00:44:06 --> 00:44:07
Right?
734
00:44:07 --> 00:44:11
And mathematics always counts
the biggest possibility too,
735
00:44:11 --> 00:44:13
which would be the whole space.
736
00:44:13 --> 00:44:15
And what's the smallest?
737
00:44:15 --> 00:44:18
So what's the smallest
subspace of R^3?
738
00:44:19 --> 00:44:21
So I have 3-dimensional
space-- you can tell me
739
00:44:21 --> 00:44:23
all the subspaces of R^3.
740
00:44:23 --> 00:44:25
So there is one, a plane.
741
00:44:25 --> 00:44:27
Yeah, tell me all the
subspaces of R^3.
742
00:44:28 --> 00:44:31
And then you'll have
that word kind of down.
743
00:44:31 --> 00:44:33
AUDIENCE: [UNINTELLIGIBLE]
744
00:44:33 --> 00:44:34
PROFESSOR STRANG: A line.
745
00:44:34 --> 00:44:38
So planes and lines, those
you could say, the real,
746
00:44:38 --> 00:44:39
the proper subspaces.
747
00:44:39 --> 00:44:41
The best, the right ones.
748
00:44:41 --> 00:44:45
But there are a couple more
possibilities which are?
749
00:44:45 --> 00:44:46
AUDIENCE: [UNINTELLIGIBLE]
750
00:44:46 --> 00:44:47
PROFESSOR STRANG: A point.
751
00:44:47 --> 00:44:49
Which point?
752
00:44:49 --> 00:44:50
The origin.
753
00:44:50 --> 00:44:50
Only the origin.
754
00:44:50 --> 00:44:53
Because if you tried
to say that point was
755
00:44:53 --> 00:44:56
a subspace, no way.
756
00:44:56 --> 00:44:56
Why not?
757
00:44:56 --> 00:45:00
Because I wouldn't be able to
multiply that vector by five
758
00:45:00 --> 00:45:03
and I would be away
from the point.
759
00:45:03 --> 00:45:09
But the zero subspace, the
really small subspace that just
760
00:45:09 --> 00:45:13
has the zero vector-- it's got
one vector in it. not empty.
761
00:45:13 --> 00:45:15
It's got that one
point but that's all.
762
00:45:15 --> 00:45:21
OK, so planes, lines, the
origin, and then the other
763
00:45:21 --> 00:45:25
possibility for a subspaces is?
764
00:45:25 --> 00:45:26
The whole space.
765
00:45:26 --> 00:45:26
Right.
766
00:45:26 --> 00:45:27
Right.
767
00:45:27 --> 00:45:30
So the dimensions could be
three for the whole space,
768
00:45:30 --> 00:45:35
two for a plane, one for
a line, zero for a point.
769
00:45:35 --> 00:45:41
It just kicks together.
770
00:45:41 --> 00:45:42
How are we for time?
771
00:45:42 --> 00:45:47
Maybe I went more than a half,
but now is a chance to just
772
00:45:47 --> 00:45:53
asked me, if you want to, like
anything about the course.
773
00:45:53 --> 00:45:54
Is at all linear algebra?
774
00:45:54 --> 00:45:55
No.
775
00:45:55 --> 00:46:03
But I think I can't do anything
more helpful to you then to for
776
00:46:03 --> 00:46:08
you to begin to see when you
look at a matrix-- begin
777
00:46:08 --> 00:46:09
to see what is it doing.
778
00:46:09 --> 00:46:11
What is it about.
779
00:46:11 --> 00:46:14
Right, and of course matrices
can be rectangular.
780
00:46:14 --> 00:46:18
So I'll give you a hint
about what's coming
781
00:46:18 --> 00:46:20
in the course itself.
782
00:46:20 --> 00:46:26
We'll have rectangular
matrix A, OK.
783
00:46:26 --> 00:46:28
They're not in invertible.
784
00:46:28 --> 00:46:31
They're taking 7-dimensional
space to three-dimensional
785
00:46:31 --> 00:46:33
space or something.
786
00:46:33 --> 00:46:35
You can't invert that.
787
00:46:35 --> 00:46:40
What comes up every time-- I
sort of got the idea finally.
788
00:46:40 --> 00:46:43
Every time I see a rectangular
matrix, maybe seven by
789
00:46:43 --> 00:46:48
three, that would be seven
rows by three columns.
790
00:46:48 --> 00:46:53
Then what comes up with a
rectangular matrix A is
791
00:46:53 --> 00:46:57
sooner or later A transpose
sticks it's nose in
792
00:46:57 --> 00:47:04
and multiplies that A.
793
00:47:04 --> 00:47:07
And we couldn't do
it for our A here.
794
00:47:07 --> 00:47:11
Actually if I did it for that
original matrix A I would get
795
00:47:11 --> 00:47:14
something you'd recognize.
796
00:47:14 --> 00:47:18
What I want to say is
that the course focuses
797
00:47:18 --> 00:47:21
on A transpose A.
798
00:47:21 --> 00:47:25
I'll just say now that that
matrix always comes out square,
799
00:47:25 --> 00:47:28
because this would be three
times seven times seven times
800
00:47:28 --> 00:47:32
three, so this would be three
by three, and it always
801
00:47:32 --> 00:47:34
comes out symmetric.
802
00:47:34 --> 00:47:36
That's the nice thing.
803
00:47:36 --> 00:47:37
And even more.
804
00:47:37 --> 00:47:39
We'll see more.
805
00:47:39 --> 00:47:41
That's like a hint.
806
00:47:41 --> 00:47:47
Watch for A transpose A.
807
00:47:47 --> 00:47:52
And watch for it in
applications of all kinds.
808
00:47:52 --> 00:47:57
In networks an A will be
associated with Kirkoff's
809
00:47:57 --> 00:48:00
voltage law, in A transpose
with Kirkoff's current law.
810
00:48:00 --> 00:48:04
They just teamed up together.
811
00:48:04 --> 00:48:06
We'll see more.
812
00:48:06 --> 00:48:10
Alright now let me give you a
chance to ask any question.
813
00:48:10 --> 00:48:14
Whatever.
814
00:48:14 --> 00:48:15
Homework.
815
00:48:15 --> 00:48:17
Did I mention homework?
816
00:48:17 --> 00:48:22
You may have said that's
a crazy homework to say
817
00:48:22 --> 00:48:24
three problems 1.1.
818
00:48:24 --> 00:48:30
I've never done this before so
essentially you can get away
819
00:48:30 --> 00:48:37
with anything this week,
and indefinitely actually.
820
00:48:37 --> 00:48:42
How many is this the first
day of MIT classes.
821
00:48:42 --> 00:48:43
Oh wow.
822
00:48:43 --> 00:48:43
OK.
823
00:48:43 --> 00:48:46
Well welcome to MIT.
824
00:48:46 --> 00:48:49
I hope you like it.
825
00:48:49 --> 00:48:56
It's not so high
pressure or whatever is
826
00:48:56 --> 00:48:57
associated with MIT.
827
00:48:57 --> 00:49:01
It's kind of tolerant.
828
00:49:01 --> 00:49:04
If my advises ask for
something I always say yes.
829
00:49:04 --> 00:49:05
It's easier that way.
830
00:49:05 --> 00:49:11
AUDIENCE: [LAUGHTER].
831
00:49:11 --> 00:49:15
PROFESSOR STRANG: And let me
just again-- and I'll say
832
00:49:15 --> 00:49:19
it often and in private.
833
00:49:19 --> 00:49:21
This is like a grown-up course.
834
00:49:21 --> 00:49:24
I'm figuring you're here
to learn, so it's not
835
00:49:24 --> 00:49:26
my job to force it.
836
00:49:26 --> 00:49:31
My job is to help it, and
hope this is some help.