1
00:00:08,711 --> 00:00:09,210
Hi.
2
00:00:09,210 --> 00:00:14,230
This is the first lecture
in MIT's course 18.06,
3
00:00:14,230 --> 00:00:19,430
linear algebra, and
I'm Gilbert Strang.
4
00:00:19,430 --> 00:00:23,440
The text for the
course is this book,
5
00:00:23,440 --> 00:00:26,070
Introduction to Linear Algebra.
6
00:00:26,070 --> 00:00:32,729
And the course web page, which
has got a lot of exercises from
7
00:00:32,729 --> 00:00:38,470
the past, MatLab codes, the
syllabus for the course,
8
00:00:38,470 --> 00:00:44,060
is web.mit.edu/18.06.
9
00:00:44,060 --> 00:00:47,350
And this is the first
lecture, lecture one.
10
00:00:49,920 --> 00:00:55,490
So, and later we'll give the
web address for viewing these,
11
00:00:55,490 --> 00:00:57,270
videotapes.
12
00:00:57,270 --> 00:00:59,200
Okay, so what's in
the first lecture?
13
00:01:02,680 --> 00:01:05,530
This is my plan.
14
00:01:05,530 --> 00:01:09,340
The fundamental problem
of linear algebra,
15
00:01:09,340 --> 00:01:13,880
which is to solve a system
of linear equations.
16
00:01:13,880 --> 00:01:16,210
So let's start
with a case when we
17
00:01:16,210 --> 00:01:21,440
have some number of equations,
say n equations and n unknowns.
18
00:01:21,440 --> 00:01:23,820
So an equal number of
equations and unknowns.
19
00:01:23,820 --> 00:01:26,860
That's the normal, nice case.
20
00:01:26,860 --> 00:01:31,870
And what I want to do is --
with examples, of course --
21
00:01:31,870 --> 00:01:37,230
to describe, first, what
I call the Row picture.
22
00:01:37,230 --> 00:01:40,870
That's the picture of
one equation at a time.
23
00:01:40,870 --> 00:01:43,980
It's the picture you've
seen before in two
24
00:01:43,980 --> 00:01:46,350
by two equations
where lines meet.
25
00:01:46,350 --> 00:01:49,460
So in a minute, you'll
see lines meeting.
26
00:01:49,460 --> 00:01:52,230
The second picture,
I'll put a star
27
00:01:52,230 --> 00:01:55,070
beside that, because that's
such an important one.
28
00:01:55,070 --> 00:02:01,600
And maybe new to you is the
picture -- a column at a time.
29
00:02:01,600 --> 00:02:05,440
And those are the rows
and columns of a matrix.
30
00:02:05,440 --> 00:02:08,970
So the third -- the algebra
way to look at the problem is
31
00:02:08,970 --> 00:02:13,790
the matrix form and using
a matrix that I'll call A.
32
00:02:13,790 --> 00:02:17,490
Okay, so can I do an example?
33
00:02:17,490 --> 00:02:23,420
The whole semester will be
examples and then see what's
34
00:02:23,420 --> 00:02:25,110
going on with the example.
35
00:02:25,110 --> 00:02:27,990
So, take an example.
36
00:02:27,990 --> 00:02:30,430
Two equations, two unknowns.
37
00:02:30,430 --> 00:02:38,290
So let me take 2x
-y =0, let's say.
38
00:02:38,290 --> 00:02:45,350
And -x 2y=3.
39
00:02:45,350 --> 00:02:46,530
Okay.
40
00:02:46,530 --> 00:02:50,680
let me -- I can even
say right away --
41
00:02:50,680 --> 00:02:57,650
what's the matrix, that is,
what's the coefficient matrix?
42
00:02:57,650 --> 00:03:00,590
The matrix that involves
these numbers --
43
00:03:00,590 --> 00:03:05,190
a matrix is just a
rectangular array of numbers.
44
00:03:05,190 --> 00:03:09,610
Here it's two rows and
two columns, so 2 and --
45
00:03:09,610 --> 00:03:14,420
minus 1 in the first row minus
1 and 2 in the second row,
46
00:03:14,420 --> 00:03:16,700
that's the matrix.
47
00:03:16,700 --> 00:03:21,610
And the right-hand
-- the, unknown --
48
00:03:21,610 --> 00:03:23,090
well, we've got two unknowns.
49
00:03:23,090 --> 00:03:27,370
So we've got a vector, with
two components, x and y,
50
00:03:27,370 --> 00:03:31,150
and we've got two right-hand
sides that go into a vector
51
00:03:31,150 --> 00:03:32,940
0 3.
52
00:03:32,940 --> 00:03:38,260
I couldn't resist writing
the matrix form, right --
53
00:03:38,260 --> 00:03:41,050
even before the pictures.
54
00:03:41,050 --> 00:03:45,030
So I always will think
of this as the matrix A,
55
00:03:45,030 --> 00:03:49,440
the matrix of coefficients,
then there's a vector of
56
00:03:49,440 --> 00:03:50,030
unknowns.
57
00:03:50,030 --> 00:03:51,790
Here we've only
got two unknowns.
58
00:03:51,790 --> 00:03:54,100
Later we'll have any
number of unknowns.
59
00:03:54,100 --> 00:03:57,620
And that vector of
unknowns, well I'll often --
60
00:03:57,620 --> 00:03:59,820
I'll make that x --
61
00:03:59,820 --> 00:04:01,150
extra bold.
62
00:04:01,150 --> 00:04:04,850
A and the right-hand
side is also a vector
63
00:04:04,850 --> 00:04:08,230
that I'll always call b.
64
00:04:08,230 --> 00:04:13,790
So linear equations are A
x equal b and the idea now
65
00:04:13,790 --> 00:04:16,820
is to solve this
particular example
66
00:04:16,820 --> 00:04:20,940
and then step back to
see the bigger picture.
67
00:04:20,940 --> 00:04:24,630
Okay, what's the picture for
this example, the Row picture?
68
00:04:24,630 --> 00:04:26,740
Okay, so here comes
the Row picture.
69
00:04:30,210 --> 00:04:32,890
So that means I take
one row at a time
70
00:04:32,890 --> 00:04:38,520
and I'm drawing
here the xy plane
71
00:04:38,520 --> 00:04:43,990
and I'm going to plot
all the points that
72
00:04:43,990 --> 00:04:47,860
satisfy that first equation.
73
00:04:47,860 --> 00:04:53,120
So I'm looking at all the
points that satisfy 2x-y =0.
74
00:04:53,120 --> 00:04:59,000
It's often good to start with
which point on the horizontal
75
00:04:59,000 --> 00:05:02,480
line -- on this horizontal
line, y is zero.
76
00:05:02,480 --> 00:05:06,510
The x axis has y as zero and
that -- in this case, actually,
77
00:05:06,510 --> 00:05:07,720
then x is zero.
78
00:05:07,720 --> 00:05:11,230
So the point, the origin --
79
00:05:11,230 --> 00:05:16,560
the point with coordinates
(0,0) is on the line.
80
00:05:16,560 --> 00:05:18,290
It solves that equation.
81
00:05:18,290 --> 00:05:22,340
Okay, tell me in -- well,
I guess I have to tell you
82
00:05:22,340 --> 00:05:26,980
another point that solves
this same equation.
83
00:05:26,980 --> 00:05:32,090
Let me suppose x is one,
so I'll take x to be one.
84
00:05:32,090 --> 00:05:34,930
Then y should be two, right?
85
00:05:34,930 --> 00:05:40,250
So there's the point one two
that also solves this equation.
86
00:05:40,250 --> 00:05:42,840
And I could put in more points.
87
00:05:42,840 --> 00:05:45,720
But, but let me put
in all the points
88
00:05:45,720 --> 00:05:50,810
at once, because they all
lie on a straight line.
89
00:05:50,810 --> 00:05:53,540
This is a linear equation
and that word linear
90
00:05:53,540 --> 00:05:54,310
got the letters
91
00:05:54,310 --> 00:05:56,760
Okay, thanks. for line in it.
92
00:05:56,760 --> 00:05:59,760
That's the equation --
this is the line that ...
93
00:05:59,760 --> 00:06:08,690
of solutions to 2x-y=0 my
first row, first equation.
94
00:06:08,690 --> 00:06:14,060
So typically, maybe, x equal
a half, y equal one will work.
95
00:06:14,060 --> 00:06:15,660
And sure enough it does.
96
00:06:15,660 --> 00:06:17,600
Okay, that's the first one.
97
00:06:17,600 --> 00:06:22,750
Now the second one is not
going to go through the origin.
98
00:06:22,750 --> 00:06:24,390
It's always important.
99
00:06:24,390 --> 00:06:26,900
Do we go through
the origin or not?
100
00:06:26,900 --> 00:06:30,020
In this case, yes, because
there's a zero over there.
101
00:06:30,020 --> 00:06:32,270
In this case we don't
go through the origin,
102
00:06:32,270 --> 00:06:36,220
because if x and y are
zero, we don't get three.
103
00:06:36,220 --> 00:06:40,350
So, let me again say
suppose y is zero,
104
00:06:40,350 --> 00:06:42,560
what x do we actually get?
105
00:06:42,560 --> 00:06:48,230
If y is zero, then I
get x is minus three.
106
00:06:48,230 --> 00:06:52,610
So if y is zero, I
go along minus three.
107
00:06:52,610 --> 00:06:56,360
So there's one point
on this second line.
108
00:06:56,360 --> 00:07:03,450
Now let me say, well,
suppose x is minus one --
109
00:07:03,450 --> 00:07:05,370
just to take another x.
110
00:07:05,370 --> 00:07:09,280
If x is minus one,
then this is a one
111
00:07:09,280 --> 00:07:15,870
and I think y should be a one,
because if x is minus one,
112
00:07:15,870 --> 00:07:19,790
then I think y should be a
one and we'll get that point.
113
00:07:19,790 --> 00:07:20,900
Is that right?
114
00:07:20,900 --> 00:07:23,330
If x is minus one, that's a one.
115
00:07:23,330 --> 00:07:25,640
If y is a one, that's
a two and the one
116
00:07:25,640 --> 00:07:28,550
and the two make three and
that point's on the equation.
117
00:07:28,550 --> 00:07:29,330
Okay.
118
00:07:29,330 --> 00:07:31,740
Now, I should just
draw the line, right,
119
00:07:31,740 --> 00:07:34,470
connecting those
two points at --
120
00:07:34,470 --> 00:07:36,410
that will give me
the whole line.
121
00:07:36,410 --> 00:07:40,480
And if I've done
this reasonably well,
122
00:07:40,480 --> 00:07:42,640
I think it's going to happen
to go through -- well,
123
00:07:42,640 --> 00:07:46,520
not happen -- it was arranged
to go through that point.
124
00:07:46,520 --> 00:07:50,600
So I think that the
second line is this one,
125
00:07:50,600 --> 00:07:56,100
and this is the all-important
point that lies on both lines.
126
00:07:56,100 --> 00:07:58,830
Shall we just check
that that point which
127
00:07:58,830 --> 00:08:03,690
is the point x equal one
and y was two, right?
128
00:08:03,690 --> 00:08:07,790
That's the point there
and that, I believe,
129
00:08:07,790 --> 00:08:10,530
solves both equations.
130
00:08:10,530 --> 00:08:11,500
Let's just check this.
131
00:08:11,500 --> 00:08:18,590
If x is one, I have a minus one
plus four equals three, okay.
132
00:08:18,590 --> 00:08:21,680
Apologies for
drawing this picture
133
00:08:21,680 --> 00:08:25,150
that you've seen before.
134
00:08:25,150 --> 00:08:29,690
But this -- seeing
the row picture --
135
00:08:29,690 --> 00:08:33,140
first of all, for n equal 2,
two equations and two unknowns,
136
00:08:33,140 --> 00:08:35,460
it's the right place to start.
137
00:08:35,460 --> 00:08:36,270
Okay.
138
00:08:36,270 --> 00:08:37,690
So we've got the solution.
139
00:08:37,690 --> 00:08:39,679
The point that
lies on both lines.
140
00:08:39,679 --> 00:08:43,600
Now can I come to
the column picture?
141
00:08:43,600 --> 00:08:47,880
Pay attention, this
is the key point.
142
00:08:47,880 --> 00:08:49,930
So the column picture.
143
00:08:49,930 --> 00:08:53,520
I'm now going to look at
the columns of the matrix.
144
00:08:53,520 --> 00:08:56,790
I'm going to look at
this part and this part.
145
00:08:56,790 --> 00:09:05,010
I'm going to say that the
x part is really x times --
146
00:09:05,010 --> 00:09:06,740
you see, I'm putting the two --
147
00:09:06,740 --> 00:09:10,160
I'm kind of getting the
two equations at once --
148
00:09:10,160 --> 00:09:15,290
that part and then I have a
y and in the first equation
149
00:09:15,290 --> 00:09:19,260
it's multiplying a minus one and
in the second equation a two,
150
00:09:19,260 --> 00:09:22,030
and on the right-hand
side, zero and three.
151
00:09:26,500 --> 00:09:29,860
You see, the columns of the
matrix, the columns of A
152
00:09:29,860 --> 00:09:33,050
are here and the
right-hand side b is there.
153
00:09:33,050 --> 00:09:36,940
And now what is the
equation asking for?
154
00:09:36,940 --> 00:09:42,750
It's asking us to find --
somehow to combine that vector
155
00:09:42,750 --> 00:09:47,470
and this one in the right
amounts to get that one.
156
00:09:47,470 --> 00:09:51,830
It's asking us to find the
right linear combination --
157
00:09:51,830 --> 00:09:55,420
this is called a
linear combination.
158
00:09:55,420 --> 00:09:58,840
And it's the most fundamental
operation in the whole course.
159
00:10:01,710 --> 00:10:06,330
It's a linear combination
of the columns.
160
00:10:06,330 --> 00:10:09,550
That's what we're
seeing on the left side.
161
00:10:09,550 --> 00:10:13,190
Again, I don't want to
write down a big definition.
162
00:10:13,190 --> 00:10:15,010
You can see what it is.
163
00:10:15,010 --> 00:10:17,470
There's column one,
there's column two.
164
00:10:17,470 --> 00:10:21,690
I multiply by some
numbers and I add.
165
00:10:21,690 --> 00:10:25,940
That's a combination -- a linear
combination and I want to make
166
00:10:25,940 --> 00:10:30,870
those numbers the right
numbers to produce zero three.
167
00:10:30,870 --> 00:10:32,070
Okay.
168
00:10:32,070 --> 00:10:37,110
Now I want to draw a picture
that, represents what this --
169
00:10:37,110 --> 00:10:38,230
this is algebra.
170
00:10:38,230 --> 00:10:41,740
What's the geometry, what's
the picture that goes with it?
171
00:10:41,740 --> 00:10:42,360
Okay.
172
00:10:42,360 --> 00:10:45,260
So again, these vectors
have two components,
173
00:10:45,260 --> 00:10:50,730
so I better draw a
picture like that.
174
00:10:50,730 --> 00:10:52,680
So can I put down these columns?
175
00:10:52,680 --> 00:10:55,470
I'll draw these
columns as they are,
176
00:10:55,470 --> 00:10:58,700
and then I'll do a
combination of them.
177
00:10:58,700 --> 00:11:04,000
So the first column is over
two and down one, right?
178
00:11:04,000 --> 00:11:06,210
So there's the first column.
179
00:11:06,210 --> 00:11:07,860
The first column.
180
00:11:07,860 --> 00:11:10,960
Column one.
181
00:11:10,960 --> 00:11:13,830
It's the vector two minus one.
182
00:11:13,830 --> 00:11:18,580
The second column is --
183
00:11:18,580 --> 00:11:22,000
minus one is the first
component and up two.
184
00:11:22,000 --> 00:11:24,960
It's here.
185
00:11:24,960 --> 00:11:27,710
There's column two.
186
00:11:27,710 --> 00:11:32,280
So this, again, you see
what its components are.
187
00:11:32,280 --> 00:11:35,530
Its components are
minus one, two.
188
00:11:35,530 --> 00:11:36,030
Good.
189
00:11:36,030 --> 00:11:36,840
That's this guy.
190
00:11:39,471 --> 00:11:40,845
Now I have to take
a combination.
191
00:11:43,740 --> 00:11:47,310
What combination shall I take?
192
00:11:47,310 --> 00:11:50,510
Why not the right
combination, what the hell?
193
00:11:50,510 --> 00:11:51,010
Okay.
194
00:11:51,010 --> 00:11:53,840
So the combination
I'm going to take
195
00:11:53,840 --> 00:11:57,250
is the right one to
produce zero three
196
00:11:57,250 --> 00:11:59,620
and then we'll see it
happen in the picture.
197
00:11:59,620 --> 00:12:04,530
So the right combination is
to take x as one of those
198
00:12:04,530 --> 00:12:06,140
and two of these.
199
00:12:09,230 --> 00:12:12,760
It's because we already know
that that's the right x and y,
200
00:12:12,760 --> 00:12:15,720
so why not take the correct
combination here and see it
201
00:12:15,720 --> 00:12:16,280
happen?
202
00:12:16,280 --> 00:12:21,840
Okay, so how do I picture
this linear combination?
203
00:12:21,840 --> 00:12:25,180
So I start with this vector
that's already here --
204
00:12:25,180 --> 00:12:28,220
so that's one of column
one, that's one times column
205
00:12:28,220 --> 00:12:30,400
one, right there.
206
00:12:30,400 --> 00:12:33,720
And now I want to add on --
so I'm going to hook the next
207
00:12:33,720 --> 00:12:38,660
vector onto the front of the
arrow will start the next
208
00:12:38,660 --> 00:12:40,590
vector and it will go this way.
209
00:12:40,590 --> 00:12:44,170
So let's see, can I do it right?
210
00:12:44,170 --> 00:12:46,240
If I added on one
of these vectors,
211
00:12:46,240 --> 00:12:50,589
it would go left one and up two,
so we'd go left one and up two,
212
00:12:50,589 --> 00:12:52,130
so it would probably
get us to there.
213
00:12:52,130 --> 00:12:54,650
Maybe I'll do dotted
line for that.
214
00:12:54,650 --> 00:12:55,610
Okay?
215
00:12:55,610 --> 00:12:59,710
That's one of column
two tucked onto the end,
216
00:12:59,710 --> 00:13:03,390
but I wanted to tuck
on two of column two.
217
00:13:03,390 --> 00:13:07,000
So that -- the second one --
we'll go up left one and up two
218
00:13:07,000 --> 00:13:07,660
also.
219
00:13:07,660 --> 00:13:09,440
It'll probably end there.
220
00:13:09,440 --> 00:13:10,520
And there's another one.
221
00:13:10,520 --> 00:13:14,350
So what I've put in here
is two of column two.
222
00:13:17,310 --> 00:13:19,340
Added on.
223
00:13:19,340 --> 00:13:20,615
And where did I end up?
224
00:13:23,280 --> 00:13:28,230
What are the coordinates
of this result?
225
00:13:28,230 --> 00:13:32,570
What do I get when I take
one of this plus two of that?
226
00:13:32,570 --> 00:13:34,810
I do get that, of course.
227
00:13:34,810 --> 00:13:38,870
There it is, x is zero,
y is three, that's b.
228
00:13:38,870 --> 00:13:41,530
That's the answer we wanted.
229
00:13:41,530 --> 00:13:43,530
And how do I do it?
230
00:13:43,530 --> 00:13:46,860
You see I do it just
like the first component.
231
00:13:46,860 --> 00:13:50,700
I have a two and a minus
two that produces a zero,
232
00:13:50,700 --> 00:13:54,680
and in the second component I
have a minus one and a four,
233
00:13:54,680 --> 00:13:57,750
they combine to give the three.
234
00:13:57,750 --> 00:14:01,190
But look at this picture.
235
00:14:01,190 --> 00:14:03,230
So here's our key picture.
236
00:14:03,230 --> 00:14:10,280
I combine this column and
this column to get this guy.
237
00:14:10,280 --> 00:14:11,360
That was the b.
238
00:14:11,360 --> 00:14:13,680
That's the zero three.
239
00:14:13,680 --> 00:14:14,540
Okay.
240
00:14:14,540 --> 00:14:18,790
So that idea of linear
combination is crucial,
241
00:14:18,790 --> 00:14:21,130
and also --
242
00:14:21,130 --> 00:14:22,915
do we want to think
about this question?
243
00:14:25,420 --> 00:14:26,470
Sure, why not.
244
00:14:26,470 --> 00:14:29,830
What are all the combinations?
245
00:14:29,830 --> 00:14:32,655
If I took -- can I
go back to xs and ys?
246
00:14:35,240 --> 00:14:38,610
This is a question for really --
247
00:14:38,610 --> 00:14:41,450
it's going to come
up over and over,
248
00:14:41,450 --> 00:14:45,350
but why don't we
see it once now?
249
00:14:45,350 --> 00:14:50,610
If I took all the xs and all
the ys, all the combinations,
250
00:14:50,610 --> 00:14:53,830
what would be all the results?
251
00:14:53,830 --> 00:14:56,200
And, actually, the
result would be
252
00:14:56,200 --> 00:14:59,400
that I could get any
right-hand side at all.
253
00:14:59,400 --> 00:15:02,680
The combinations
of this and this
254
00:15:02,680 --> 00:15:06,250
would fill the whole plane.
255
00:15:06,250 --> 00:15:08,510
You can tuck that away.
256
00:15:08,510 --> 00:15:14,080
We'll, explore it further.
257
00:15:14,080 --> 00:15:19,610
But this idea of what linear
combination gives b and what do
258
00:15:19,610 --> 00:15:21,650
all the linear
combinations give,
259
00:15:21,650 --> 00:15:25,070
what are all the possible,
achievable right-hand sides be
260
00:15:25,070 --> 00:15:26,610
-- that's going to be basic.
261
00:15:26,610 --> 00:15:27,220
Okay.
262
00:15:27,220 --> 00:15:31,980
Can I move to three
equations and three unknowns?
263
00:15:31,980 --> 00:15:38,610
Because it's easy to
picture the two by two case.
264
00:15:38,610 --> 00:15:40,980
Let me do a three
by three example.
265
00:15:40,980 --> 00:15:43,330
Okay, I'll sort of
start it the same way,
266
00:15:43,330 --> 00:15:50,210
say maybe 2x-y and maybe I'll
take no zs as a zero and maybe
267
00:15:50,210 --> 00:15:55,940
a -x 2y and maybe
a -z as a -- oh,
268
00:15:55,940 --> 00:16:01,130
let me make that a minus one
and, just for variety let me
269
00:16:01,130 --> 00:16:09,930
take, -3z, -3ys, I should
keep the ys in that line,
270
00:16:09,930 --> 00:16:14,400
and 4zs is, say, 4.
271
00:16:14,400 --> 00:16:16,480
Okay.
272
00:16:16,480 --> 00:16:19,100
That's three equations.
273
00:16:19,100 --> 00:16:22,380
I'm in three
dimensions, x, y, z.
274
00:16:22,380 --> 00:16:27,210
And, I don't have
a solution yet.
275
00:16:27,210 --> 00:16:32,230
So I want to understand the
equations and then solve them.
276
00:16:32,230 --> 00:16:32,780
Okay.
277
00:16:32,780 --> 00:16:35,450
So how do I you understand them?
278
00:16:35,450 --> 00:16:37,350
The row picture one way.
279
00:16:37,350 --> 00:16:41,120
The column picture is
another very important way.
280
00:16:41,120 --> 00:16:43,720
Just let's remember
the matrix form, here,
281
00:16:43,720 --> 00:16:45,020
because that's easy.
282
00:16:45,020 --> 00:16:49,140
The matrix form --
what's our matrix A?
283
00:16:49,140 --> 00:16:54,280
Our matrix A is this right-hand
side, the two and the minus one
284
00:16:54,280 --> 00:16:58,620
and the zero from the first
row, the minus one and the two
285
00:16:58,620 --> 00:17:00,780
and the minus one
from the second row,
286
00:17:00,780 --> 00:17:08,530
the zero, the minus three and
the four from the third row.
287
00:17:08,530 --> 00:17:11,050
So it's a three by three matrix.
288
00:17:11,050 --> 00:17:12,970
Three equations, three unknowns.
289
00:17:12,970 --> 00:17:14,680
And what's our right-hand side?
290
00:17:14,680 --> 00:17:20,190
Of course, it's the vector,
zero minus one, four.
291
00:17:20,190 --> 00:17:21,180
Okay.
292
00:17:21,180 --> 00:17:26,950
So that's the way, well, that's
the short-hand to write out
293
00:17:26,950 --> 00:17:28,690
the three equations.
294
00:17:28,690 --> 00:17:31,429
But it's the picture that
I'm looking for today.
295
00:17:31,429 --> 00:17:32,470
Okay, so the row picture.
296
00:17:32,470 --> 00:17:38,370
All right, so I'm in
three dimensions, x,
297
00:17:38,370 --> 00:17:42,680
find out when there
isn't a solution.
298
00:17:42,680 --> 00:17:45,950
y and z.
299
00:17:45,950 --> 00:17:51,150
And I want to take those
equations one at a time and ask
300
00:17:51,150 --> 00:17:51,820
--
301
00:17:51,820 --> 00:17:55,280
and make a picture of all
the points that satisfy --
302
00:17:55,280 --> 00:17:57,960
let's take equation number two.
303
00:17:57,960 --> 00:18:00,820
If I make a picture of all
the points that satisfy --
304
00:18:00,820 --> 00:18:05,900
all the x, y, z points
that solve this equation --
305
00:18:05,900 --> 00:18:09,550
well, first of all, the
origin is not one of them.
306
00:18:09,550 --> 00:18:14,620
x, y, z -- it being 0, 0, 0
would not solve that equation.
307
00:18:14,620 --> 00:18:17,790
So what are some points
that do solve the equation?
308
00:18:17,790 --> 00:18:23,750
Let's see, maybe if x is
one, y and z could be zero.
309
00:18:23,750 --> 00:18:24,750
That would work, right?
310
00:18:24,750 --> 00:18:26,870
So there's one point.
311
00:18:26,870 --> 00:18:29,520
I'm looking at this
second equation,
312
00:18:29,520 --> 00:18:33,120
here, just, to start with.
313
00:18:33,120 --> 00:18:33,670
Let's see.
314
00:18:33,670 --> 00:18:36,430
Also, I guess, if
z could be one,
315
00:18:36,430 --> 00:18:38,710
x and y could be zero,
so that would just
316
00:18:38,710 --> 00:18:41,930
go straight up that axis.
317
00:18:41,930 --> 00:18:46,880
And, probably I'd want
a third point here.
318
00:18:46,880 --> 00:18:55,390
Let me take x to be
zero, z to be zero,
319
00:18:55,390 --> 00:19:00,400
then y would be
minus a half, right?
320
00:19:00,400 --> 00:19:06,120
So there's a third point,
somewhere -- oh my -- okay.
321
00:19:06,120 --> 00:19:07,900
Let's see.
322
00:19:07,900 --> 00:19:13,080
I want to put in all the points
that satisfy that equation.
323
00:19:13,080 --> 00:19:17,930
Do you know what that
bunch of points will be?
324
00:19:17,930 --> 00:19:19,350
It's a plane.
325
00:19:19,350 --> 00:19:22,430
If we have a linear
equation, then, fortunately,
326
00:19:22,430 --> 00:19:27,070
the graph of the thing, the plot
of all the points that solve it
327
00:19:27,070 --> 00:19:30,390
are a plane.
328
00:19:30,390 --> 00:19:32,220
These three points
determine a plane,
329
00:19:32,220 --> 00:19:37,570
but your lecturer
is not Rembrandt
330
00:19:37,570 --> 00:19:43,380
and the art is going to
be the weak point here.
331
00:19:43,380 --> 00:19:46,460
So I'm just going to
draw a plane, right?
332
00:19:46,460 --> 00:19:48,310
There's a plane somewhere.
333
00:19:48,310 --> 00:19:50,800
That's my plane.
334
00:19:50,800 --> 00:19:54,960
That plane is all the
points that solves this guy.
335
00:19:54,960 --> 00:19:59,880
Then, what about this one?
336
00:19:59,880 --> 00:20:02,760
Two x minus y plus zero z.
337
00:20:02,760 --> 00:20:05,410
So z actually can be anything.
338
00:20:05,410 --> 00:20:08,390
Again, it's going
to be another plane.
339
00:20:08,390 --> 00:20:11,290
Each row in a three
by three problem
340
00:20:11,290 --> 00:20:14,480
gives us a plane in
three dimensions.
341
00:20:14,480 --> 00:20:17,730
So this one is going to
be some other plane --
342
00:20:17,730 --> 00:20:20,250
maybe I'll try to
draw it like this.
343
00:20:20,250 --> 00:20:25,240
And those two planes
meet in a line.
344
00:20:25,240 --> 00:20:29,280
So if I have two equations,
just the first two
345
00:20:29,280 --> 00:20:33,380
equations in three dimensions,
those give me a line.
346
00:20:33,380 --> 00:20:35,290
The line where those
two planes meet.
347
00:20:35,290 --> 00:20:42,210
And now, the third
guy is a third plane.
348
00:20:42,210 --> 00:20:49,290
And it goes somewhere.
349
00:20:49,290 --> 00:20:51,820
Okay, those three
things meet in a point.
350
00:20:51,820 --> 00:20:55,090
Now I don't know where
that point is, frankly.
351
00:20:55,090 --> 00:20:58,140
But -- linear
algebra will find it.
352
00:20:58,140 --> 00:21:05,580
The main point is that the three
planes, because they're not
353
00:21:05,580 --> 00:21:08,100
parallel, they're not special.
354
00:21:08,100 --> 00:21:11,780
They do meet in one point
and that's the solution.
355
00:21:11,780 --> 00:21:16,910
But, maybe you can see
that this row picture is
356
00:21:16,910 --> 00:21:19,120
getting a little hard to see.
357
00:21:19,120 --> 00:21:24,520
The row picture was a cinch
when we looked at two lines
358
00:21:24,520 --> 00:21:25,220
meeting.
359
00:21:25,220 --> 00:21:27,740
When we look at
three planes meeting,
360
00:21:27,740 --> 00:21:32,890
it's not so clear and in
four dimensions probably
361
00:21:32,890 --> 00:21:34,780
a little less clear.
362
00:21:34,780 --> 00:21:37,530
So, can I quit on
the row picture?
363
00:21:37,530 --> 00:21:41,870
Or quit on the row picture
before I've successfully
364
00:21:41,870 --> 00:21:45,940
found the point where
the three planes meet?
365
00:21:45,940 --> 00:21:51,510
All I really want to see is
that the row picture consists
366
00:21:51,510 --> 00:21:55,780
of three planes and, if
everything works right,
367
00:21:55,780 --> 00:21:59,340
three planes meet in one
point and that's a solution.
368
00:21:59,340 --> 00:22:04,730
Now, you can tell I
prefer the column picture.
369
00:22:04,730 --> 00:22:06,990
Okay, so let me take
the column picture.
370
00:22:06,990 --> 00:22:09,820
That's x times --
371
00:22:09,820 --> 00:22:14,400
so there were two xs in the
first equation minus one x is,
372
00:22:14,400 --> 00:22:16,690
and no xs in the third.
373
00:22:16,690 --> 00:22:19,000
It's just the first
column of that.
374
00:22:19,000 --> 00:22:21,300
And how many ys are there?
375
00:22:21,300 --> 00:22:24,730
There's minus one in the first
equations, two in the second
376
00:22:24,730 --> 00:22:27,430
and maybe minus
three in the third.
377
00:22:27,430 --> 00:22:29,830
Just the second
column of my matrix.
378
00:22:29,830 --> 00:22:37,740
And z times no zs minus
one zs and four zs.
379
00:22:37,740 --> 00:22:41,270
And it's those three
columns, right,
380
00:22:41,270 --> 00:22:46,640
that I have to combine to
produce the right-hand side,
381
00:22:46,640 --> 00:22:48,945
which is zero minus one four.
382
00:22:52,830 --> 00:22:54,910
Okay.
383
00:22:54,910 --> 00:22:57,940
So what have we got on
this left-hand side?
384
00:22:57,940 --> 00:22:59,470
A linear combination.
385
00:22:59,470 --> 00:23:02,540
It's a linear combination
now of three vectors,
386
00:23:02,540 --> 00:23:05,720
and they happen to be -- each
one is a three dimensional
387
00:23:05,720 --> 00:23:10,000
vector, so we want to know
what combination of those three
388
00:23:10,000 --> 00:23:12,090
vectors produces that one.
389
00:23:12,090 --> 00:23:15,820
Shall I try to draw the
column picture, then?
390
00:23:15,820 --> 00:23:18,830
So, since these vectors
have three components --
391
00:23:18,830 --> 00:23:21,660
so it's some multiple -- let
me draw in the first column
392
00:23:21,660 --> 00:23:23,340
as before --
393
00:23:23,340 --> 00:23:27,287
x is two and y is minus one.
394
00:23:27,287 --> 00:23:28,620
Maybe there is the first column.
395
00:23:32,320 --> 00:23:38,570
y -- the second column has maybe
a minus one and a two and the y
396
00:23:38,570 --> 00:23:44,180
is a minus three, somewhere,
there possibly, column two.
397
00:23:44,180 --> 00:23:46,950
And the third column has --
398
00:23:46,950 --> 00:23:52,760
no zero minus one four,
so how shall I draw that?
399
00:23:52,760 --> 00:23:57,000
So this was the first component.
400
00:23:57,000 --> 00:23:59,230
The second component
was a minus one.
401
00:23:59,230 --> 00:24:01,270
Maybe up here.
402
00:24:01,270 --> 00:24:08,921
That's column three, that's the
column zero minus one and four.
403
00:24:08,921 --> 00:24:09,420
This guy.
404
00:24:12,050 --> 00:24:15,430
So, again, what's my problem?
405
00:24:15,430 --> 00:24:18,260
What this equation
is asking me to do
406
00:24:18,260 --> 00:24:21,340
is to combine
these three vectors
407
00:24:21,340 --> 00:24:27,900
with a right combination
to produce this one.
408
00:24:27,900 --> 00:24:33,620
Well, you can see what the
right combination is, because
409
00:24:33,620 --> 00:24:38,250
in this special problem,
specially chosen
410
00:24:38,250 --> 00:24:42,780
by the lecturer, that right-hand
side that I'm trying to get
411
00:24:42,780 --> 00:24:45,060
is actually one
of these columns.
412
00:24:45,060 --> 00:24:46,660
So I know how to get that one.
413
00:24:46,660 --> 00:24:48,420
So what's the solution?
414
00:24:48,420 --> 00:24:50,910
What combination will work?
415
00:24:50,910 --> 00:24:53,940
I just want one of
these and none of these.
416
00:24:53,940 --> 00:24:59,070
So x should be zero, y
should be zero and z should
417
00:24:59,070 --> 00:25:00,040
be one.
418
00:25:02,860 --> 00:25:04,520
That's the combination.
419
00:25:04,520 --> 00:25:07,600
One of those is
obviously the right one.
420
00:25:07,600 --> 00:25:09,870
Column three is
actually the same
421
00:25:09,870 --> 00:25:12,090
as b in this particular problem.
422
00:25:15,640 --> 00:25:17,860
I made it work
that way just so we
423
00:25:17,860 --> 00:25:21,770
would get an answer,
(0,0,1), so somehow that's
424
00:25:21,770 --> 00:25:25,890
the point where those
three planes met
425
00:25:25,890 --> 00:25:28,710
and I couldn't see it before.
426
00:25:28,710 --> 00:25:31,830
Of course, I won't always be
able to see it from the column
427
00:25:31,830 --> 00:25:33,170
picture, either.
428
00:25:33,170 --> 00:25:39,390
It's the next lecture, actually,
which is about elimination,
429
00:25:39,390 --> 00:25:46,670
which is the systematic
way that everybody --
430
00:25:46,670 --> 00:25:51,950
every bit of software, too --
431
00:25:51,950 --> 00:25:56,830
production, large-scale software
would solve the equations.
432
00:25:56,830 --> 00:25:59,020
So the lecture that's coming up.
433
00:25:59,020 --> 00:26:02,450
If I was to add that
to the syllabus,
434
00:26:02,450 --> 00:26:09,050
will be about how to find
x, y, z in all cases.
435
00:26:09,050 --> 00:26:15,110
Can I just think again,
though, about the big picture?
436
00:26:15,110 --> 00:26:18,910
By the big picture I mean
let's keep this same matrix
437
00:26:18,910 --> 00:26:21,990
on the left but
imagine that we have
438
00:26:21,990 --> 00:26:24,250
a different right-hand side.
439
00:26:24,250 --> 00:26:27,430
Oh, let me take a
different right-hand side.
440
00:26:27,430 --> 00:26:29,100
So I'll change that
right-hand side
441
00:26:29,100 --> 00:26:34,610
to something that actually
is also pretty special.
442
00:26:34,610 --> 00:26:36,810
Let me change it to --
443
00:26:36,810 --> 00:26:38,940
if I add those
first two columns,
444
00:26:38,940 --> 00:26:43,220
that would give me a one
and a one and a minus three.
445
00:26:43,220 --> 00:26:46,210
There's a very special
right-hand side.
446
00:26:46,210 --> 00:26:51,420
I just cooked it up by
adding this one to this one.
447
00:26:51,420 --> 00:26:54,640
Now, what's the solution with
this new right-hand side?
448
00:26:54,640 --> 00:26:58,960
The solution with this new
right-hand side is clear.
449
00:26:58,960 --> 00:27:04,880
took one of these
and none of those.
450
00:27:04,880 --> 00:27:08,030
So actually, it just
changed around to this
451
00:27:08,030 --> 00:27:11,150
when I took this
new right-hand side.
452
00:27:11,150 --> 00:27:12,010
Okay.
453
00:27:12,010 --> 00:27:19,070
So in the row picture, I
have three different planes,
454
00:27:19,070 --> 00:27:23,800
three new planes meeting
now at this point.
455
00:27:23,800 --> 00:27:27,000
In the column picture, I
have the same three columns,
456
00:27:27,000 --> 00:27:30,640
but now I'm combining
them to produce this guy,
457
00:27:30,640 --> 00:27:34,100
and it turned out that column
one plus column two which would
458
00:27:34,100 --> 00:27:38,500
be somewhere -- there
is the right column --
459
00:27:38,500 --> 00:27:42,260
one of this and one of this
would give me the new b.
460
00:27:45,190 --> 00:27:45,690
Okay.
461
00:27:45,690 --> 00:27:48,730
So we squeezed in
an extra example.
462
00:27:48,730 --> 00:27:55,970
But now think about all
bs, all right-hand sides.
463
00:27:55,970 --> 00:27:58,920
Can I solve these equations
for every right-hand side?
464
00:28:02,030 --> 00:28:04,980
Can I ask that question?
465
00:28:04,980 --> 00:28:07,020
So that's the algebra question.
466
00:28:07,020 --> 00:28:11,870
Can I solve A x=b for every b?
467
00:28:11,870 --> 00:28:13,490
Let me write that down.
468
00:28:13,490 --> 00:28:24,320
Can I solve A x =b for
every right-hand side b?
469
00:28:24,320 --> 00:28:26,420
I mean, is there a solution?
470
00:28:26,420 --> 00:28:29,670
And then, if there
is, elimination
471
00:28:29,670 --> 00:28:31,940
will give me a way to find it.
472
00:28:31,940 --> 00:28:34,520
I really wanted to ask,
is there a solution
473
00:28:34,520 --> 00:28:36,850
for every right-hand side?
474
00:28:36,850 --> 00:28:40,390
So now, can I put that
in different words --
475
00:28:40,390 --> 00:28:43,140
in this linear
combination words?
476
00:28:43,140 --> 00:28:52,510
So in linear combination words,
do the linear combinations
477
00:28:52,510 --> 00:29:03,060
of the columns fill
three dimensional space?
478
00:29:05,740 --> 00:29:12,500
Every b means all the bs
in three dimensional space.
479
00:29:12,500 --> 00:29:16,050
Do you see that I'm just
asking the same question
480
00:29:16,050 --> 00:29:19,600
in different words?
481
00:29:19,600 --> 00:29:21,040
Solving A x --
482
00:29:21,040 --> 00:29:25,020
A x -- that's very important.
483
00:29:25,020 --> 00:29:31,790
A times x -- when I multiply
a matrix by a vector,
484
00:29:31,790 --> 00:29:34,940
I get a combination
of the columns.
485
00:29:34,940 --> 00:29:38,650
I'll write that
down in a moment.
486
00:29:38,650 --> 00:29:43,550
But in my column picture,
that's really what I'm doing.
487
00:29:43,550 --> 00:29:47,080
I'm taking linear combinations
of these three columns
488
00:29:47,080 --> 00:29:50,620
and I'm trying to find b.
489
00:29:50,620 --> 00:29:57,750
And, actually, the answer
for this matrix will be yes.
490
00:29:57,750 --> 00:30:04,330
For this matrix A -- for these
columns, the answer is yes.
491
00:30:04,330 --> 00:30:19,590
This matrix -- that I chose for
an example is a good matrix.
492
00:30:19,590 --> 00:30:21,780
A non-singular matrix.
493
00:30:21,780 --> 00:30:23,180
An invertible matrix.
494
00:30:23,180 --> 00:30:26,700
Those will be the matrices
that we like best.
495
00:30:26,700 --> 00:30:29,490
There could be other --
496
00:30:29,490 --> 00:30:35,070
and we will see other matrices
where the answer becomes, no --
497
00:30:35,070 --> 00:30:38,320
oh, actually, you can see
when it would become no.
498
00:30:38,320 --> 00:30:43,890
What could go wrong? find out
-- because if elimination fails,
499
00:30:43,890 --> 00:30:46,820
How could it go wrong
that out of these --
500
00:30:46,820 --> 00:30:51,310
out of three columns and
all their combinations --
501
00:30:51,310 --> 00:30:58,390
when would I not be able
to produce some b off here?
502
00:30:58,390 --> 00:31:00,610
When could it go wrong?
503
00:31:00,610 --> 00:31:04,350
Do you see that
the combinations --
504
00:31:04,350 --> 00:31:06,370
let me say when it goes wrong.
505
00:31:06,370 --> 00:31:12,310
If these three columns
all lie in the same plane,
506
00:31:12,310 --> 00:31:18,110
then their combinations
will lie in that same plane.
507
00:31:18,110 --> 00:31:19,640
So then we're in trouble.
508
00:31:19,640 --> 00:31:23,880
If the three columns
of my matrix --
509
00:31:23,880 --> 00:31:28,180
if those three vectors happen
to lie in the same plane --
510
00:31:28,180 --> 00:31:31,090
for example, if
column three is just
511
00:31:31,090 --> 00:31:36,360
the sum of column one and column
two, I would be in trouble.
512
00:31:36,360 --> 00:31:40,060
That would be a matrix A
where the answer would be no,
513
00:31:40,060 --> 00:31:44,360
because the combinations --
514
00:31:44,360 --> 00:31:48,260
if column three is in the same
plane as column one and two,
515
00:31:48,260 --> 00:31:50,250
I don't get anything
new from that.
516
00:31:50,250 --> 00:31:54,610
All the combinations are in the
plane and only right-hand sides
517
00:31:54,610 --> 00:32:00,010
b that I could get would
be the ones in that plane.
518
00:32:00,010 --> 00:32:03,680
So I could solve it for
some right-hand sides, when
519
00:32:03,680 --> 00:32:08,320
b is in the plane, but
most right-hand sides
520
00:32:08,320 --> 00:32:11,270
would be out of the
plane and unreachable.
521
00:32:11,270 --> 00:32:14,190
So that would be
a singular case.
522
00:32:14,190 --> 00:32:16,830
The matrix would
be not invertible.
523
00:32:16,830 --> 00:32:19,810
There would not be a
solution for every b.
524
00:32:19,810 --> 00:32:22,511
The answer would
become no for that.
525
00:32:22,511 --> 00:32:23,010
Okay.
526
00:32:25,740 --> 00:32:27,040
I don't know --
527
00:32:27,040 --> 00:32:29,190
shall we take just a
little shot at thinking
528
00:32:29,190 --> 00:32:34,510
about nine dimensions?
529
00:32:34,510 --> 00:32:38,645
Imagine that we have vectors
with nine components.
530
00:32:41,170 --> 00:32:44,920
Well, it's going to be
hard to visualize those.
531
00:32:44,920 --> 00:32:46,800
I don't pretend to do it.
532
00:32:46,800 --> 00:32:51,950
But somehow, pretend you do.
533
00:32:51,950 --> 00:32:55,610
Pretend we have -- if this
was nine equations and nine
534
00:32:55,610 --> 00:32:59,260
unknowns, then we would
have nine columns,
535
00:32:59,260 --> 00:33:02,720
and each one would be a vector
in nine-dimensional space
536
00:33:02,720 --> 00:33:06,930
and we would be looking at
their linear combinations.
537
00:33:06,930 --> 00:33:09,340
So we would be having
the linear combinations
538
00:33:09,340 --> 00:33:12,320
of nine vectors in
nine-dimensional space,
539
00:33:12,320 --> 00:33:15,380
and we would be trying to
find the combination that hit
540
00:33:15,380 --> 00:33:18,000
the correct right-hand side b.
541
00:33:18,000 --> 00:33:22,990
And we might also ask the
question can we always do it?
542
00:33:22,990 --> 00:33:25,380
Can we get every
right-hand side b?
543
00:33:25,380 --> 00:33:29,840
And certainly it will depend
on those nine columns.
544
00:33:29,840 --> 00:33:32,480
Sometimes the answer
will be yes --
545
00:33:32,480 --> 00:33:35,500
if I picked a random matrix,
it would be yes, actually.
546
00:33:35,500 --> 00:33:39,630
If I used MatLab and just used
the random command, picked
547
00:33:39,630 --> 00:33:44,820
out a nine by nine matrix,
I guarantee it would be
548
00:33:44,820 --> 00:33:45,320
good.
549
00:33:45,320 --> 00:33:47,140
It would be
non-singular, it would
550
00:33:47,140 --> 00:33:49,270
be invertible, all beautiful.
551
00:33:49,270 --> 00:34:00,750
But if I choose those columns
so that they're not independent,
552
00:34:00,750 --> 00:34:05,790
so that the ninth column is
the same as the eighth column,
553
00:34:05,790 --> 00:34:09,020
then it contributes
nothing new and there
554
00:34:09,020 --> 00:34:13,080
would be right-hand sides
b that I couldn't get.
555
00:34:13,080 --> 00:34:18,210
Can you sort of think
about nine vectors
556
00:34:18,210 --> 00:34:22,050
in nine-dimensional space
an take their combinations?
557
00:34:22,050 --> 00:34:26,460
That's really the
central thought --
558
00:34:26,460 --> 00:34:30,920
that you get kind of used
to in linear algebra.
559
00:34:30,920 --> 00:34:33,340
Even though you can't
really visualize it,
560
00:34:33,340 --> 00:34:36,139
you sort of think you
can after a while.
561
00:34:36,139 --> 00:34:40,830
Those nine columns and
all their combinations
562
00:34:40,830 --> 00:34:45,030
may very well fill out the
whole nine-dimensional space.
563
00:34:45,030 --> 00:34:48,239
But if the ninth column happened
to be the same as the eighth
564
00:34:48,239 --> 00:34:51,090
column and gave nothing new,
then probably what it would
565
00:34:51,090 --> 00:34:53,102
fill out would be --
566
00:34:55,820 --> 00:35:03,070
I hesitate even to say this --
it would be a sort of a plane
567
00:35:03,070 --> 00:35:04,320
--
568
00:35:04,320 --> 00:35:10,030
an eight dimensional plane
inside nine-dimensional space.
569
00:35:10,030 --> 00:35:12,890
And it's those eight
dimensional planes
570
00:35:12,890 --> 00:35:16,040
inside nine-dimensional
space that we
571
00:35:16,040 --> 00:35:18,830
have to work with eventually.
572
00:35:18,830 --> 00:35:25,970
For now, let's stay with a nice
case where the matrices work,
573
00:35:25,970 --> 00:35:29,980
we can get every
right-hand side b and here
574
00:35:29,980 --> 00:35:32,350
we see how to do
it with columns.
575
00:35:32,350 --> 00:35:33,160
Okay.
576
00:35:33,160 --> 00:35:36,520
There was one step
which I realized
577
00:35:36,520 --> 00:35:41,210
I was saying in words that I
now want to write in letters.
578
00:35:41,210 --> 00:35:45,750
Because I'm coming back to the
matrix form of the equation,
579
00:35:45,750 --> 00:35:49,210
so let me write it here.
580
00:35:49,210 --> 00:35:54,270
The matrix form of my
equation, of my system
581
00:35:54,270 --> 00:35:57,970
is some matrix A
times some vector x
582
00:35:57,970 --> 00:36:00,920
equals some right-hand side b.
583
00:36:00,920 --> 00:36:01,950
Okay.
584
00:36:01,950 --> 00:36:03,450
So this is a multiplication.
585
00:36:03,450 --> 00:36:04,520
A times x.
586
00:36:04,520 --> 00:36:07,430
Matrix times vector,
and I just want to say
587
00:36:07,430 --> 00:36:11,620
how do you multiply
a matrix by a vector?
588
00:36:11,620 --> 00:36:16,550
Okay, so I'm just going
to create a matrix --
589
00:36:16,550 --> 00:36:21,640
let me take two
five one three --
590
00:36:21,640 --> 00:36:28,140
and let me take a vector
x to be, say, 1and 2.
591
00:36:28,140 --> 00:36:31,500
How do I multiply a
matrix by a vector?
592
00:36:34,150 --> 00:36:40,530
But just think a little
bit about matrix notation
593
00:36:40,530 --> 00:36:42,120
and how to do that
in multiplication.
594
00:36:42,120 --> 00:36:45,390
So let me say how I multiply
a matrix by a vector.
595
00:36:45,390 --> 00:36:47,760
Actually, there are
two ways to do it.
596
00:36:47,760 --> 00:36:50,430
Let me tell you my favorite way.
597
00:36:50,430 --> 00:36:52,770
It's columns again.
598
00:36:52,770 --> 00:36:54,720
It's a column at a time.
599
00:36:54,720 --> 00:36:57,980
For me, this matrix
multiplication
600
00:36:57,980 --> 00:37:03,290
says I take one of that column
and two of that column and add.
601
00:37:03,290 --> 00:37:06,770
So this is the way
I would think of it
602
00:37:06,770 --> 00:37:12,820
is one of the first column
and two of the second column
603
00:37:12,820 --> 00:37:17,300
and let's just see what we get.
604
00:37:17,300 --> 00:37:21,510
So in the first component
I'm getting a two and a ten.
605
00:37:21,510 --> 00:37:23,470
I'm getting a twelve there.
606
00:37:23,470 --> 00:37:26,220
In the second component I'm
getting a one and a six,
607
00:37:26,220 --> 00:37:27,860
I'm getting a seven.
608
00:37:27,860 --> 00:37:35,460
So that matrix times that
vector is twelve seven.
609
00:37:35,460 --> 00:37:39,207
Now, you could do
that another way.
610
00:37:39,207 --> 00:37:40,540
You could do it a row at a time.
611
00:37:40,540 --> 00:37:43,760
And you would get this twelve --
and actually I pretty much did
612
00:37:43,760 --> 00:37:44,970
it here --
613
00:37:44,970 --> 00:37:45,710
this way.
614
00:37:45,710 --> 00:37:48,470
Two -- I could take that
row times my vector.
615
00:37:48,470 --> 00:37:53,180
This is the idea
of a dot product.
616
00:37:53,180 --> 00:37:58,140
This vector times this vector,
two times one plus five times
617
00:37:58,140 --> 00:38:00,720
two is the twelve.
618
00:38:00,720 --> 00:38:04,610
This vector times this vector --
one times one plus three times
619
00:38:04,610 --> 00:38:06,000
two is the seven.
620
00:38:06,000 --> 00:38:11,810
So I can do it by rows,
and in each row times
621
00:38:11,810 --> 00:38:16,360
my x is what I'll later
call a dot product.
622
00:38:16,360 --> 00:38:19,430
But I also like to
see it by columns.
623
00:38:19,430 --> 00:38:22,490
I see this as a linear
combination of a column.
624
00:38:22,490 --> 00:38:24,130
So here's my point.
625
00:38:24,130 --> 00:38:36,180
A times x is a combination
of the columns of A.
626
00:38:36,180 --> 00:38:43,650
That's how I hope you will
think of A times x when we need
627
00:38:43,650 --> 00:38:44,150
it.
628
00:38:44,150 --> 00:38:47,230
Right now we've got
-- with small ones,
629
00:38:47,230 --> 00:38:51,390
we can always do it in
different ways, but later,
630
00:38:51,390 --> 00:38:53,930
think of it that way.
631
00:38:53,930 --> 00:38:54,480
Okay.
632
00:38:54,480 --> 00:39:02,020
So that's the picture
for a two by two system.
633
00:39:02,020 --> 00:39:05,970
And if the right-hand side B
happened to be twelve seven,
634
00:39:05,970 --> 00:39:12,110
then of course the correct
solution would be one two.
635
00:39:12,110 --> 00:39:12,610
Okay.
636
00:39:12,610 --> 00:39:17,950
So let me come back next
time to a systematic way,
637
00:39:17,950 --> 00:39:23,470
using elimination,
to find the solution,
638
00:39:23,470 --> 00:39:29,810
if there is one, to a
system of any size and