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PROFESSOR STRANG: This is
the one and only review, you
9
00:00:27,190 --> 00:00:30,740
could say, of linear algebra.
10
00:00:30,740 --> 00:00:33,660
I just think linear
algebra is very important.
11
00:00:33,660 --> 00:00:35,420
You may have got that idea.
12
00:00:35,420 --> 00:00:43,430
And my website even has a little
essay called Too Much Calculus.
13
00:00:43,430 --> 00:00:49,660
Because I think it's crazy
for all the U.S. universities
14
00:00:49,660 --> 00:00:51,890
do this pretty much,
you get semester
15
00:00:51,890 --> 00:00:55,460
after semester in differential
calculus, integral calculus,
16
00:00:55,460 --> 00:00:57,130
ultimately
differential equations.
17
00:00:57,130 --> 00:01:00,050
You run out of steam
before the good stuff,
18
00:01:00,050 --> 00:01:03,340
before you run out of time.
19
00:01:03,340 --> 00:01:08,010
And anybody who computes,
who's living in the real world,
20
00:01:08,010 --> 00:01:10,830
is using linear algebra.
21
00:01:10,830 --> 00:01:13,030
You're taking a
differential equation,
22
00:01:13,030 --> 00:01:16,230
you're taking your
model, making it discrete
23
00:01:16,230 --> 00:01:20,830
and computing with matrices.
24
00:01:20,830 --> 00:01:27,080
The world's digital
now, not analog.
25
00:01:27,080 --> 00:01:33,870
I hope it's okay to start the
course with linear algebra.
26
00:01:33,870 --> 00:01:38,200
But many engineering
curricula don't fully
27
00:01:38,200 --> 00:01:40,170
recognize that and
so if you haven't
28
00:01:40,170 --> 00:01:46,770
had an official course, linear
algebra, stay with 18.085.
29
00:01:46,770 --> 00:01:48,280
This is a good way to learn it.
30
00:01:48,280 --> 00:01:51,330
You're sort of learning
what's important.
31
00:01:51,330 --> 00:01:58,340
So my review would be-- And
then this is-- Future Wednesdays
32
00:01:58,340 --> 00:02:02,190
will be in our regular
room for homework, review,
33
00:02:02,190 --> 00:02:06,830
questions of all kinds,
and today questions too.
34
00:02:06,830 --> 00:02:10,990
Shall I just fire away for
the first half of the time
35
00:02:10,990 --> 00:02:15,350
to give you a sense of
how I see the subject,
36
00:02:15,350 --> 00:02:17,950
or at least within
that limited time.
37
00:02:17,950 --> 00:02:22,440
And then questions
are totally welcome.
38
00:02:22,440 --> 00:02:24,031
Always welcome, actually.
39
00:02:24,031 --> 00:02:24,530
Right?
40
00:02:24,530 --> 00:02:26,590
So I'll just start right up.
41
00:02:26,590 --> 00:02:31,130
So essentially linear
algebra progresses
42
00:02:31,130 --> 00:02:41,040
starting with vectors to
matrices and then finally
43
00:02:41,040 --> 00:02:42,510
to subspaces.
44
00:02:42,510 --> 00:02:45,240
So that's, like,
the abstraction.
45
00:02:45,240 --> 00:02:49,440
You could say abstraction,
but it's not difficult,
46
00:02:49,440 --> 00:02:50,680
that you want to see.
47
00:02:50,680 --> 00:02:53,290
Until you see the
idea of a subspace,
48
00:02:53,290 --> 00:02:56,130
you haven't really
got linear algebra.
49
00:02:56,130 --> 00:02:58,460
Okay, so I'll start
at the beginning.
50
00:02:58,460 --> 00:03:01,190
What do you do with vectors?
51
00:03:01,190 --> 00:03:04,190
Answer: you take their
linear combinations.
52
00:03:04,190 --> 00:03:06,160
That's what you can
do with a vector.
53
00:03:06,160 --> 00:03:11,760
You can multiply it by a number
and you can add or subtract.
54
00:03:11,760 --> 00:03:13,790
So that's the key operation.
55
00:03:13,790 --> 00:03:18,120
Suppose I have
vectors u, v and w.
56
00:03:18,120 --> 00:03:20,850
Let me take three of them.
57
00:03:20,850 --> 00:03:22,710
So I can take
their combinations.
58
00:03:22,710 --> 00:03:28,170
So some combination will
be, say some number times
59
00:03:28,170 --> 00:03:33,230
u plus some number times v
plus some number times w.
60
00:03:33,230 --> 00:03:36,490
So these numbers
are called scalars.
61
00:03:36,490 --> 00:03:39,640
So these would be
called scalars.
62
00:03:39,640 --> 00:03:44,270
And the whole thing is
a linear combination.
63
00:03:44,270 --> 00:03:48,180
Let me abbreviate those
words, linear combination.
64
00:03:48,180 --> 00:03:54,390
And you get some answer, say b.
65
00:03:54,390 --> 00:04:00,600
But let's put it down, make
this whole discussion specific.
66
00:04:00,600 --> 00:04:05,120
Yeah, I started a
little early, I think.
67
00:04:05,120 --> 00:04:08,640
I'm going to take three
vectors, u, v and w,
68
00:04:08,640 --> 00:04:12,640
and take their combinations.
69
00:04:12,640 --> 00:04:14,920
They're carefully chosen.
70
00:04:14,920 --> 00:04:20,440
My u is going to be [1, -1, 0].
71
00:04:20,440 --> 00:04:23,580
And I'll take vectors
in three dimensions.
72
00:04:23,580 --> 00:04:26,550
So that means their combinations
will be in three dimensions,
73
00:04:26,550 --> 00:04:29,920
R^3, three-dimensional space.
74
00:04:29,920 --> 00:04:37,900
So that'll be u and then
v, let's take 0, I think,
75
00:04:37,900 --> 00:04:40,530
1 and -1.
76
00:04:40,530 --> 00:04:41,240
Okay.
77
00:04:41,240 --> 00:04:45,170
Suppose I stopped there and
took their linear combinations.
78
00:04:45,170 --> 00:04:47,820
It's very helpful
to see a picture
79
00:04:47,820 --> 00:04:49,940
in three-dimensional space.
80
00:04:49,940 --> 00:04:52,100
I mean the great thing
about linear algebra,
81
00:04:52,100 --> 00:04:55,000
it moves into
n-dimensional space,
82
00:04:55,000 --> 00:04:58,320
10-dimensional, 100-dimensional,
where we can't visualize,
83
00:04:58,320 --> 00:05:02,020
but yet, our instinct is
right if we just follow.
84
00:05:02,020 --> 00:05:06,980
So what's your instinct if
I took those two vectors,
85
00:05:06,980 --> 00:05:09,290
and notice they're
not on the same line,
86
00:05:09,290 --> 00:05:11,180
one isn't a multiple
of the other,
87
00:05:11,180 --> 00:05:13,220
they go in different directions.
88
00:05:13,220 --> 00:05:21,340
If I took their combinations,
say x_1*u + x_2*v. Oh now,
89
00:05:21,340 --> 00:05:23,930
let me push, this is
a serious question.
90
00:05:23,930 --> 00:05:26,630
If I took all
their combinations.
91
00:05:26,630 --> 00:05:28,630
So let me try to
draw a little bit.
92
00:05:28,630 --> 00:05:29,130
Okay.
93
00:05:29,130 --> 00:05:33,540
I'm in three-dimensional
space and u goes somewhere,
94
00:05:33,540 --> 00:05:39,030
maybe there and v goes
somewhere, maybe here.
95
00:05:39,030 --> 00:05:41,980
Now suppose I take
all the combinations,
96
00:05:41,980 --> 00:05:44,040
so I could multiply
that first guy
97
00:05:44,040 --> 00:05:46,740
by any number, that
would fill the line.
98
00:05:46,740 --> 00:05:50,780
I can multiply that second
guy, v. So this was u
99
00:05:50,780 --> 00:05:54,360
and this was v. I can multiply
that by any number x_2, that
100
00:05:54,360 --> 00:05:56,290
would fill its line.
101
00:05:56,290 --> 00:05:58,070
Each of those
lines I would later
102
00:05:58,070 --> 00:06:01,500
call a one-dimensional
subspace, just a line.
103
00:06:01,500 --> 00:06:06,730
But now, what happens if I take
all combinations of the two?
104
00:06:06,730 --> 00:06:08,550
What do you think?
105
00:06:08,550 --> 00:06:09,880
You got a plane.
106
00:06:09,880 --> 00:06:11,060
Get a plane.
107
00:06:11,060 --> 00:06:14,460
If I take anything on this
line and anything on this line
108
00:06:14,460 --> 00:06:19,460
and add them up you can see
that I'm not going to fill 3-D.
109
00:06:19,460 --> 00:06:23,690
But I'm going to fill
a plane and that maybe
110
00:06:23,690 --> 00:06:25,640
takes a little thinking.
111
00:06:25,640 --> 00:06:28,450
It just, then it
becomes sort of,
112
00:06:28,450 --> 00:06:31,230
you see that that's
what it has to be.
113
00:06:31,230 --> 00:06:34,670
Okay, now I'm going to
have a third vector.
114
00:06:34,670 --> 00:06:38,680
Okay, my third vector
will be [0, 0, 1].
115
00:06:38,680 --> 00:06:43,210
Okay, so that [0, 0, 1]
is 0 in the x, 0 in the y
116
00:06:43,210 --> 00:06:44,660
and 1 in the z direction.
117
00:06:44,660 --> 00:06:48,740
So there's w.
118
00:06:48,740 --> 00:06:51,810
Now I want to take
their combinations.
119
00:06:51,810 --> 00:06:54,530
So let me do that
very specifically.
120
00:06:54,530 --> 00:06:57,310
How do I take combinations?
121
00:06:57,310 --> 00:06:58,540
This is important.
122
00:06:58,540 --> 00:07:01,550
Seems it's very
simple, but important.
123
00:07:01,550 --> 00:07:07,770
I like to think of taking the
combinations of some vectors,
124
00:07:07,770 --> 00:07:11,920
I'm always putting vectors
into the columns of a matrix.
125
00:07:11,920 --> 00:07:15,120
So now I'm going to move
to step two: matrix.
126
00:07:15,120 --> 00:07:19,090
I'm going to move to step
two and maybe I'll put it--
127
00:07:19,090 --> 00:07:20,770
well no, I better put it here.
128
00:07:20,770 --> 00:07:26,410
Okay, step two is the matrix has
those vectors in its columns.
129
00:07:26,410 --> 00:07:31,790
So in this case,
it's three by three.
130
00:07:31,790 --> 00:07:38,240
That's my matrix and
I'm going to call it A.
131
00:07:38,240 --> 00:07:44,870
How do I take
combinations of vectors?
132
00:07:44,870 --> 00:07:47,870
I should have maybe
done it in detail here,
133
00:07:47,870 --> 00:07:52,790
but I'll just do it
with a matrix here.
134
00:07:52,790 --> 00:07:54,850
Watch this now.
135
00:07:54,850 --> 00:08:05,030
If I multiply A by the vector
of x's, what that does,
136
00:08:05,030 --> 00:08:08,080
so this is now A times
x, so very important,
137
00:08:08,080 --> 00:08:10,090
a matrix times a vector.
138
00:08:10,090 --> 00:08:12,510
What does it do?
139
00:08:12,510 --> 00:08:15,730
The output is just what I want.
140
00:08:15,730 --> 00:08:17,840
This is the output.
141
00:08:17,840 --> 00:08:21,510
It takes x_1 times
the first column
142
00:08:21,510 --> 00:08:24,460
plus x_2 times the second
plus x_3 times the third.
143
00:08:24,460 --> 00:08:27,390
That's the way matrix
multiplication works,
144
00:08:27,390 --> 00:08:28,870
by columns.
145
00:08:28,870 --> 00:08:30,870
And you don't always see that.
146
00:08:30,870 --> 00:08:32,140
Because what do you see?
147
00:08:32,140 --> 00:08:34,870
You probably know how
to multiply that matrix
148
00:08:34,870 --> 00:08:35,950
by that vector.
149
00:08:35,950 --> 00:08:38,700
Let me ask you to do it.
150
00:08:38,700 --> 00:08:40,520
What do you get?
151
00:08:40,520 --> 00:08:43,770
So everybody does it
a component at a time.
152
00:08:43,770 --> 00:08:48,730
So what's the first component
of the answer? x_1, yeah.
153
00:08:48,730 --> 00:08:49,800
How do you get that?
154
00:08:49,800 --> 00:08:52,850
It's row times the vector.
155
00:08:52,850 --> 00:08:56,610
And when I say "times", I
really mean that dot product.
156
00:08:56,610 --> 00:09:00,000
This plus this plus this is x_1.
157
00:09:00,000 --> 00:09:05,130
And what about the second row?
158
00:09:05,130 --> 00:09:06,520
-x_1 + x_2.
159
00:09:06,520 --> 00:09:09,730
Or I'll just say x_2 - x_1.
160
00:09:09,730 --> 00:09:12,480
And the third guy,
the third component
161
00:09:12,480 --> 00:09:22,050
would be x_3 - x_2, right?
162
00:09:22,050 --> 00:09:25,050
So right away I'm
going to say, I'm
163
00:09:25,050 --> 00:09:28,100
going to call this matrix
A a difference matrix.
164
00:09:28,100 --> 00:09:31,370
It always helps to
give names to things.
165
00:09:31,370 --> 00:09:35,880
So this A is a difference matrix
because it takes differences
166
00:09:35,880 --> 00:09:37,430
of the x's.
167
00:09:37,430 --> 00:09:40,440
And I would even say a
first difference matrix
168
00:09:40,440 --> 00:09:43,850
because it's just the
straightforward difference
169
00:09:43,850 --> 00:09:48,770
and we'll see second
differences in class Friday.
170
00:09:48,770 --> 00:09:50,120
So that's what A does.
171
00:09:50,120 --> 00:09:52,520
But you remember
my first point was
172
00:09:52,520 --> 00:09:55,610
that when a matrix
multiplies a vector,
173
00:09:55,610 --> 00:10:01,560
the result is a
combination of the columns.
174
00:10:01,560 --> 00:10:05,250
And that's not always, because
see, I'm looking at the picture
175
00:10:05,250 --> 00:10:07,040
not just by numbers.
176
00:10:07,040 --> 00:10:10,880
You know, with numbers
I'm just doing this stuff.
177
00:10:10,880 --> 00:10:12,750
But now I'm stepping
back a little bit
178
00:10:12,750 --> 00:10:14,830
and saying I'm combining.
179
00:10:14,830 --> 00:10:18,120
It's this vector times x_1.
180
00:10:18,120 --> 00:10:22,030
That vector times x_1 plus
this vector times x_2 plus that
181
00:10:22,030 --> 00:10:25,790
one times x_3 added
together gives me this.
182
00:10:25,790 --> 00:10:28,780
Saying nothing complicated here.
183
00:10:28,780 --> 00:10:36,960
It's just look at
it by vectors, also.
184
00:10:36,960 --> 00:10:39,080
It's a little
interesting, already.
185
00:10:39,080 --> 00:10:47,400
Here we multiplied these vectors
by numbers. x_1, x_2, x_3.
186
00:10:47,400 --> 00:10:49,130
That was our thinking here.
187
00:10:49,130 --> 00:10:50,770
Now our thinking
here is a little--
188
00:10:50,770 --> 00:10:52,420
we've switched slightly.
189
00:10:52,420 --> 00:10:57,870
Now I'm multiplying the
matrix times the numbers in x.
190
00:10:57,870 --> 00:11:01,220
Just a slight switch, multiply
the matrix times the number.
191
00:11:01,220 --> 00:11:03,200
And I get some answer, b.
192
00:11:03,200 --> 00:11:07,880
Which is this, this is b.
193
00:11:07,880 --> 00:11:11,960
And of course, I can do
a specific example like,
194
00:11:11,960 --> 00:11:19,850
suppose I take, well, I could
take the squares to be in x.
195
00:11:19,850 --> 00:11:27,820
So suppose I take A times the
first three squares, [1, 4, 9].
196
00:11:27,820 --> 00:11:30,330
What answer would I get?
197
00:11:30,330 --> 00:11:35,130
Just to keep it clear that
we're very specific here.
198
00:11:35,130 --> 00:11:38,650
So what would be the output be?
199
00:11:38,650 --> 00:11:42,260
I think of this as the input,
the [1, 4, 9], the x's.
200
00:11:42,260 --> 00:11:48,140
Now the machine is multiply
by A and here's the output.
201
00:11:48,140 --> 00:11:49,680
And what would be the output?
202
00:11:49,680 --> 00:11:51,270
What numbers am I
going to get there?
203
00:11:51,270 --> 00:11:52,710
Yeah?
204
00:11:52,710 --> 00:11:57,700
One, three,
something? [1, 3, 5].
205
00:11:57,700 --> 00:12:01,020
Which is actually a
little neat that you
206
00:12:01,020 --> 00:12:06,000
find the differences of the
squares are the odd numbers.
207
00:12:06,000 --> 00:12:11,280
That appealed to me
in school somehow.
208
00:12:11,280 --> 00:12:13,430
That was already
a bad sign, right?
209
00:12:13,430 --> 00:12:18,790
This dumb kid notices that you
take differences of squares
210
00:12:18,790 --> 00:12:21,780
and get odd numbers, whatever.
211
00:12:21,780 --> 00:12:25,170
So now is a big step.
212
00:12:25,170 --> 00:12:29,310
This was the forward
direction, right?
213
00:12:29,310 --> 00:12:31,960
Input, and there's the output.
214
00:12:31,960 --> 00:12:36,460
But now the real reality--
That's easy and important,
215
00:12:36,460 --> 00:12:43,590
but the more deep
problem is, what
216
00:12:43,590 --> 00:12:48,250
if I give you b and ask for x?
217
00:12:48,250 --> 00:12:52,450
So again, we're switching
the direction here.
218
00:12:52,450 --> 00:12:55,990
We're solving an equation now,
or three equations and three
219
00:12:55,990 --> 00:12:58,080
unknowns, Ax=b.
220
00:12:58,080 --> 00:13:02,960
So if I give you this
b, can you get x?
221
00:13:02,960 --> 00:13:06,270
How do you solve
three equations?
222
00:13:06,270 --> 00:13:07,960
We're looking backwards.
223
00:13:07,960 --> 00:13:14,760
Now that won't be too hard
for this particular matrix
224
00:13:14,760 --> 00:13:18,350
that I chose; because
it's triangular,
225
00:13:18,350 --> 00:13:23,180
we'll be able to go backwards.
226
00:13:23,180 --> 00:13:25,930
So let me do that.
227
00:13:25,930 --> 00:13:29,780
Let me take b to
be-- It's a vector,
228
00:13:29,780 --> 00:13:31,630
it's got three components.
229
00:13:31,630 --> 00:13:36,190
And now I'm going to
go backwards to find x.
230
00:13:36,190 --> 00:13:39,400
Or we will.
231
00:13:39,400 --> 00:13:41,950
So do you see the
three equations?
232
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.
233
00:13:47,890 --> 00:13:49,290
Those are my three equations.
234
00:13:49,290 --> 00:13:53,180
Three unknown x's, three
known right-hand sides.
235
00:13:53,180 --> 00:13:57,230
Or I think of it as A
times x, as a matrix
236
00:13:57,230 --> 00:14:00,500
times x giving a vector b.
237
00:14:00,500 --> 00:14:02,430
What's the answer?
238
00:14:02,430 --> 00:14:05,660
As I said, we're going
to be able to do this.
239
00:14:05,660 --> 00:14:09,300
We're going to be able to solve
this system easily because it's
240
00:14:09,300 --> 00:14:11,810
already triangular.
241
00:14:11,810 --> 00:14:14,290
And it's actually
lower triangular
242
00:14:14,290 --> 00:14:17,890
so that means we'll
start from the top.
243
00:14:17,890 --> 00:14:24,750
So the answers, the
solution will be what?
244
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.
245
00:14:30,490 --> 00:14:33,860
And what's the answer?
246
00:14:33,860 --> 00:14:37,450
Can we just go from
top to bottom now?
247
00:14:37,450 --> 00:14:42,120
What's x_1? b_1, great.
248
00:14:42,120 --> 00:14:44,790
What's x_2?
249
00:14:44,790 --> 00:14:47,010
So x_2 - x_1.
250
00:14:47,010 --> 00:14:48,280
These are my equations.
251
00:14:48,280 --> 00:14:50,620
So what's x_2 - x_1?
252
00:14:50,620 --> 00:14:59,720
Well, it's b_2, so what
is x_2? b_1 + b_2, right?
253
00:14:59,720 --> 00:15:03,030
And what's x_3?
254
00:15:03,030 --> 00:15:05,870
What do we need there for x_3?
255
00:15:05,870 --> 00:15:08,550
So I'm looking at
the third equation.
256
00:15:08,550 --> 00:15:11,090
That'll determine x_3.
257
00:15:11,090 --> 00:15:13,670
When I see it this
way, I see those ones
258
00:15:13,670 --> 00:15:16,190
and I see it multiplying x_3.
259
00:15:16,190 --> 00:15:20,750
And what do I get?
260
00:15:20,750 --> 00:15:25,010
Yeah, so x_3 minus
this guy is b_3,
261
00:15:25,010 --> 00:15:28,040
so I have to add in
another b_3, right?
262
00:15:28,040 --> 00:15:31,570
I'm doing sort of
substitution down as I go.
263
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.
264
00:15:38,240 --> 00:15:40,700
And now I'll use
x_2 to find x_3.
265
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.
266
00:15:48,520 --> 00:15:50,810
I think you've got
it. b_1 + b_2 + b_3.
267
00:15:55,410 --> 00:15:59,760
So that's the solution.
268
00:15:59,760 --> 00:16:02,570
Not difficult because the
matrix was triangular.
269
00:16:02,570 --> 00:16:06,700
But let's think
about that solution.
270
00:16:06,700 --> 00:16:13,520
That solution is
a matrix times b.
271
00:16:13,520 --> 00:16:18,540
When you look at that-- So
this is like a good early step
272
00:16:18,540 --> 00:16:19,700
in linear algebra.
273
00:16:19,700 --> 00:16:26,090
When I look at that I see
a matrix multiplying b.
274
00:16:26,090 --> 00:16:29,000
You take that step up
to seeing a matrix.
275
00:16:29,000 --> 00:16:30,560
And you can just read it off.
276
00:16:30,560 --> 00:16:34,930
So let me say, what's the
matrix there that's multiplying
277
00:16:34,930 --> 00:16:41,550
b to give that answer?
278
00:16:41,550 --> 00:16:46,050
Remember the columns
of this matrix-- well,
279
00:16:46,050 --> 00:16:47,960
I don't know how you
want to read it off.
280
00:16:47,960 --> 00:16:52,810
But one way is, the think
the columns of that matrix
281
00:16:52,810 --> 00:16:57,250
are multiplying b_1, b_2,
and b_3 to give this.
282
00:16:57,250 --> 00:17:01,630
So what's the first
column of the matrix?
283
00:17:01,630 --> 00:17:05,550
It's whatever I'm reading
off, the coefficients, really,
284
00:17:05,550 --> 00:17:09,150
of b_1 here: [1, 1, 1].
285
00:17:09,150 --> 00:17:13,101
And what's the second column
of the matrix? [0, 1, 1].
286
00:17:13,101 --> 00:17:13,600
Good.
287
00:17:13,600 --> 00:17:16,640
Zero b_2's, one, one.
288
00:17:16,640 --> 00:17:19,830
And the third is? [0, 0, 1].
289
00:17:19,830 --> 00:17:23,200
Good.
290
00:17:23,200 --> 00:17:27,780
Now, so lots of things
to comment here.
291
00:17:27,780 --> 00:17:30,700
Let me write up again
here, this is x.
292
00:17:30,700 --> 00:17:34,390
That was the answer.
293
00:17:34,390 --> 00:17:38,630
It's a matrix times b.
294
00:17:38,630 --> 00:17:42,580
And what's the name
of that matrix?
295
00:17:42,580 --> 00:17:44,470
It's the inverse matrix.
296
00:17:44,470 --> 00:17:48,120
If Ax gives b,
the inverse matrix
297
00:17:48,120 --> 00:17:50,930
does it the other way
around, x is A inverse b.
298
00:17:50,930 --> 00:17:53,550
Let me just put that over here.
299
00:17:53,550 --> 00:18:01,070
If Ax is b, then x
should be A inverse b.
300
00:18:01,070 --> 00:18:04,680
So we had inverse, I wrote
down inverse this morning
301
00:18:04,680 --> 00:18:10,150
but without saying the point,
but so you see how that comes?
302
00:18:10,150 --> 00:18:11,690
I mean, if I want
to go formally,
303
00:18:11,690 --> 00:18:15,100
I multiply both
sides by A inverse.
304
00:18:15,100 --> 00:18:18,010
If there is an A inverse.
305
00:18:18,010 --> 00:18:20,180
That's a critical
thing as we saw.
306
00:18:20,180 --> 00:18:21,740
Is the matrix invertible?
307
00:18:21,740 --> 00:18:24,430
The answer here is, yes,
there is an inverse.
308
00:18:24,430 --> 00:18:26,410
And what does that really mean?
309
00:18:26,410 --> 00:18:32,140
The inverse is the thing that
takes us from b back to x.
310
00:18:32,140 --> 00:18:34,820
Think of A as kind of
a-- multiplying by A
311
00:18:34,820 --> 00:18:39,150
is kind of a mapping,
mathematicians use the word,
312
00:18:39,150 --> 00:18:40,930
or transform.
313
00:18:40,930 --> 00:18:42,670
Transform would be good.
314
00:18:42,670 --> 00:18:45,850
Transform from x to b.
315
00:18:45,850 --> 00:18:49,100
And this is the
inverse transform.
316
00:18:49,100 --> 00:18:52,490
So it doesn't happen to
be the discrete Fourier
317
00:18:52,490 --> 00:18:55,940
transform or a wavelet
transform, it's a-- well,
318
00:18:55,940 --> 00:18:57,830
actually we could
give it a name.
319
00:18:57,830 --> 00:19:00,740
This is kind of a
difference transform, right?
320
00:19:00,740 --> 00:19:03,570
That's what A did,
took differences.
321
00:19:03,570 --> 00:19:06,780
So what does A inverse do?
322
00:19:06,780 --> 00:19:09,090
It takes sums.
323
00:19:09,090 --> 00:19:10,030
It takes sums.
324
00:19:10,030 --> 00:19:15,550
That's why you see 1, 1
and 1, 1, 1 along the rows
325
00:19:15,550 --> 00:19:18,160
because it's just
adding, and you see it
326
00:19:18,160 --> 00:19:20,130
here in fully display.
327
00:19:20,130 --> 00:19:21,730
It's a sum matrix.
328
00:19:21,730 --> 00:19:25,170
I might as well
call it S for sum.
329
00:19:25,170 --> 00:19:28,610
So that matrix, that sum
matrix is the inverse
330
00:19:28,610 --> 00:19:33,750
of the different matrix.
331
00:19:33,750 --> 00:19:40,220
And maybe, since I hit
on calculus earlier,
332
00:19:40,220 --> 00:19:43,590
you could say that
calculus is all about one
333
00:19:43,590 --> 00:19:45,420
thing and its inverse.
334
00:19:45,420 --> 00:19:50,170
The derivative is
A, and what's S?
335
00:19:50,170 --> 00:19:51,990
In calculus.
336
00:19:51,990 --> 00:19:53,510
The integral.
337
00:19:53,510 --> 00:19:57,880
The whole subject is
about one operation,
338
00:19:57,880 --> 00:20:00,330
now admittedly
it's not a matrix,
339
00:20:00,330 --> 00:20:04,810
it operates on functions
instead of just little vectors,
340
00:20:04,810 --> 00:20:06,870
but that's the main point.
341
00:20:06,870 --> 00:20:09,390
The fundamental
theorem of calculus
342
00:20:09,390 --> 00:20:14,380
is telling us that integration's
the inverse of differentiation.
343
00:20:14,380 --> 00:20:20,390
So this is good and if
I put in b = [1, 3, 5]
344
00:20:20,390 --> 00:20:24,660
for example just to
put in some numbers,
345
00:20:24,660 --> 00:20:31,340
if I put in b = [1, 3, 5], what
would the x that comes out be?
346
00:20:31,340 --> 00:20:33,170
[1, 4, 9].
347
00:20:33,170 --> 00:20:33,870
Right?
348
00:20:33,870 --> 00:20:35,580
Because it takes us back.
349
00:20:35,580 --> 00:20:39,080
Here, previously we started, we
took differences of [1, 4, 9],
350
00:20:39,080 --> 00:20:40,690
got [1, 3, 5].
351
00:20:40,690 --> 00:20:46,300
Now if we take sums of
[1, 3, 5], we get [1, 4, 9].
352
00:20:46,300 --> 00:20:51,240
Now we have a system
of linear equations.
353
00:20:51,240 --> 00:20:53,760
Now I want to step
back and see what
354
00:20:53,760 --> 00:20:56,570
was good about this matrix.
355
00:20:56,570 --> 00:20:59,520
Somehow it has an inverse.
356
00:20:59,520 --> 00:21:03,040
Ax=b has a solution,
in other words.
357
00:21:03,040 --> 00:21:05,510
And it has only one
solution, right?
358
00:21:05,510 --> 00:21:07,190
Because we worked it out.
359
00:21:07,190 --> 00:21:08,120
We had no choice.
360
00:21:08,120 --> 00:21:10,340
That was it.
361
00:21:10,340 --> 00:21:12,300
So there's just one solution.
362
00:21:12,300 --> 00:21:14,830
There's always one
and only one solution.
363
00:21:14,830 --> 00:21:18,430
It's like a perfect
transform from the x's
364
00:21:18,430 --> 00:21:20,400
to the b's and back again.
365
00:21:20,400 --> 00:21:24,200
Yeah so that's what an
invertible matrix is.
366
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
367
00:21:29,810 --> 00:21:33,580
can get back again.
368
00:21:33,580 --> 00:21:36,120
Questions always.
369
00:21:36,120 --> 00:21:39,850
Now I think I'm ready
for another example.
370
00:21:39,850 --> 00:21:41,140
There are only two examples.
371
00:21:41,140 --> 00:21:47,550
And actually these two examples
are on the 18.06 web page.
372
00:21:47,550 --> 00:21:51,480
If some people asked
after class how
373
00:21:51,480 --> 00:21:55,650
to get sort of a review
of linear algebra,
374
00:21:55,650 --> 00:22:06,220
well the 18.06 website would
be definitely a possibility.
375
00:22:06,220 --> 00:22:12,070
Well, I'll put down the
OpenCourseWare website, mit.edu
376
00:22:12,070 --> 00:22:14,780
and then you would look
at the linear algebra
377
00:22:14,780 --> 00:22:20,640
course or the math one.
378
00:22:20,640 --> 00:22:26,130
What is it?
web.math.edu, is that it?
379
00:22:26,130 --> 00:22:34,930
No, maybe that's an
MIT-- so is it math?
380
00:22:34,930 --> 00:22:39,190
I can't live without
edu at the end, right?
381
00:22:39,190 --> 00:22:41,700
Is it just edu?
382
00:22:41,700 --> 00:22:49,690
Whatever!
383
00:22:49,690 --> 00:22:54,070
So that website has, well, all
the old exams you could ever
384
00:22:54,070 --> 00:22:55,730
want if you wanted any.
385
00:22:55,730 --> 00:23:03,630
And it has this
example and you click
386
00:23:03,630 --> 00:23:06,720
on Starting With Two Matrices.
387
00:23:06,720 --> 00:23:09,050
And this is one of them.
388
00:23:09,050 --> 00:23:11,240
Okay, ready for the other.
389
00:23:11,240 --> 00:23:14,100
So here comes the second
matrix, second example
390
00:23:14,100 --> 00:23:16,560
that you can contrast.
391
00:23:16,560 --> 00:23:20,350
Second example is going
to have the same u.
392
00:23:20,350 --> 00:23:24,996
Let me put-- our matrix,
I'm going to call it,
393
00:23:24,996 --> 00:23:26,120
what am I going to call it?
394
00:23:26,120 --> 00:23:33,080
Maybe C. So it'll
have the same u.
395
00:23:33,080 --> 00:23:39,580
And the same v. But
I'm going to change w.
396
00:23:39,580 --> 00:23:41,430
And that's going to
make all the difference.
397
00:23:41,430 --> 00:23:48,630
My w, I'm going to
make that into w.
398
00:23:48,630 --> 00:23:52,700
So now I have three vectors.
399
00:23:52,700 --> 00:23:55,260
I can take their combinations.
400
00:23:55,260 --> 00:23:58,920
I can look at the
equation Cx=b .
401
00:23:58,920 --> 00:24:00,710
I can try to solve it.
402
00:24:00,710 --> 00:24:06,020
All the normal stuff
with those combinations
403
00:24:06,020 --> 00:24:09,200
of those three vectors.
404
00:24:09,200 --> 00:24:12,770
And we'll see a difference.
405
00:24:12,770 --> 00:24:15,880
So now, what happens
if I do-- Could
406
00:24:15,880 --> 00:24:24,160
I even like do just a little
erase to deal with C now?
407
00:24:24,160 --> 00:24:26,010
How does C differ?
408
00:24:26,010 --> 00:24:31,620
If I change this
multiplication from A
409
00:24:31,620 --> 00:24:34,420
to C, to this new matrix.
410
00:24:34,420 --> 00:24:38,690
Then what we've done is to put
in a minus one there, right?
411
00:24:38,690 --> 00:24:41,390
That's the only change we made.
412
00:24:41,390 --> 00:24:49,710
And what's the change in Cx?
413
00:24:49,710 --> 00:24:51,350
I've changed the
first row, so I'm
414
00:24:51,350 --> 00:24:56,720
going to change the first row of
the answer to what? x_1 - x_3.
415
00:25:04,110 --> 00:25:06,270
You could say again,
as I said this morning,
416
00:25:06,270 --> 00:25:09,050
you've sort of changed the
boundary condition maybe.
417
00:25:09,050 --> 00:25:14,540
You've made this difference
equation somehow circular.
418
00:25:14,540 --> 00:25:23,370
That's why I'm
using that letter C.
419
00:25:23,370 --> 00:25:25,560
Is it different?
420
00:25:25,560 --> 00:25:26,920
Ah, yes!
421
00:25:26,920 --> 00:25:29,240
I didn't get it right here.
422
00:25:29,240 --> 00:25:33,820
Thank you, thank you very much.
423
00:25:33,820 --> 00:25:35,147
Absolutely.
424
00:25:35,147 --> 00:25:37,480
I mean that would have been
another matrix that we could
425
00:25:37,480 --> 00:25:39,021
think about but it
wouldn't have made
426
00:25:39,021 --> 00:25:42,560
the point I wanted, so thanks,
that's absolutely great.
427
00:25:42,560 --> 00:25:48,260
So now it's correct
here and this is correct
428
00:25:48,260 --> 00:25:53,970
and I can look at equations
but can I solve them?
429
00:25:53,970 --> 00:25:56,320
Can I solve them?
430
00:25:56,320 --> 00:26:00,170
And you're guessing
already, no we can't do it.
431
00:26:00,170 --> 00:26:02,380
Right?
432
00:26:02,380 --> 00:26:07,480
So now let me maybe
go to a board, work
433
00:26:07,480 --> 00:26:11,440
below, because I'd hate to
erase, that was so great,
434
00:26:11,440 --> 00:26:14,490
that being able to solve
it in a nice clear solution
435
00:26:14,490 --> 00:26:17,010
and some matrix coming in.
436
00:26:17,010 --> 00:26:19,090
But now, how about this one?
437
00:26:19,090 --> 00:26:24,370
Okay.
438
00:26:24,370 --> 00:26:27,480
One comment I should
have made here.
439
00:26:27,480 --> 00:26:30,640
Suppose the b's were zero.
440
00:26:30,640 --> 00:26:32,610
Suppose I was looking
at, originally
441
00:26:32,610 --> 00:26:38,290
at A times x equal
all zeroes, What's x?
442
00:26:38,290 --> 00:26:41,600
If all the b's
were zero in this,
443
00:26:41,600 --> 00:26:46,220
this was the one that dealt with
the matrix A. If all the b's
444
00:26:46,220 --> 00:26:50,280
are zero then the x's are zero.
445
00:26:50,280 --> 00:26:53,930
The only way to get
zero right-hand sides,
446
00:26:53,930 --> 00:26:58,630
b's, was to have zero x's.
447
00:26:58,630 --> 00:27:01,670
Right?
448
00:27:01,670 --> 00:27:06,080
If you wanted to get zero
out, you had to put zero in.
449
00:27:06,080 --> 00:27:08,360
Well, you can always put
zero in and get zero out,
450
00:27:08,360 --> 00:27:12,250
but here you can put other
vectors in and get zero out.
451
00:27:12,250 --> 00:27:17,320
So I want to say there's
a solution with zeroes
452
00:27:17,320 --> 00:27:22,230
out, coming out of C, but
some non-zeroes going in.
453
00:27:22,230 --> 00:27:25,630
And of course we know
from this morning
454
00:27:25,630 --> 00:27:29,900
that that's a signal that it's
a different sort of matrix,
455
00:27:29,900 --> 00:27:36,510
there won't be an inverse,
we've got questions.
456
00:27:36,510 --> 00:27:39,260
Tell me all the solutions.
457
00:27:39,260 --> 00:27:42,780
All the solutions, so actually
not just one, well you could
458
00:27:42,780 --> 00:27:44,177
tell me one, tell me one first.
459
00:27:44,177 --> 00:27:45,260
AUDIENCE: [UNINTELLIGIBLE]
460
00:27:45,260 --> 00:27:46,426
PROFESSOR STRANG: [1, 1, 1].
461
00:27:46,426 --> 00:27:47,260
Okay.
462
00:27:47,260 --> 00:27:48,290
Now tell me all.
463
00:27:48,290 --> 00:27:50,754
AUDIENCE: C, C, C.
464
00:27:50,754 --> 00:27:51,920
PROFESSOR STRANG: [C, C, C].
465
00:27:51,920 --> 00:27:52,450
Yeah.
466
00:27:52,450 --> 00:27:55,230
That whole line
through [1, 1, 1].
467
00:27:55,230 --> 00:27:57,910
And that would be normal.
468
00:27:57,910 --> 00:28:01,380
So this is a line of solutions.
469
00:28:01,380 --> 00:28:01,880
Right.
470
00:28:01,880 --> 00:28:03,110
A line of a solutions.
471
00:28:03,110 --> 00:28:08,190
I think of [1, 1, 1] as in some
solution space, and then all
472
00:28:08,190 --> 00:28:08,710
multiples.
473
00:28:08,710 --> 00:28:10,550
That whole line.
474
00:28:10,550 --> 00:28:13,470
Later I would say
it's a subspace.
475
00:28:13,470 --> 00:28:16,570
When I say what that
word subspace means
476
00:28:16,570 --> 00:28:20,780
it's just this--
linear algebra's
477
00:28:20,780 --> 00:28:23,810
done its job beyond
just [1, 1, 1].
478
00:28:23,810 --> 00:28:25,030
Okay.
479
00:28:25,030 --> 00:28:33,510
So, again, it's this
fact of-- if we only
480
00:28:33,510 --> 00:28:38,290
know the differences-- Yeah.
481
00:28:38,290 --> 00:28:42,040
You can see different ways
that this has got problems.
482
00:28:42,040 --> 00:28:43,980
So that's C times x.
483
00:28:43,980 --> 00:28:48,520
Now one way to see
a problem is to say
484
00:28:48,520 --> 00:28:52,020
we can get the answer of all
zeroes by putting constants.
485
00:28:52,020 --> 00:28:55,480
All that's saying in
words the differences
486
00:28:55,480 --> 00:28:58,720
of a constant factor
are all zeroes, right?
487
00:28:58,720 --> 00:29:00,610
That's all that happened.
488
00:29:00,610 --> 00:29:06,410
Another way to see a problem if
I had this system of equations,
489
00:29:06,410 --> 00:29:08,470
how would you see that
there's a problem,
490
00:29:08,470 --> 00:29:10,960
and how would you see that
there is sometimes an answer
491
00:29:10,960 --> 00:29:13,270
and even decide when?
492
00:29:13,270 --> 00:29:17,360
I don't know if you
can take a quick look.
493
00:29:17,360 --> 00:29:20,990
If I had three equations,
x_1-x_3 is b_1,
494
00:29:20,990 --> 00:29:23,150
this equals b_2,
this equals b_3.
495
00:29:27,330 --> 00:29:32,010
Do you see something that I
can do to the left sides that's
496
00:29:32,010 --> 00:29:36,070
important somehow?
497
00:29:36,070 --> 00:29:39,510
Suppose I add those
left-hand sides.
498
00:29:39,510 --> 00:29:41,040
What do I get?
499
00:29:41,040 --> 00:29:42,510
And I'm allowed
to do that, right?
500
00:29:42,510 --> 00:29:45,510
If I've got three equations
I'm allowed to add them,
501
00:29:45,510 --> 00:29:50,970
and I would get zero, if I
add, I get zero equals --
502
00:29:50,970 --> 00:29:53,370
I have to add the right-sides
of course -- b_1+b_2+b_3.
503
00:29:57,810 --> 00:30:00,890
I hesitate to say a fourth
equation because it's not
504
00:30:00,890 --> 00:30:02,990
independent of those
three, but it's
505
00:30:02,990 --> 00:30:04,930
a consequence of those three.
506
00:30:04,930 --> 00:30:11,470
So actually this is telling
me when I could get an answer
507
00:30:11,470 --> 00:30:14,390
and when I couldn't.
508
00:30:14,390 --> 00:30:16,500
If I get zero on
the left side I have
509
00:30:16,500 --> 00:30:19,540
to have zero on the
right side or I'm lost.
510
00:30:19,540 --> 00:30:23,640
So I could actually solve
this when b_1+b_2+b_3=0.
511
00:30:30,920 --> 00:30:33,530
So I've taken a step there.
512
00:30:33,530 --> 00:30:37,210
I've said that okay,
we're in trouble often,
513
00:30:37,210 --> 00:30:42,300
but in case the right-side
adds up to zero then we're not.
514
00:30:42,300 --> 00:30:47,670
And if you'll allow me to
jump to a mechanical meaning
515
00:30:47,670 --> 00:30:53,940
of this, if these were
springs or something, masses,
516
00:30:53,940 --> 00:30:58,210
and these were forces on
them -- so I'm solving
517
00:30:58,210 --> 00:31:02,150
for displacements of masses
that we'll see very soon,
518
00:31:02,150 --> 00:31:07,790
and these are forces -- what
that equation is saying is,
519
00:31:07,790 --> 00:31:12,200
because they're sorta cyclical,
it's somehow saying that
520
00:31:12,200 --> 00:31:17,230
if the forces add up to zero,
if the resulting force is zero,
521
00:31:17,230 --> 00:31:19,160
then you're okay.
522
00:31:19,160 --> 00:31:23,060
The springs and masses don't
like take off, or start
523
00:31:23,060 --> 00:31:25,750
spinning or whatever.
524
00:31:25,750 --> 00:31:30,030
So there's a physical
meaning for that condition
525
00:31:30,030 --> 00:31:35,730
that it's okay provided,
if, the b's add up to zero.
526
00:31:35,730 --> 00:31:38,850
But of course, if the b's don't
add up to zero we're lost.
527
00:31:38,850 --> 00:31:40,790
Right yeah.
528
00:31:40,790 --> 00:31:42,070
Okay.
529
00:31:42,070 --> 00:31:52,200
So Cx=b could be solved
sometimes, but not always.
530
00:31:52,200 --> 00:31:55,520
The difficulty with C is
showing up several ways.
531
00:31:55,520 --> 00:32:00,320
It's showing up in C times
a vector x giving zero.
532
00:32:00,320 --> 00:32:02,830
That's bad news.
533
00:32:02,830 --> 00:32:05,430
Because no C inverse
can bring you back.
534
00:32:05,430 --> 00:32:07,650
I mean it's like you
can't come back from zero.
535
00:32:07,650 --> 00:32:11,600
Once you get to zero, C inverse
can never bring you back
536
00:32:11,600 --> 00:32:14,010
to x, right?
537
00:32:14,010 --> 00:32:21,220
A took x into b up there, and
then A inverse brought back x.
538
00:32:21,220 --> 00:32:23,150
But here there's no
way to bring back
539
00:32:23,150 --> 00:32:25,620
that x because I can't
multiply zero by anything
540
00:32:25,620 --> 00:32:27,380
and get back to x.
541
00:32:27,380 --> 00:32:30,150
So that's why I see
it's got troubles here.
542
00:32:30,150 --> 00:32:33,410
Here I see it's got troubles
because if I add the left sides
543
00:32:33,410 --> 00:32:35,210
I get zero.
544
00:32:35,210 --> 00:32:37,590
And therefore the right
sides must add to zero.
545
00:32:37,590 --> 00:32:41,220
So you've got
trouble several ways.
546
00:32:41,220 --> 00:32:45,290
Ah, let's see another way,
let's see geometrically
547
00:32:45,290 --> 00:32:46,890
why were in trouble.
548
00:32:46,890 --> 00:32:53,930
Okay, so let me draw a picture
to go with that picture.
549
00:32:53,930 --> 00:32:56,880
So there's
three-dimensional space.
550
00:32:56,880 --> 00:33:00,940
I didn't change u,
I didn't change v,
551
00:33:00,940 --> 00:33:06,450
but I changed w to minus
one, what does that mean?
552
00:33:06,450 --> 00:33:09,840
Minus one sort of going
this way maybe, zero,
553
00:33:09,840 --> 00:33:13,670
one is the z direction,
somehow I changed it to there.
554
00:33:13,670 --> 00:33:17,630
So this is w star
maybe, a different w.
555
00:33:17,630 --> 00:33:23,330
This is the w that
gave me problems.
556
00:33:23,330 --> 00:33:26,160
What's the problem?
557
00:33:26,160 --> 00:33:37,170
How does the picture
show the problem?
558
00:33:37,170 --> 00:33:41,030
What's the problem with those
three vectors, those three
559
00:33:41,030 --> 00:33:45,730
columns of C?
560
00:33:45,730 --> 00:33:46,230
Yeah?
561
00:33:46,230 --> 00:33:47,380
AUDIENCE: [UNINTELLIGIBLE]
562
00:33:47,380 --> 00:33:49,580
PROFESSOR STRANG: There
in the same plane.
563
00:33:49,580 --> 00:33:53,560
There in the same plane.
w* gave us nothing new.
564
00:33:53,560 --> 00:33:56,690
We had the combinations
of u and v made a plane,
565
00:33:56,690 --> 00:34:00,280
and w* happened to
fall in that plane.
566
00:34:00,280 --> 00:34:06,360
So this is a plane
here somehow, and goes
567
00:34:06,360 --> 00:34:09,340
through the origin of course.
568
00:34:09,340 --> 00:34:10,310
What is that plane?
569
00:34:10,310 --> 00:34:17,830
This is all combinations,
all combinations of u, v,
570
00:34:17,830 --> 00:34:21,370
and the third guy, w*.
571
00:34:21,370 --> 00:34:22,040
Right.
572
00:34:22,040 --> 00:34:23,950
It's a plane, and
I drew a triangle,
573
00:34:23,950 --> 00:34:28,200
but of course, I should draw
the plane goes out to infinity.
574
00:34:28,200 --> 00:34:32,060
But the point is there
are lots of b's, lots
575
00:34:32,060 --> 00:34:36,590
of right-hand sides
not on that plane.
576
00:34:36,590 --> 00:34:37,190
Okay.
577
00:34:37,190 --> 00:34:43,830
Now if I drew all combinations
of u, v, w, the original w,
578
00:34:43,830 --> 00:34:45,180
what have I got?
579
00:34:45,180 --> 00:34:48,750
So let me bring that
picture back for a moment.
580
00:34:48,750 --> 00:34:51,020
If I took all
combinations of those
581
00:34:51,020 --> 00:34:55,560
does w lie in the
plane of u and v?
582
00:34:55,560 --> 00:34:56,580
No, right?
583
00:34:56,580 --> 00:34:58,760
I would call it independent.
584
00:34:58,760 --> 00:35:00,900
These three vectors
are independent.
585
00:35:00,900 --> 00:35:05,370
These three, u, v, and w*
I would call dependent.
586
00:35:05,370 --> 00:35:09,670
Because the third guy was a
combination of the first two.
587
00:35:09,670 --> 00:35:13,310
Okay, so tell me
what do I get now?
588
00:35:13,310 --> 00:35:16,610
So now you're really
up to 3-D. What
589
00:35:16,610 --> 00:35:20,580
do you get if you take all
combinations of u, v, and w?
590
00:35:20,580 --> 00:35:23,640
AUDIENCE: [INAUDIBLE].
591
00:35:23,640 --> 00:35:25,500
PROFESSOR STRANG: Say it again.
592
00:35:25,500 --> 00:35:29,130
The whole space.
593
00:35:29,130 --> 00:35:31,495
If taking all
combinations of u, v, w
594
00:35:31,495 --> 00:35:33,390
will give you the whole space.
595
00:35:33,390 --> 00:35:34,550
Why is that?
596
00:35:34,550 --> 00:35:38,620
Well we just showed--
when it was A we
597
00:35:38,620 --> 00:35:43,500
showed that we
could get every b.
598
00:35:43,500 --> 00:35:48,330
We wanted the combination
that gave b and we found it.
599
00:35:48,330 --> 00:35:53,990
So in the beginning when we
were working with u, v, w,
600
00:35:53,990 --> 00:36:01,330
we found -- and this
was short hand here --
601
00:36:01,330 --> 00:36:04,200
this said find a
combination to give b,
602
00:36:04,200 --> 00:36:06,830
and this says that
combination will work.
603
00:36:06,830 --> 00:36:09,310
And we wrote out what x was.
604
00:36:09,310 --> 00:36:13,170
Now what's the
difference-- Okay, here.
605
00:36:13,170 --> 00:36:21,080
So those were dependent,
sorry, those were independent.
606
00:36:21,080 --> 00:36:24,580
I would even call those
three vectors a basis
607
00:36:24,580 --> 00:36:25,860
for three-dimensional space.
608
00:36:25,860 --> 00:36:28,510
That word "basis" is a big deal.
609
00:36:28,510 --> 00:36:31,065
So a basis for
five-dimensional space
610
00:36:31,065 --> 00:36:35,610
is five vectors that
are independent.
611
00:36:35,610 --> 00:36:37,190
That's one way to say it.
612
00:36:37,190 --> 00:36:39,690
The second way to say it
would be their combinations
613
00:36:39,690 --> 00:36:42,390
give the whole
five-dimensional space.
614
00:36:42,390 --> 00:36:45,590
A third way to say it-- See if
you can finish this sentence.
615
00:36:45,590 --> 00:36:48,140
This is for the
independent, the good guys.
616
00:36:48,140 --> 00:36:54,000
If I put those five vectors
into a five by five matrix,
617
00:36:54,000 --> 00:37:01,530
that matrix will
be... invertible.
618
00:37:01,530 --> 00:37:04,060
That matrix will be invertible.
619
00:37:04,060 --> 00:37:06,650
So an invertible matrix
is one with a basis
620
00:37:06,650 --> 00:37:09,240
sitting in its columns.
621
00:37:09,240 --> 00:37:12,810
It's a transform that
has an inverse transform.
622
00:37:12,810 --> 00:37:16,300
This matrix is not invertible,
those three vectors
623
00:37:16,300 --> 00:37:17,970
are not a basis.
624
00:37:17,970 --> 00:37:21,980
Their combinations
are only in a plane.
625
00:37:21,980 --> 00:37:24,390
By the way, a plane
as a subspace.
626
00:37:24,390 --> 00:37:27,490
A plane would be a
typical subspace.
627
00:37:27,490 --> 00:37:29,030
It's like fill it out.
628
00:37:29,030 --> 00:37:31,900
You took all the combinations,
you did your job,
629
00:37:31,900 --> 00:37:37,460
but in that case the whole space
would count as a subspace too.
630
00:37:37,460 --> 00:37:39,540
That's the way
you get subspaces,
631
00:37:39,540 --> 00:37:42,010
by taking all combinations.
632
00:37:42,010 --> 00:37:46,180
Okay, now I'm even going to
push you one more step and then
633
00:37:46,180 --> 00:37:50,900
this example is complete.
634
00:37:50,900 --> 00:37:56,250
Can you tell me what
vectors do you get?
635
00:37:56,250 --> 00:37:58,190
All combinations of u, v, w.
636
00:37:58,190 --> 00:37:59,710
Let me try to write something.
637
00:37:59,710 --> 00:38:08,190
This gives only a plane.
638
00:38:08,190 --> 00:38:10,470
Because we've got two
independent vectors but not
639
00:38:10,470 --> 00:38:12,180
the third.
640
00:38:12,180 --> 00:38:15,370
Okay.
641
00:38:15,370 --> 00:38:17,570
I don't know if I
should even ask.
642
00:38:17,570 --> 00:38:20,960
Do we know an equation
for that plane?
643
00:38:20,960 --> 00:38:25,880
Well I think we do if we
think about it correctly.
644
00:38:25,880 --> 00:38:33,640
All combinations of u, v, w* is
the same as saying all vectors
645
00:38:33,640 --> 00:38:41,610
C times x, right?
646
00:38:41,610 --> 00:38:48,780
Do you agree that those two
are exactly the same thing?
647
00:38:48,780 --> 00:38:51,940
This is the key,
because we're moving up
648
00:38:51,940 --> 00:38:56,670
to vectors, combinations,
and now comes subspaces.
649
00:38:56,670 --> 00:38:59,590
If I take all
combinations of u, v, w*,
650
00:38:59,590 --> 00:39:02,610
I say that that's the same
as all vectors C times x,
651
00:39:02,610 --> 00:39:07,520
why's that?
652
00:39:07,520 --> 00:39:12,670
It's what I said in the very
first sentence at 4 o'clock.
653
00:39:12,670 --> 00:39:17,210
The combinations of u, v,
w*, how do I produce them?
654
00:39:17,210 --> 00:39:21,260
I create the matrix
with those columns.
655
00:39:21,260 --> 00:39:28,150
I multiply them by x's, and
I get all the combinations.
656
00:39:28,150 --> 00:39:31,510
And this is just C times x.
657
00:39:31,510 --> 00:39:35,870
So what I've said there
is just another way
658
00:39:35,870 --> 00:39:38,670
of saying how does matrix
multiplication work.
659
00:39:38,670 --> 00:39:46,440
Put the guys in its columns
and multiply by a vector.
660
00:39:46,440 --> 00:39:49,190
So we're getting all
vectors C times x,
661
00:39:49,190 --> 00:39:55,870
and now I was going to stretch
it that little bit further.
662
00:39:55,870 --> 00:39:58,120
Can we describe
what vectors we get?
663
00:39:58,120 --> 00:40:02,220
So that's my question.
664
00:40:02,220 --> 00:40:09,550
What b's -- so this is b =
[b 1, b 2, b 3] -- do we get?
665
00:40:09,550 --> 00:40:14,912
We don't get them all.
666
00:40:14,912 --> 00:40:16,120
Right, we don't get them all.
667
00:40:16,120 --> 00:40:21,050
That's the trouble with C.
We only get a plane of them.
668
00:40:21,050 --> 00:40:25,440
And now can you
tell me which b's
669
00:40:25,440 --> 00:40:33,480
we do get when we look at all
combinations of these three
670
00:40:33,480 --> 00:40:36,910
dependent vectors.
671
00:40:36,910 --> 00:40:39,850
Well we've done a lot today.
672
00:40:39,850 --> 00:40:42,850
Let me just tell you the
answer because it's here.
673
00:40:42,850 --> 00:40:45,560
The b's have to add to zero.
674
00:40:45,560 --> 00:40:49,400
That's the equation that
the b's have to satisfy.
675
00:40:49,400 --> 00:40:54,820
Because when we wrote out Cx
we noticed that the components
676
00:40:54,820 --> 00:40:58,550
always added to zero.
677
00:40:58,550 --> 00:41:00,270
Which b's do we get?
678
00:41:00,270 --> 00:41:05,260
We get the ones where the
components add to zero.
679
00:41:05,260 --> 00:41:10,170
In other words that's the
equation of the plane,
680
00:41:10,170 --> 00:41:11,000
you could say.
681
00:41:11,000 --> 00:41:11,500
Yeah.
682
00:41:11,500 --> 00:41:13,780
Actually that's a good
way to look at it.
683
00:41:13,780 --> 00:41:19,720
All these vectors
are on the plane.
684
00:41:19,720 --> 00:41:23,520
Do the components
of u add to zero?
685
00:41:23,520 --> 00:41:25,190
Look at u.
686
00:41:25,190 --> 00:41:26,750
Yes.
687
00:41:26,750 --> 00:41:30,930
Do the components
of v add to zero?
688
00:41:30,930 --> 00:41:31,510
Yes.
689
00:41:31,510 --> 00:41:32,580
Add them up.
690
00:41:32,580 --> 00:41:37,010
Do the components of w*, now
that you've fixed it correctly,
691
00:41:37,010 --> 00:41:38,001
do they add to zero?
692
00:41:38,001 --> 00:41:38,500
Yes.
693
00:41:38,500 --> 00:41:40,470
So all the combinations
will add to zero.
694
00:41:40,470 --> 00:41:42,800
That's the plane.
695
00:41:42,800 --> 00:41:44,370
That's the plane.
696
00:41:44,370 --> 00:41:47,880
You see there are so many
different ways to see,
697
00:41:47,880 --> 00:41:50,610
and none of this is
difficult, but it's
698
00:41:50,610 --> 00:41:55,100
coming fast because we're
seeing the same thing
699
00:41:55,100 --> 00:41:56,310
in different languages.
700
00:41:56,310 --> 00:41:59,620
We're seeing it geometrically
in a picture of a plane.
701
00:41:59,620 --> 00:42:02,190
We're seeing it as a
combination of vectors.
702
00:42:02,190 --> 00:42:05,310
We're seeing it as a
multiplication by a matrix.
703
00:42:05,310 --> 00:42:11,280
And we saw it sort
of here by operation,
704
00:42:11,280 --> 00:42:16,140
operating and simplifying,
and getting the key fact out
705
00:42:16,140 --> 00:42:21,000
of the equations.
706
00:42:21,000 --> 00:42:22,880
Well, okay.
707
00:42:22,880 --> 00:42:28,580
I wanted to give you this
example, the two examples,
708
00:42:28,580 --> 00:42:31,380
because they bring out
so many of the key ideas.
709
00:42:31,380 --> 00:42:34,990
The key idea of a subspace.
710
00:42:34,990 --> 00:42:38,670
Shall I just say a little
about what that word means?
711
00:42:38,670 --> 00:42:41,430
A subspace.
712
00:42:41,430 --> 00:42:43,560
What's a subspace?
713
00:42:43,560 --> 00:42:48,360
Well, what's a vector
space first of all?
714
00:42:48,360 --> 00:42:51,570
A vector space is
a bunch of vectors.
715
00:42:51,570 --> 00:42:54,770
And the rule is you
have to be able to take
716
00:42:54,770 --> 00:42:56,130
their combinations.
717
00:42:56,130 --> 00:42:57,760
That what linear algebra does.
718
00:42:57,760 --> 00:42:59,370
Takes combinations.
719
00:42:59,370 --> 00:43:05,810
So a vector space is one where
you take all combinations.
720
00:43:05,810 --> 00:43:09,690
So if I only took just this
triangle that would not
721
00:43:09,690 --> 00:43:14,380
be a subspace because one
combination would be 2u
722
00:43:14,380 --> 00:43:16,190
and it would be out
of the triangle.
723
00:43:16,190 --> 00:43:22,210
So a subspace, just
think of it as a plane,
724
00:43:22,210 --> 00:43:25,370
but then of course it could
be in higher dimensions.
725
00:43:25,370 --> 00:43:28,010
You know it could be a
7-dimensional subspace
726
00:43:28,010 --> 00:43:30,820
inside a 15-dimensional space.
727
00:43:30,820 --> 00:43:37,780
And I don't know if you're good
at visualizing that, I'm not.
728
00:43:37,780 --> 00:43:38,730
Never mind.
729
00:43:38,730 --> 00:43:41,730
You you've got seven
vectors, you think okay,
730
00:43:41,730 --> 00:43:44,800
their combinations give us
seven-dimensional subspace.
731
00:43:44,800 --> 00:43:47,250
Each factor has 15 components.
732
00:43:47,250 --> 00:43:48,300
No problem.
733
00:43:48,300 --> 00:43:50,110
I mean no problem
for MATLAB certainly.
734
00:43:50,110 --> 00:43:53,080
It's got what, a matrix
with a 105 entries.
735
00:43:53,080 --> 00:43:55,770
It deals with that instantly.
736
00:43:55,770 --> 00:44:02,480
Okay, so a subspace is like a
vector space inside a bigger
737
00:44:02,480 --> 00:44:03,200
one.
738
00:44:03,200 --> 00:44:06,460
That's why the prefix
"sub-" is there.
739
00:44:06,460 --> 00:44:07,160
Right?
740
00:44:07,160 --> 00:44:11,720
And mathematics always counts
the biggest possibility too,
741
00:44:11,720 --> 00:44:13,640
which would be the whole space.
742
00:44:13,640 --> 00:44:15,680
And what's the smallest?
743
00:44:15,680 --> 00:44:19,160
So what's the smallest
subspace of R^3?
744
00:44:19,160 --> 00:44:21,880
So I have 3-dimensional
space-- you can tell me all
745
00:44:21,880 --> 00:44:23,880
the subspaces of R^3.
746
00:44:23,880 --> 00:44:25,980
So there is one, a plane.
747
00:44:25,980 --> 00:44:28,180
Yeah, tell me all
the subspaces of R^3.
748
00:44:28,180 --> 00:44:31,450
And then you'll have
that word kind of down.
749
00:44:31,450 --> 00:44:33,050
AUDIENCE: [UNINTELLIGIBLE]
750
00:44:33,050 --> 00:44:34,660
PROFESSOR STRANG: A line.
751
00:44:34,660 --> 00:44:37,420
So planes and lines,
those you could say,
752
00:44:37,420 --> 00:44:39,490
the real, the proper subspaces.
753
00:44:39,490 --> 00:44:41,330
The best, the right ones.
754
00:44:41,330 --> 00:44:45,230
But there are a couple more
possibilities which are?
755
00:44:45,230 --> 00:44:46,337
AUDIENCE: [UNINTELLIGIBLE]
756
00:44:46,337 --> 00:44:47,420
PROFESSOR STRANG: A point.
757
00:44:47,420 --> 00:44:49,140
Which point?
758
00:44:49,140 --> 00:44:50,010
The origin.
759
00:44:50,010 --> 00:44:50,980
Only the origin.
760
00:44:50,980 --> 00:44:56,020
Because if you tried to say that
point was a subspace, no way.
761
00:44:56,020 --> 00:44:56,910
Why not?
762
00:44:56,910 --> 00:44:58,520
Because I wouldn't
be able to multiply
763
00:44:58,520 --> 00:45:03,870
that vector by five and I
would be away from the point.
764
00:45:03,870 --> 00:45:08,790
But the zero subspace,
the really small subspace
765
00:45:08,790 --> 00:45:12,670
that just has the zero vector,
it's got one vector in it.
766
00:45:12,670 --> 00:45:13,330
Not empty.
767
00:45:13,330 --> 00:45:15,520
It's got that one
point but that's all.
768
00:45:15,520 --> 00:45:20,090
Okay, so planes,
lines, the origin,
769
00:45:20,090 --> 00:45:25,030
and then the other possibility
for a subspaces is?
770
00:45:25,030 --> 00:45:26,000
The whole space.
771
00:45:26,000 --> 00:45:26,520
Right.
772
00:45:26,520 --> 00:45:27,080
Right.
773
00:45:27,080 --> 00:45:30,390
So the dimensions could be
three for the whole space,
774
00:45:30,390 --> 00:45:35,420
two for a plane, one for
a line, zero for a point.
775
00:45:35,420 --> 00:45:41,300
It just kicks together.
776
00:45:41,300 --> 00:45:42,300
How are we for time?
777
00:45:42,300 --> 00:45:48,120
Maybe I went more than half, but
now is a chance to just ask me,
778
00:45:48,120 --> 00:45:53,160
if you want to, like
anything about the course.
779
00:45:53,160 --> 00:45:54,540
Is at all linear algebra?
780
00:45:54,540 --> 00:45:55,380
No.
781
00:45:55,380 --> 00:46:02,520
But I think I can't do
anything more helpful to you
782
00:46:02,520 --> 00:46:06,230
then to for you
to begin to see--
783
00:46:06,230 --> 00:46:09,960
when you look at a matrix,
begin to see what is it doing.
784
00:46:09,960 --> 00:46:11,640
What is it about.
785
00:46:11,640 --> 00:46:14,910
Right, and of course
matrices can be rectangular.
786
00:46:14,910 --> 00:46:17,970
So I'll give you a
hint about what's
787
00:46:17,970 --> 00:46:20,780
coming in the course itself.
788
00:46:20,780 --> 00:46:26,340
We'll have rectangular
matrices A, okay.
789
00:46:26,340 --> 00:46:28,310
They're not invertible.
790
00:46:28,310 --> 00:46:30,760
They're taking
seven-dimensional space
791
00:46:30,760 --> 00:46:33,720
to three-dimensional
space or something.
792
00:46:33,720 --> 00:46:35,750
You can't invert that.
793
00:46:35,750 --> 00:46:40,070
What comes up every time-- I
sort of got the idea finally.
794
00:46:40,070 --> 00:46:45,510
Every time I see a rectangular
matrix, maybe seven by three,
795
00:46:45,510 --> 00:46:48,890
that would be seven
rows by three columns.
796
00:46:48,890 --> 00:46:53,170
Then what comes up with
a rectangular matrix
797
00:46:53,170 --> 00:46:57,820
A is sooner or later A
transpose sticks its nose in
798
00:46:57,820 --> 00:47:07,970
and multiplies that A. And we
couldn't do it for our A here.
799
00:47:07,970 --> 00:47:10,730
Actually if I did it for
that original matrix A
800
00:47:10,730 --> 00:47:14,770
I would get something
you'd recognize.
801
00:47:14,770 --> 00:47:18,790
What I want to say is
that the course focuses
802
00:47:18,790 --> 00:47:23,860
on A transpose A. I'll just
say now that that matrix always
803
00:47:23,860 --> 00:47:27,390
comes out square, because this
would be three times seven
804
00:47:27,390 --> 00:47:31,380
times seven times three, so
this would be three by three,
805
00:47:31,380 --> 00:47:34,300
and it always comes
out symmetric.
806
00:47:34,300 --> 00:47:36,300
That's the nice thing.
807
00:47:36,300 --> 00:47:37,190
And even more.
808
00:47:37,190 --> 00:47:39,760
We'll see more.
809
00:47:39,760 --> 00:47:41,590
That's like a hint.
810
00:47:41,590 --> 00:47:48,320
Watch for A transpose
A. And watch for it
811
00:47:48,320 --> 00:47:52,740
in applications of all kinds.
812
00:47:52,740 --> 00:47:56,740
In networks an A
will be associated
813
00:47:56,740 --> 00:47:59,270
with Kirchhoff's voltage
law, and A transpose
814
00:47:59,270 --> 00:48:00,530
with Kirchhoff's current law.
815
00:48:00,530 --> 00:48:04,720
They just teamed up together.
816
00:48:04,720 --> 00:48:06,780
We'll see more.
817
00:48:06,780 --> 00:48:10,230
Alright now let me give you
a chance to ask any question.
818
00:48:10,230 --> 00:48:14,230
Whatever.
819
00:48:14,230 --> 00:48:15,370
Homework.
820
00:48:15,370 --> 00:48:17,460
Did I mention homework?
821
00:48:17,460 --> 00:48:21,450
You may have said
that's a crazy homework
822
00:48:21,450 --> 00:48:24,730
to say three problems 1.1.
823
00:48:24,730 --> 00:48:30,200
I've never done this
before so essentially you
824
00:48:30,200 --> 00:48:34,380
can get away with
anything this week,
825
00:48:34,380 --> 00:48:37,780
and indefinitely actually.
826
00:48:37,780 --> 00:48:42,300
How many are-- Is this the
first day of MIT classes?
827
00:48:42,300 --> 00:48:43,000
Oh wow.
828
00:48:43,000 --> 00:48:43,840
Okay.
829
00:48:43,840 --> 00:48:46,380
Well, welcome to MIT.
830
00:48:46,380 --> 00:48:49,930
I hope you like it.
831
00:48:49,930 --> 00:48:56,050
It's not so high
pressure or whatever
832
00:48:56,050 --> 00:48:57,560
is associated with MIT.
833
00:48:57,560 --> 00:49:01,450
It's kind of tolerant.
834
00:49:01,450 --> 00:49:04,160
If my advisees ask for
something I always say yes.
835
00:49:04,160 --> 00:49:05,610
It's easier that way.
836
00:49:05,610 --> 00:49:11,030
AUDIENCE: [LAUGHTER].
837
00:49:11,030 --> 00:49:15,400
PROFESSOR STRANG: And
let me just again--
838
00:49:15,400 --> 00:49:19,000
and I'll say it
often and in private.
839
00:49:19,000 --> 00:49:21,170
This is like a grown-up course.
840
00:49:21,170 --> 00:49:23,430
I'm figuring you're
here to learn,
841
00:49:23,430 --> 00:49:26,090
so it's not my job to force it.
842
00:49:26,090 --> 00:49:31,570
My job is to help it, and
hope this is some help.