1
00:00:05,870 --> 00:00:08,750
OK, when the camera
says, we'll start.
2
00:00:08,750 --> 00:00:13,320
You want to give me a signal?
3
00:00:13,320 --> 00:00:17,290
OK, this is lecture
eight in linear algebra,
4
00:00:17,290 --> 00:00:20,020
and this is the lecture
where we completely
5
00:00:20,020 --> 00:00:22,820
solve linear equations.
6
00:00:22,820 --> 00:00:23,810
So Ax=b.
7
00:00:27,080 --> 00:00:29,040
That's our goal.
8
00:00:29,040 --> 00:00:32,159
If it has a solution.
9
00:00:32,159 --> 00:00:35,990
It certainly can happen
that there is no solution.
10
00:00:35,990 --> 00:00:40,370
We have to identify that
possibility by elimination.
11
00:00:40,370 --> 00:00:45,110
And then if there is a solution
we want to find out is there
12
00:00:45,110 --> 00:00:48,650
only one solution or are
-- is there a whole family
13
00:00:48,650 --> 00:00:51,190
of solutions, and
then find them all.
14
00:00:51,190 --> 00:00:52,370
OK.
15
00:00:52,370 --> 00:00:57,420
Can I use as an
example the same matrix
16
00:00:57,420 --> 00:01:01,010
that I had last
time when we were
17
00:01:01,010 --> 00:01:02,890
looking for the null space.
18
00:01:02,890 --> 00:01:10,060
So the, the matrix has
rows 1 2 2 2, 2 4 6 8,
19
00:01:10,060 --> 00:01:13,150
and the third row -- you
remember the main point was
20
00:01:13,150 --> 00:01:21,890
the third row, 3 6 8 10, is the
sum of row one plus row two.
21
00:01:21,890 --> 00:01:25,710
In other words, if I add
those left-hand sides,
22
00:01:25,710 --> 00:01:28,710
I get the third left-hand side.
23
00:01:28,710 --> 00:01:31,530
So you can tell
me right away what
24
00:01:31,530 --> 00:01:35,460
elimination is going to discover
about the right-hand sides.
25
00:01:35,460 --> 00:01:41,570
What's -- there is a
condition on b1, b2,
26
00:01:41,570 --> 00:01:44,690
and b3 for this system
to have a solution.
27
00:01:44,690 --> 00:01:50,240
Most cases -- if I took
these numbers to be one --
28
00:01:50,240 --> 00:01:53,950
5, and 17, there would
not be a solution.
29
00:01:53,950 --> 00:01:58,700
In fact, if I took those
first numbers to be 1 and 5,
30
00:01:58,700 --> 00:02:03,600
what is the only b3
that would be OK?
31
00:02:03,600 --> 00:02:05,560
Six.
32
00:02:05,560 --> 00:02:09,860
If the left-hand -- if these
left-hand sides add up to that,
33
00:02:09,860 --> 00:02:10,729
then B --
34
00:02:10,729 --> 00:02:15,000
I need b1 plus b2 to equal b3.
35
00:02:15,000 --> 00:02:19,470
Let's just see how
elimination discovers that.
36
00:02:19,470 --> 00:02:23,430
But we can see it coming, right?
37
00:02:23,430 --> 00:02:26,450
That if -- let me say
it in other words.
38
00:02:26,450 --> 00:02:29,300
If some combination
on the left-hand side
39
00:02:29,300 --> 00:02:33,090
gives all 0s then
the same combination
40
00:02:33,090 --> 00:02:35,580
on the right-hand
side must give 0.
41
00:02:35,580 --> 00:02:36,080
OK.
42
00:02:36,080 --> 00:02:43,670
So let me take that
example and write down
43
00:02:43,670 --> 00:02:46,670
instead of copying out
all the plus signs,
44
00:02:46,670 --> 00:02:49,830
let me write down the matrix.
45
00:02:49,830 --> 00:02:59,130
1 2 2 2, 2 4 6 8,
and that 6 3 8 10,
46
00:02:59,130 --> 00:03:02,930
where the third row is the
sum of the first two rows.
47
00:03:02,930 --> 00:03:06,700
Now how do we deal with
the right-hand side?
48
00:03:06,700 --> 00:03:09,840
That's -- we want to do the same
thing to the right-hand side
49
00:03:09,840 --> 00:03:12,700
that we're doing to these
rows on the left side,
50
00:03:12,700 --> 00:03:17,920
so we just tack on the
right-hand side as another
51
00:03:17,920 --> 00:03:20,660
vector, another column.
52
00:03:20,660 --> 00:03:26,220
So this is the augmented matrix.
53
00:03:29,640 --> 00:03:35,840
It's, it's the matrix A
with the vector b tacked on.
54
00:03:35,840 --> 00:03:38,440
In Matlab, that's all
you would need to type.
55
00:03:38,440 --> 00:03:39,050
OK.
56
00:03:39,050 --> 00:03:41,470
So we do elimination on that.
57
00:03:41,470 --> 00:03:43,580
Can we just do
elimination quickly?
58
00:03:43,580 --> 00:03:46,790
The first pivot is fine,
I subtract two of this
59
00:03:46,790 --> 00:03:49,470
away from this, three
of this away from this,
60
00:03:49,470 --> 00:03:54,870
so I have 1 2 2 2 b1.
61
00:03:54,870 --> 00:04:00,840
Two of those away will
give me 0 0 2 and 4,
62
00:04:00,840 --> 00:04:03,570
and that was b2 minus two b1.
63
00:04:03,570 --> 00:04:07,130
I, I have to do the same
thing to that third,
64
00:04:07,130 --> 00:04:08,630
that last column.
65
00:04:08,630 --> 00:04:10,700
And then three of
these away from this
66
00:04:10,700 --> 00:04:17,980
gave me 0 0 2 4 b3
minus three b1s.
67
00:04:17,980 --> 00:04:21,660
So that's the,
that's elimination
68
00:04:21,660 --> 00:04:25,830
with the first column completed.
69
00:04:25,830 --> 00:04:26,750
We move on.
70
00:04:26,750 --> 00:04:29,570
There's the first pivot still.
71
00:04:29,570 --> 00:04:31,500
Here is the second pivot.
72
00:04:31,500 --> 00:04:34,660
We're always remembering,
now, these are then
73
00:04:34,660 --> 00:04:36,650
going to be the pivot columns.
74
00:04:41,810 --> 00:04:47,640
And let me get the final
result -- well, let me --
75
00:04:47,640 --> 00:04:51,615
can I do it by eraser?
76
00:04:55,550 --> 00:05:00,680
We're capable of subtracting
this row from this row,
77
00:05:00,680 --> 00:05:05,280
just by -- that'll knock this
out completely and give me
78
00:05:05,280 --> 00:05:08,090
the row of 0s, and on
the right-hand side,
79
00:05:08,090 --> 00:05:12,660
when I subtract this away
from this, what do I have?
80
00:05:16,260 --> 00:05:22,550
I think I have b3 minus a b2,
and I had minus three b1s.
81
00:05:22,550 --> 00:05:25,380
This is going to, it's
going to be a minus a b1.
82
00:05:25,380 --> 00:05:28,000
Oh yeah that's
exactly what I expect.
83
00:05:31,490 --> 00:05:34,430
So now the -- what's
the last equation?
84
00:05:34,430 --> 00:05:38,930
The last equation, this
represented by this zero row,
85
00:05:38,930 --> 00:05:45,750
that last equation is, says 0
equals b3 minus b2 minus b1.
86
00:05:45,750 --> 00:05:51,519
So that's the condition
for solvability.
87
00:05:51,519 --> 00:05:53,310
That's the condition
on the right-hand side
88
00:05:53,310 --> 00:05:54,450
that we expected.
89
00:05:54,450 --> 00:05:57,900
It says that b1+b2
has to match b3,
90
00:05:57,900 --> 00:06:02,350
and if our numbers happen
to have been 1, 5, and 6 --
91
00:06:02,350 --> 00:06:07,220
so let me take,
suppose b is 1 5 6.
92
00:06:07,220 --> 00:06:09,920
That's an OK b.
93
00:06:09,920 --> 00:06:13,120
And when I do this
elimination, what will I have?
94
00:06:13,120 --> 00:06:16,390
The b1 will still be a 1.
95
00:06:16,390 --> 00:06:18,950
b2 would be 5 minus
2, this would be a 3.
96
00:06:18,950 --> 00:06:24,860
5 -- my 6 minus 5 minus
1, this will be --
97
00:06:24,860 --> 00:06:29,800
this is the main point --
this will be a 0, thanks.
98
00:06:29,800 --> 00:06:30,300
OK.
99
00:06:30,300 --> 00:06:32,720
So the last equation is OK now.
100
00:06:37,080 --> 00:06:41,880
And I can proceed to solve the
two equations that are really
101
00:06:41,880 --> 00:06:44,160
there with four unknowns.
102
00:06:44,160 --> 00:06:48,660
OK, I, I, I want to do
that, so this, this b is OK.
103
00:06:48,660 --> 00:06:51,700
It allows a solution.
104
00:06:51,700 --> 00:06:56,850
We're going to be,
naturally, interested
105
00:06:56,850 --> 00:07:04,200
to keep track what are
the conditions on b that
106
00:07:04,200 --> 00:07:06,550
make the equation solvable.
107
00:07:06,550 --> 00:07:11,020
So let me write down
what we already see
108
00:07:11,020 --> 00:07:14,490
before I continue to solve it.
109
00:07:14,490 --> 00:07:17,286
Let me first --
solvability, solvability.
110
00:07:23,450 --> 00:07:31,015
So which -- so this is the
condition on the right-hand
111
00:07:31,015 --> 00:07:31,515
sides.
112
00:07:34,170 --> 00:07:36,010
And what is that condition?
113
00:07:36,010 --> 00:07:39,440
This is solvability
always of Ax=b.
114
00:07:39,440 --> 00:07:45,150
So Ax=b is solvable --
115
00:07:45,150 --> 00:07:50,870
well, actually, we had an answer
in the language of the column
116
00:07:50,870 --> 00:07:52,640
space.
117
00:07:52,640 --> 00:07:54,530
Can you remind me
what that answer is?
118
00:07:54,530 --> 00:07:58,510
That, that was like our
answer from earlier lecture.
119
00:07:58,510 --> 00:08:00,980
b had to be in the column space.
120
00:08:00,980 --> 00:08:10,140
Solvable if -- when -- exactly
when b is in the column space
121
00:08:10,140 --> 00:08:13,560
of A.
122
00:08:13,560 --> 00:08:14,110
Right?
123
00:08:14,110 --> 00:08:17,840
That just says that b has to be
a combination of the columns,
124
00:08:17,840 --> 00:08:21,890
and of course that's exactly
what the equation is looking
125
00:08:21,890 --> 00:08:22,660
for.
126
00:08:22,660 --> 00:08:25,800
So that -- now I
want to answer it --
127
00:08:25,800 --> 00:08:30,050
the same answer but
in different language.
128
00:08:30,050 --> 00:08:33,990
Another way to answer this --
129
00:08:33,990 --> 00:08:52,980
if a combination of the rows
of A gives the zero row,
130
00:08:52,980 --> 00:08:57,010
and this was an example
where it happened,
131
00:08:57,010 --> 00:09:00,940
some combination of the rows
of A produced the zero row --
132
00:09:00,940 --> 00:09:04,560
then what's the
requirement on b?
133
00:09:04,560 --> 00:09:07,130
Since we're going to do the
same thing to both sides of all
134
00:09:07,130 --> 00:09:08,520
equations --
135
00:09:08,520 --> 00:09:16,190
the same combination of the
components of b has to give 0.
136
00:09:16,190 --> 00:09:16,740
Right?
137
00:09:16,740 --> 00:09:19,590
That's -- so if there's a
combination of the rows that
138
00:09:19,590 --> 00:09:29,920
gives the zero row, then the
same combination of the entries
139
00:09:29,920 --> 00:09:34,700
of b must give 0.
140
00:09:37,950 --> 00:09:40,475
And this isn't the zero
row, that's the zero number.
141
00:09:43,650 --> 00:09:47,860
Tthis is another way of saying
-- and it is not immediate,
142
00:09:47,860 --> 00:09:54,582
OK. right, that these two
statements are equivalent.
143
00:09:54,582 --> 00:09:56,290
But somehow they must
be, because they're
144
00:09:56,290 --> 00:09:59,961
both equivalent to the
solvability of the system.
145
00:09:59,961 --> 00:10:00,460
OK.
146
00:10:00,460 --> 00:10:05,030
So we've got this, this sort
of -- like question zero is,
147
00:10:05,030 --> 00:10:08,180
does the system have a solution?
148
00:10:08,180 --> 00:10:12,670
OK, I'll come back to
discuss that further.
149
00:10:12,670 --> 00:10:17,310
Let's go forward when it does.
150
00:10:17,310 --> 00:10:19,470
When there is a solution.
151
00:10:19,470 --> 00:10:22,684
And so what's our job now?
152
00:10:22,684 --> 00:10:24,600
Abstractly we sit back
and we say, OK, there's
153
00:10:24,600 --> 00:10:26,660
a solution, finished.
154
00:10:26,660 --> 00:10:27,570
It exists.
155
00:10:27,570 --> 00:10:29,450
But we want to construct it.
156
00:10:29,450 --> 00:10:34,500
So what's the
algorithm, the sequence
157
00:10:34,500 --> 00:10:37,330
of steps to find the solution?
158
00:10:37,330 --> 00:10:38,670
That's what I --
159
00:10:38,670 --> 00:10:42,130
and of course the
quiz and the final,
160
00:10:42,130 --> 00:10:45,440
I'm going to give you a system
Ax=b and I'm going to ask you
161
00:10:45,440 --> 00:10:48,350
for the solution,
if there is one.
162
00:10:48,350 --> 00:10:54,430
And so this algorithm
that you want to follow.
163
00:10:54,430 --> 00:10:58,290
OK, let's see.
164
00:10:58,290 --> 00:11:13,190
So what's the -- so now to find
the complete solution to Ax=b.
165
00:11:13,190 --> 00:11:14,000
OK.
166
00:11:14,000 --> 00:11:17,150
Let me start by
finding one solution,
167
00:11:17,150 --> 00:11:19,410
one particular solution.
168
00:11:22,030 --> 00:11:26,060
I'm expecting that I can,
because my system of equations
169
00:11:26,060 --> 00:11:30,730
now, that last equation
is zero equals zero,
170
00:11:30,730 --> 00:11:33,850
so that's all fine.
171
00:11:33,850 --> 00:11:36,720
I really have two equations --
172
00:11:36,720 --> 00:11:38,910
actually I've got
four unknowns, so I'm
173
00:11:38,910 --> 00:11:41,410
expecting to find
not only a solution
174
00:11:41,410 --> 00:11:44,230
but a whole bunch of them.
175
00:11:44,230 --> 00:11:46,020
But let's just find one.
176
00:11:46,020 --> 00:11:50,930
So step one, a particular
solution, x particular.
177
00:11:54,430 --> 00:11:57,010
How do I find one
particular solution?
178
00:11:57,010 --> 00:12:00,740
Well, let me tell you
how I, how I find it.
179
00:12:00,740 --> 00:12:02,160
So this is --
180
00:12:02,160 --> 00:12:04,080
since there are
lots of solutions,
181
00:12:04,080 --> 00:12:07,100
you could have your own way
to find a particular one.
182
00:12:07,100 --> 00:12:10,780
But this is a
pretty natural way.
183
00:12:10,780 --> 00:12:20,880
Set all free variables to zero.
184
00:12:20,880 --> 00:12:25,810
Since those free variables are
the guys that can be anything,
185
00:12:25,810 --> 00:12:28,790
the most convenient
choice is zero.
186
00:12:28,790 --> 00:12:37,975
And then solve Ax=b for
the pivot variables.
187
00:12:41,170 --> 00:12:44,240
So what does that
mean in this example?
188
00:12:44,240 --> 00:12:46,490
Which are the free variables?
189
00:12:46,490 --> 00:12:49,500
Which, which are the variables
that we can assign freely
190
00:12:49,500 --> 00:12:52,500
and then there's
one and only one way
191
00:12:52,500 --> 00:12:55,070
to find the pivot variables?
192
00:12:55,070 --> 00:13:01,120
They're x2 and -- so x2 is
zero, because that's in a column
193
00:13:01,120 --> 00:13:04,280
without a pivot, the
second column has no pivot.
194
00:13:04,280 --> 00:13:08,010
And the -- what's the other one?
195
00:13:08,010 --> 00:13:11,620
The fourth, x4 is zero.
196
00:13:11,620 --> 00:13:16,330
Because that, those
are the, the free ones.
197
00:13:16,330 --> 00:13:18,870
Those are in the
columns with no pivots.
198
00:13:18,870 --> 00:13:21,300
So you see what my
-- so when I knock --
199
00:13:21,300 --> 00:13:28,660
when x2 and x4 are zero,
I'm left with the --
200
00:13:28,660 --> 00:13:31,010
what I left with here?
201
00:13:31,010 --> 00:13:33,460
I'm just left with --
202
00:13:33,460 --> 00:13:36,500
see, now I'm not
using the two free
203
00:13:36,500 --> 00:13:37,020
columns.
204
00:13:37,020 --> 00:13:39,380
I'm only using
the pivot columns.
205
00:13:39,380 --> 00:13:42,370
So I'm really left with x1 --
206
00:13:42,370 --> 00:13:45,360
the first equation
is just x1 and two
207
00:13:45,360 --> 00:13:48,720
x3s should be the
right-hand side, which
208
00:13:48,720 --> 00:13:50,700
we picked to be a one.
209
00:13:50,700 --> 00:13:54,130
And the second
equation is two x3s,
210
00:13:54,130 --> 00:13:57,735
as it happened, turned
out to be, three.
211
00:14:02,090 --> 00:14:06,680
I just write it again here
with the x2 and the x4
212
00:14:06,680 --> 00:14:09,420
knocked out, since
we're set them to zero.
213
00:14:09,420 --> 00:14:14,150
And you see that we're back in
the normal case of having back
214
00:14:14,150 --> 00:14:16,030
-- where back
substitution will do it.
215
00:14:16,030 --> 00:14:21,640
So x3 is three halves,
and then we go back up
216
00:14:21,640 --> 00:14:25,490
and x1 is one minus two x3.
217
00:14:25,490 --> 00:14:29,270
That's probably minus two.
218
00:14:29,270 --> 00:14:30,400
Good.
219
00:14:30,400 --> 00:14:34,210
So now we have the
solution, x particular
220
00:14:34,210 --> 00:14:41,940
is the vector minus two
zero three halves zero.
221
00:14:44,710 --> 00:14:46,790
OK, good.
222
00:14:46,790 --> 00:14:52,200
That's one particular solution,
and we should and could plug it
223
00:14:52,200 --> 00:14:54,600
into the original system.
224
00:14:54,600 --> 00:14:57,010
Really if -- on
the quiz, please,
225
00:14:57,010 --> 00:14:59,230
it's a good thing to do.
226
00:14:59,230 --> 00:15:03,650
So we did all this,
these, row operations,
227
00:15:03,650 --> 00:15:06,960
but this is supposed to
solve the original system,
228
00:15:06,960 --> 00:15:09,430
and I think it does.
229
00:15:09,430 --> 00:15:10,120
OK.
230
00:15:10,120 --> 00:15:14,810
So that's x particular
which we've got.
231
00:15:14,810 --> 00:15:19,320
So that's like what's new today.
232
00:15:19,320 --> 00:15:23,780
The particular solution comes
-- first you check that you have
233
00:15:23,780 --> 00:15:26,700
zero equals zero, so
you're OK on the last
234
00:15:26,700 --> 00:15:27,920
equations.
235
00:15:27,920 --> 00:15:31,980
And then you set the
free variables to zero,
236
00:15:31,980 --> 00:15:34,830
solve for the pivot
variables, and you've
237
00:15:34,830 --> 00:15:38,500
got a particular solution,
the particular solution that
238
00:15:38,500 --> 00:15:41,150
has zero free variables.
239
00:15:41,150 --> 00:15:42,040
OK.
240
00:15:42,040 --> 00:15:45,000
Now -- but that's
only one solution,
241
00:15:45,000 --> 00:15:46,270
and now I'm looking for all.
242
00:15:49,020 --> 00:15:51,480
So how do I find the rest?
243
00:15:51,480 --> 00:15:58,770
The point is I can add on x --
anything out of the null space.
244
00:16:03,320 --> 00:16:06,190
We know how to find the
vectors in the null space --
245
00:16:06,190 --> 00:16:08,950
because we did it last time,
but I'll remind you what we
246
00:16:08,950 --> 00:16:09,710
got.
247
00:16:09,710 --> 00:16:12,620
And then I'll add.
248
00:16:15,630 --> 00:16:20,650
So the final result will be
that the complete solution --
249
00:16:20,650 --> 00:16:23,620
this is now the complete guy --
250
00:16:23,620 --> 00:16:27,980
the complete solution is
this one particular solution
251
00:16:27,980 --> 00:16:34,930
plus any, any vector,
all different vectors out
252
00:16:34,930 --> 00:16:37,600
of the null space.
253
00:16:37,600 --> 00:16:39,050
xn, OK.
254
00:16:39,050 --> 00:16:42,630
Well why, why this pattern,
because this pattern shows up
255
00:16:42,630 --> 00:16:46,560
through all of mathematics,
because it shows up everywhere
256
00:16:46,560 --> 00:16:48,690
we have linear equations.
257
00:16:48,690 --> 00:16:52,050
Let me just put here
the, the reason.
258
00:16:52,050 --> 00:17:01,830
A xp, so that's x particular,
so what does Ax particular give?
259
00:17:01,830 --> 00:17:05,410
That gives the correct
right-hand side b.
260
00:17:05,410 --> 00:17:10,520
And what does A times an
x in the null space give?
261
00:17:10,520 --> 00:17:11,710
Zero.
262
00:17:11,710 --> 00:17:17,920
So I add, and I
put in parentheses.
263
00:17:17,920 --> 00:17:25,420
So xp plus xn is b
plus zero, which is b.
264
00:17:25,420 --> 00:17:27,540
So -- oh, what I saying?
265
00:17:27,540 --> 00:17:30,450
Let me just say it in words.
266
00:17:30,450 --> 00:17:36,800
If I have one solution,
I can add on anything
267
00:17:36,800 --> 00:17:40,600
in the null space, because
anything in the null space
268
00:17:40,600 --> 00:17:43,910
has a zero right-hand
side, and I still
269
00:17:43,910 --> 00:17:46,670
have the correct
right-hand side B.
270
00:17:46,670 --> 00:17:47,850
So that's my system.
271
00:17:47,850 --> 00:17:50,290
That's my complete solution.
272
00:17:50,290 --> 00:17:54,620
Now let me write out what
that will be for this example.
273
00:17:54,620 --> 00:18:02,070
So in this example, x
general, x complete,
274
00:18:02,070 --> 00:18:07,440
the complete solution,
is x particular,
275
00:18:07,440 --> 00:18:12,230
which is minus two
zero three halves zero,
276
00:18:12,230 --> 00:18:15,900
with those zeroes in the
free variable, plus --
277
00:18:15,900 --> 00:18:18,410
you remember there were the
special solutions in the null
278
00:18:18,410 --> 00:18:21,680
space that had a one in
the free variables --
279
00:18:21,680 --> 00:18:24,220
or one and zero in
the free variables,
280
00:18:24,220 --> 00:18:29,880
and then we filled in to find
I've forgotten what they were,
281
00:18:29,880 --> 00:18:32,020
but maybe it was
that. the others?
282
00:18:32,020 --> 00:18:34,010
That was a special
solution, and then
283
00:18:34,010 --> 00:18:36,950
there was another
special solution that
284
00:18:36,950 --> 00:18:41,820
had that free variable zero and
this free variable equal one,
285
00:18:41,820 --> 00:18:46,260
and I have to fill those in.
286
00:18:46,260 --> 00:18:48,420
Let's see, can I remember
how those fill in?
287
00:18:48,420 --> 00:18:51,570
Maybe this was a minus
two and this was a two,
288
00:18:51,570 --> 00:18:53,070
possibly?
289
00:18:53,070 --> 00:18:57,700
I think probably that's right.
290
00:18:57,700 --> 00:18:59,290
I'm not -- yeah.
291
00:18:59,290 --> 00:19:05,230
Does that look write to you?
292
00:19:05,230 --> 00:19:08,480
I would have to remember
what are my equations.
293
00:19:08,480 --> 00:19:11,930
Can I, rather than go
way back to that board,
294
00:19:11,930 --> 00:19:14,680
let me remember the
first equation was
295
00:19:14,680 --> 00:19:19,520
two x3 plus two x4
equaling zero now,
296
00:19:19,520 --> 00:19:22,450
because I'm looking for
the guys in the null space.
297
00:19:22,450 --> 00:19:28,510
So I set x4 to be one
and the second equation,
298
00:19:28,510 --> 00:19:32,850
that I didn't copy again, gave
me minus two for this and then
299
00:19:32,850 --> 00:19:35,090
-- yeah, so I
think that's right.
300
00:19:35,090 --> 00:19:40,131
Two minus four and
two gives zero, check.
301
00:19:40,131 --> 00:19:40,630
OK.
302
00:19:40,630 --> 00:19:43,830
Those were the
special solutions.
303
00:19:43,830 --> 00:19:46,060
What do we do to get
the complete solution?
304
00:19:49,860 --> 00:19:52,270
How do I get the
complete solution now?
305
00:19:52,270 --> 00:19:57,020
I multiply this by
anything, c1, say,
306
00:19:57,020 --> 00:19:58,930
and I multiply
this by anything --
307
00:19:58,930 --> 00:20:00,960
I take any combination.
308
00:20:00,960 --> 00:20:04,140
Remember that's how we
described the null space?
309
00:20:04,140 --> 00:20:09,060
The null space consists
of all combinations of --
310
00:20:09,060 --> 00:20:10,760
so this is xn --
311
00:20:10,760 --> 00:20:15,430
all combinations of
the special solutions.
312
00:20:15,430 --> 00:20:18,410
There were two special
solutions because there
313
00:20:18,410 --> 00:20:20,730
were two free variables.
314
00:20:20,730 --> 00:20:24,560
And we want to
make that count --
315
00:20:24,560 --> 00:20:26,660
carefully now.
316
00:20:26,660 --> 00:20:27,857
Just while I'm up here.
317
00:20:27,857 --> 00:20:30,190
So there's, that's what the
-- that's the kind of answer
318
00:20:30,190 --> 00:20:31,230
I'm looking for.
319
00:20:31,230 --> 00:20:34,810
Is there a constant
multiplying this guy?
320
00:20:34,810 --> 00:20:38,790
Is there a free constant
that multiplies x particular?
321
00:20:38,790 --> 00:20:40,180
No way.
322
00:20:40,180 --> 00:20:44,430
Right? x particular
solves A xp=b.
323
00:20:44,430 --> 00:20:47,450
I'm not allowed to
multiply that by three.
324
00:20:47,450 --> 00:20:51,120
But Axn, I'm allowed to
multiply xn by three,
325
00:20:51,120 --> 00:20:56,790
or add to another xn, because I
keep getting zero on the right.
326
00:20:56,790 --> 00:20:57,290
OK.
327
00:20:57,290 --> 00:21:02,250
So, so again, xp is
one particular guy.
328
00:21:02,250 --> 00:21:04,630
xn is a whole subspace.
329
00:21:04,630 --> 00:21:05,590
Right?
330
00:21:05,590 --> 00:21:09,380
It's one guy plus, plus
anything from a subspace.
331
00:21:09,380 --> 00:21:11,120
Let me draw it.
332
00:21:11,120 --> 00:21:14,740
Let me try to -- oh.
333
00:21:14,740 --> 00:21:19,950
I want to draw, I want
to graph all this --
334
00:21:19,950 --> 00:21:25,910
I want to, I want to
plot all solutions.
335
00:21:25,910 --> 00:21:29,020
Now x.
336
00:21:29,020 --> 00:21:32,850
So what dimension I in?
337
00:21:32,850 --> 00:21:35,950
This is a unfortunate point.
338
00:21:35,950 --> 00:21:38,650
How many components does x have?
339
00:21:38,650 --> 00:21:39,150
Four.
340
00:21:39,150 --> 00:21:40,280
There are four unknowns.
341
00:21:40,280 --> 00:21:47,150
So I have to draw a four
dimensional picture on this MIT
342
00:21:47,150 --> 00:21:48,450
cheap blackboard.
343
00:21:48,450 --> 00:21:48,950
OK.
344
00:21:48,950 --> 00:21:50,970
So here we go.
345
00:21:50,970 --> 00:21:58,080
x1 -- Einstein could do
it, but, this, this is --
346
00:21:58,080 --> 00:22:06,090
those are four
perpendicular axes in --
347
00:22:06,090 --> 00:22:08,740
representing four
dimensional space.
348
00:22:08,740 --> 00:22:09,720
OK.
349
00:22:09,720 --> 00:22:12,520
Where are my solutions?
350
00:22:12,520 --> 00:22:16,580
Do my solutions form a subspace?
351
00:22:16,580 --> 00:22:20,470
Does the set of solutions
to Ax=b form a subspace?
352
00:22:20,470 --> 00:22:21,900
No way.
353
00:22:21,900 --> 00:22:23,890
What does it actually
look like, though?
354
00:22:23,890 --> 00:22:26,530
A subspace is in this picture.
355
00:22:26,530 --> 00:22:30,250
This part is a subspace, right?
356
00:22:30,250 --> 00:22:33,060
That part is some,
like, two dimensional,
357
00:22:33,060 --> 00:22:35,860
because I've got two
parameters, so it's --
358
00:22:35,860 --> 00:22:41,120
I'm thinking of this null space
as a two dimensional subspace
359
00:22:41,120 --> 00:22:42,890
inside R^4.
360
00:22:42,890 --> 00:22:46,410
Now I have to tell you and
will tell you next time,
361
00:22:46,410 --> 00:22:49,760
what does it mean to say a
subspace, what's the dimension
362
00:22:49,760 --> 00:22:50,580
of a subspace.
363
00:22:50,580 --> 00:22:52,680
But you see what
it's going to be.
364
00:22:52,680 --> 00:22:58,180
It's the number of free
independent constants
365
00:22:58,180 --> 00:22:59,750
that we can choose.
366
00:22:59,750 --> 00:23:03,560
So somehow there'll be a two
dimensional subspace, not
367
00:23:03,560 --> 00:23:07,860
a line, and not a three
dimensional plane, but only
368
00:23:07,860 --> 00:23:10,110
a two dimensional guy.
369
00:23:10,110 --> 00:23:12,710
But it's doesn't go
through the origin
370
00:23:12,710 --> 00:23:15,600
because it goes
through this point.
371
00:23:15,600 --> 00:23:17,350
So there's x particular.
372
00:23:17,350 --> 00:23:19,970
x particular is somewhere here.
373
00:23:19,970 --> 00:23:21,580
x particular.
374
00:23:21,580 --> 00:23:25,750
So it's somehow a subspace --
can I try to draw it that way?
375
00:23:28,950 --> 00:23:36,070
It's a two dimensional subspace
that goes through x particular
376
00:23:36,070 --> 00:23:39,970
and then onwards by --
so there's x particular,
377
00:23:39,970 --> 00:23:44,090
and I added on
xn, and there's x.
378
00:23:44,090 --> 00:23:46,420
There's x=xp+xn.
379
00:23:46,420 --> 00:23:51,430
But the xn was anywhere
in this subspace,
380
00:23:51,430 --> 00:23:56,700
so that filled out a plane.
381
00:23:56,700 --> 00:23:58,930
It's a subspace --
382
00:23:58,930 --> 00:24:02,320
it's not a subspace,
what I saying?
383
00:24:02,320 --> 00:24:05,220
It's like a flat thing,
it's like a subspace,
384
00:24:05,220 --> 00:24:08,450
but it's been shifted,
away from the origin.
385
00:24:08,450 --> 00:24:11,450
It doesn't contain zero.
386
00:24:11,450 --> 00:24:12,200
Thanks.
387
00:24:12,200 --> 00:24:13,150
OK.
388
00:24:13,150 --> 00:24:16,340
That's the picture, and
that's the algorithm.
389
00:24:16,340 --> 00:24:20,940
So the algorithm is just
go through elimination
390
00:24:20,940 --> 00:24:25,150
and, find the
particular solution,
391
00:24:25,150 --> 00:24:27,290
and then find those
special solutions.
392
00:24:27,290 --> 00:24:30,010
You can do that.
393
00:24:30,010 --> 00:24:34,620
Let me take our time here
in the lecture to think,
394
00:24:34,620 --> 00:24:39,240
about the bigger picture.
395
00:24:39,240 --> 00:24:43,030
So let me think about --
396
00:24:43,030 --> 00:24:46,050
so this is my pattern.
397
00:24:46,050 --> 00:24:47,040
Now I want to think --
398
00:24:47,040 --> 00:24:54,410
I want to ask you
about a question --
399
00:24:54,410 --> 00:24:57,260
I want to ask you
some questions.
400
00:24:57,260 --> 00:25:01,070
So when I mean think bigger,
I mean I'll think about an m
401
00:25:01,070 --> 00:25:09,495
by n matrix A of rank r.
402
00:25:12,640 --> 00:25:13,140
OK.
403
00:25:15,820 --> 00:25:17,630
What's our definition of rank?
404
00:25:17,630 --> 00:25:22,910
Our current definition of
rank is number of pivots.
405
00:25:22,910 --> 00:25:23,410
OK.
406
00:25:23,410 --> 00:25:26,400
First of all, how are
these numbers related?
407
00:25:26,400 --> 00:25:31,050
Can you tell me a
relation between r and m?
408
00:25:31,050 --> 00:25:35,860
If I have m rows in the
matrix and R pivots, --
409
00:25:35,860 --> 00:25:42,240
then I certainly know, always --
410
00:25:42,240 --> 00:25:46,890
what relation do I
know between r and m?
411
00:25:46,890 --> 00:25:49,810
r is less or equal, right?
412
00:25:49,810 --> 00:25:53,620
Because I've got m rows, I
can't have more than m pivots,
413
00:25:53,620 --> 00:25:56,720
I might have m and
I might have fewer.
414
00:25:56,720 --> 00:26:01,980
Also, I've got n columns.
415
00:26:01,980 --> 00:26:04,630
So what's the relation
between r and n?
416
00:26:04,630 --> 00:26:10,480
It's the same, less or
equal, because a column
417
00:26:10,480 --> 00:26:14,150
can't have more than one pivot.
418
00:26:14,150 --> 00:26:17,450
So I can't have more
than n pivots altogether.
419
00:26:17,450 --> 00:26:18,740
OK, OK.
420
00:26:18,740 --> 00:26:22,360
So I have an m by
n matrix of rank r.
421
00:26:22,360 --> 00:26:25,420
And I always know r less than
or equal to m, r less than
422
00:26:25,420 --> 00:26:26,560
or equal to n.
423
00:26:26,560 --> 00:26:29,870
Now I'm specially
interested in the case
424
00:26:29,870 --> 00:26:35,540
of full rank, when the rank
r is as big as it can be.
425
00:26:35,540 --> 00:26:40,840
Well, I guess I've got two
separate possibilities here,
426
00:26:40,840 --> 00:26:44,310
depending on what these
numbers m and n are.
427
00:26:44,310 --> 00:26:49,970
So let me talk about the
case of full column rank.
428
00:26:53,200 --> 00:26:54,958
And by that I mean r=n.
429
00:27:02,330 --> 00:27:11,960
And I want to ask you, what does
that imply about our solutions?
430
00:27:11,960 --> 00:27:16,190
What does that tell us
about the null space?
431
00:27:16,190 --> 00:27:21,240
What does that tell us
about, the complete solution?
432
00:27:21,240 --> 00:27:22,860
OK, so what does that mean?
433
00:27:22,860 --> 00:27:28,520
So I want to ask you,
well, OK, if the rank is
434
00:27:28,520 --> 00:27:31,580
n, what does that mean?
435
00:27:31,580 --> 00:27:35,570
That means there's a
pivot in every column.
436
00:27:35,570 --> 00:27:39,280
So how many pivot
variables are there?
437
00:27:39,280 --> 00:27:41,160
n.
438
00:27:41,160 --> 00:27:45,150
All the columns have
pivots in this case.
439
00:27:45,150 --> 00:27:48,150
So how many free
variables are there?
440
00:27:48,150 --> 00:27:50,460
None at all.
441
00:27:50,460 --> 00:27:52,770
So no free variables.
442
00:27:52,770 --> 00:27:54,770
r=n, no free variables.
443
00:27:57,820 --> 00:28:00,270
So what does that
tell us about what's
444
00:28:00,270 --> 00:28:04,610
going to happen then in our,
in our little algorithms?
445
00:28:04,610 --> 00:28:07,440
What will be in the null space?
446
00:28:07,440 --> 00:28:13,570
The null space of A
has got what in it?
447
00:28:13,570 --> 00:28:15,990
Only the zero vector.
448
00:28:15,990 --> 00:28:20,590
There are no free variables
to give other values to.
449
00:28:20,590 --> 00:28:23,780
So the null space is
only the zero vector.
450
00:28:29,770 --> 00:28:33,580
And what about our
solution to Ax=b?
451
00:28:33,580 --> 00:28:38,510
Solution to Ax=b?
452
00:28:38,510 --> 00:28:41,790
What, what's the
story on that one?
453
00:28:41,790 --> 00:28:43,950
So now that's coming
from today's lecture.
454
00:28:47,270 --> 00:28:51,020
The solution x is --
455
00:28:51,020 --> 00:28:52,290
what's the complete solution?
456
00:28:55,920 --> 00:28:59,570
It's just x particular, right?
457
00:28:59,570 --> 00:29:02,830
If, if, if there is an x,
if there is a solution.
458
00:29:02,830 --> 00:29:05,190
It's x equal x particular.
459
00:29:05,190 --> 00:29:08,900
There's nothing -- you know,
there's just one solution.
460
00:29:08,900 --> 00:29:11,050
If there's one at all.
461
00:29:11,050 --> 00:29:13,920
So it's unique solution --
462
00:29:13,920 --> 00:29:16,900
unique means only one --
463
00:29:16,900 --> 00:29:22,945
unique solution if it
exists, if it exists.
464
00:29:26,430 --> 00:29:29,880
In other words, I would say --
let me put it a different way.
465
00:29:29,880 --> 00:29:32,910
There're either zero
or one solutions.
466
00:29:38,940 --> 00:29:40,920
This is all in this case r=n.
467
00:29:45,140 --> 00:29:50,400
So I'm -- because many, many
applications in reality,
468
00:29:50,400 --> 00:29:55,693
the columns will be what
I'll later call independent.
469
00:29:58,710 --> 00:30:04,340
And we'll have, nothing to
look for in the null space,
470
00:30:04,340 --> 00:30:06,611
and we'll only have
particular solutions.
471
00:30:06,611 --> 00:30:07,110
OK.
472
00:30:09,970 --> 00:30:13,500
Everybody see that possibility?
473
00:30:13,500 --> 00:30:15,920
But I need an example, right?
474
00:30:15,920 --> 00:30:18,590
So let me create an example.
475
00:30:18,590 --> 00:30:23,170
What sort of a matrix -- what's
the shape of a matrix that has
476
00:30:23,170 --> 00:30:25,390
full column rank?
477
00:30:25,390 --> 00:30:30,140
So can I squeeze in
an, an example here?
478
00:30:30,140 --> 00:30:35,080
If it exists.
479
00:30:35,080 --> 00:30:38,150
Let me put in an example,
and it's just the right space
480
00:30:38,150 --> 00:30:41,140
to put in an example.
481
00:30:41,140 --> 00:30:45,330
Because the example will
be like tall and thin.
482
00:30:45,330 --> 00:30:47,940
It will have --
483
00:30:47,940 --> 00:30:54,140
well, I mean, here's an example,
one two six five, three one
484
00:30:54,140 --> 00:30:55,130
one one.
485
00:30:55,130 --> 00:30:56,610
Brilliant example.
486
00:30:56,610 --> 00:30:57,590
OK.
487
00:30:57,590 --> 00:31:06,039
So there's a matrix A,
and what's its rank?
488
00:31:06,039 --> 00:31:07,330
What's the rank of that matrix?
489
00:31:10,000 --> 00:31:12,235
How many pivots will I
find if I do elimination?
490
00:31:14,810 --> 00:31:15,810
Two, right?
491
00:31:15,810 --> 00:31:16,990
Two.
492
00:31:16,990 --> 00:31:20,320
I see a pivot there --
493
00:31:20,320 --> 00:31:23,730
oh certainly those two
columns are headed off
494
00:31:23,730 --> 00:31:26,810
in different directions.
495
00:31:26,810 --> 00:31:29,870
When I do elimination, I'll
certainly get another pivot
496
00:31:29,870 --> 00:31:35,270
here, fine, and I can use those
to clean out below and above.
497
00:31:35,270 --> 00:31:43,960
So the -- actually, tell me
what its row reduced row echelon
498
00:31:43,960 --> 00:31:45,360
form would be.
499
00:31:45,360 --> 00:31:49,410
Can you carry that,
that elimination
500
00:31:49,410 --> 00:31:52,360
process to the bitter end?
501
00:31:52,360 --> 00:31:54,590
So what do, what does that mean?
502
00:31:54,590 --> 00:31:57,610
I subtract a multiple of
this row from these rows.
503
00:31:57,610 --> 00:32:00,910
So I clean up, all zeros there.
504
00:32:00,910 --> 00:32:02,540
Then I've got some pivot here.
505
00:32:02,540 --> 00:32:04,140
What do I do with that?
506
00:32:04,140 --> 00:32:07,330
I go subtract it
below and above,
507
00:32:07,330 --> 00:32:11,860
and then I divide through,
and what's R for that example?
508
00:32:11,860 --> 00:32:14,450
Maybe I can -- you'll allow
me to put that just here
509
00:32:14,450 --> 00:32:16,580
in the next board.
510
00:32:16,580 --> 00:32:21,150
What's the row reduced echelon
form, just out of practice,
511
00:32:21,150 --> 00:32:25,300
for that matrix?
512
00:32:25,300 --> 00:32:28,800
It's got ones in the pivots.
513
00:32:28,800 --> 00:32:31,750
It's got the identity matrix,
a little two by two identity
514
00:32:31,750 --> 00:32:34,270
matrix, and below it all zeros.
515
00:32:37,850 --> 00:32:43,940
That's a matrix that really
has two independent rows,
516
00:32:43,940 --> 00:32:45,510
and they're the
first two, actually.
517
00:32:45,510 --> 00:32:47,280
The first two rows
are independent.
518
00:32:47,280 --> 00:32:49,190
They're not in the
same direction.
519
00:32:49,190 --> 00:32:52,660
But the other rows are
combinations of the first two,
520
00:32:52,660 --> 00:32:55,970
so --
521
00:32:55,970 --> 00:32:59,850
is there always a
solution to Ax=b?
522
00:32:59,850 --> 00:33:02,050
Tell me what's the picture here?
523
00:33:02,050 --> 00:33:06,880
For this matrix A, this is
a case of full column rank.
524
00:33:06,880 --> 00:33:11,320
The two columns are
-- give two pivots.
525
00:33:11,320 --> 00:33:13,090
There's nothing
in the null space.
526
00:33:13,090 --> 00:33:15,540
There's no combination
of those columns
527
00:33:15,540 --> 00:33:19,001
that gives the zero column
except the zero zero
528
00:33:19,001 --> 00:33:19,500
combination.
529
00:33:22,430 --> 00:33:25,400
So there's nothing
in the null space.
530
00:33:25,400 --> 00:33:29,830
But is there always a
solution to A X equal B?
531
00:33:29,830 --> 00:33:31,530
What's up with A X equal B?
532
00:33:34,390 --> 00:33:38,220
I've got four, four equations
here, and only two Xs.
533
00:33:40,840 --> 00:33:42,460
So the answer is certainly no.
534
00:33:42,460 --> 00:33:45,240
There's not always a solution.
535
00:33:45,240 --> 00:33:49,660
I may have zero solutions,
and if I make a random choice,
536
00:33:49,660 --> 00:33:51,590
I'll have zero solutions.
537
00:33:51,590 --> 00:33:55,690
Or if I make a great particular
choice of the right-hand side,
538
00:33:55,690 --> 00:33:59,160
which just happens to be a
combination of those two guys
539
00:33:59,160 --> 00:34:01,500
-- like tell me one right-hand
side that would have
540
00:34:01,500 --> 00:34:03,540
a solution.
541
00:34:03,540 --> 00:34:07,190
Tell me a right-hand side
that would have a solution.
542
00:34:07,190 --> 00:34:09,800
Well, 0 0 0 0, OK.
543
00:34:09,800 --> 00:34:12,880
No prize for that one.
544
00:34:12,880 --> 00:34:14,250
Tell me another one.
545
00:34:14,250 --> 00:34:18,850
Another right-hand side that
has a solution would be 4 3 7 6.
546
00:34:18,850 --> 00:34:21,900
I could add the two columns.
547
00:34:21,900 --> 00:34:25,070
What would be the total
complete solution if the
548
00:34:25,070 --> 00:34:28,360
Right? right-hand
side was 4 3 7 6?
549
00:34:28,360 --> 00:34:31,489
There would be the
particular solution one
550
00:34:31,489 --> 00:34:34,067
one, one of that column
plus one of that,
551
00:34:34,067 --> 00:34:35,150
and that would be the only
552
00:34:35,150 --> 00:34:36,429
solution.
553
00:34:36,429 --> 00:34:39,770
So there would be -- x
particular would be one one
554
00:34:39,770 --> 00:34:43,560
in the case when the right
side is the sum of those two
555
00:34:43,560 --> 00:34:46,850
columns, and that's it.
556
00:34:46,850 --> 00:34:50,670
So that would be a
case with one solution.
557
00:34:50,670 --> 00:34:51,250
OK.
558
00:34:51,250 --> 00:34:55,469
That, this is the typical
setup with full column rank.
559
00:34:55,469 --> 00:35:00,000
Now I go to full row rank.
560
00:35:00,000 --> 00:35:04,260
You see the sort of natural
symmetry of this discussion.
561
00:35:04,260 --> 00:35:14,523
Full row rank means r=m.
562
00:35:17,390 --> 00:35:19,940
So this is what I'm
interested in now, r=m.
563
00:35:23,420 --> 00:35:24,710
OK, what's up with that?
564
00:35:29,830 --> 00:35:31,010
How many pivots?
565
00:35:31,010 --> 00:35:33,000
m.
566
00:35:33,000 --> 00:35:40,060
So what happens when we do
elimination in that case?
567
00:35:40,060 --> 00:35:42,520
I'm going to get m pivots.
568
00:35:42,520 --> 00:35:47,520
So every row has a pivot, right?
569
00:35:47,520 --> 00:35:48,855
Every row has a pivot.
570
00:35:52,120 --> 00:35:55,950
Then what about solvability?
571
00:35:55,950 --> 00:35:59,880
What about this business of --
for which right-hand sides can
572
00:35:59,880 --> 00:36:01,120
I solve it?
573
00:36:01,120 --> 00:36:02,970
So that's my question.
574
00:36:02,970 --> 00:36:14,180
I can solve Ax=b for
which right-hand sides?
575
00:36:14,180 --> 00:36:18,450
Do you see what's coming?
576
00:36:18,450 --> 00:36:23,990
I do elimination, I
don't get any zero rows.
577
00:36:23,990 --> 00:36:26,890
So there aren't any
requirements on b.
578
00:36:26,890 --> 00:36:29,730
I can solve Ax=b for every b.
579
00:36:36,450 --> 00:36:39,990
I can solve Ax=b for
every right-hand side.
580
00:36:39,990 --> 00:36:46,705
So this is the existence,
exists a solution.
581
00:36:49,260 --> 00:36:57,180
Now tell me, so the, u- u- so
every row has a pivot in it.
582
00:36:57,180 --> 00:37:00,820
So how many free
variables are there?
583
00:37:00,820 --> 00:37:04,780
How many free
variables in this case?
584
00:37:04,780 --> 00:37:07,560
If I had n variables
to start with,
585
00:37:07,560 --> 00:37:11,180
how many are used up
by pivot variables?
586
00:37:11,180 --> 00:37:13,890
r, which is m.
587
00:37:13,890 --> 00:37:25,600
So I'm left with, left
with n-r free variables.
588
00:37:31,050 --> 00:37:31,910
OK.
589
00:37:31,910 --> 00:37:37,220
So this case of full row
rank I can always solve,
590
00:37:37,220 --> 00:37:41,440
and then this tells me how
many variables are free,
591
00:37:41,440 --> 00:37:43,400
and this is of course n-m.
592
00:37:43,400 --> 00:37:48,040
This is n-m free variables.
593
00:37:48,040 --> 00:37:48,980
Can I do an example?
594
00:37:52,310 --> 00:37:54,750
You know, the best way for
me to do an example is just
595
00:37:54,750 --> 00:37:58,140
to transpose that example.
596
00:37:58,140 --> 00:38:01,950
So let me take, let me take
that matrix that had column one
597
00:38:01,950 --> 00:38:05,970
two six five and make it a row.
598
00:38:05,970 --> 00:38:11,470
And let me take three one
one one as the second row.
599
00:38:11,470 --> 00:38:18,700
And let me ask you, this is
my matrix A, what's its rank?
600
00:38:18,700 --> 00:38:20,230
What's the rank of that matrix?
601
00:38:20,230 --> 00:38:24,560
Sorry to ask, but
not sorry really,
602
00:38:24,560 --> 00:38:27,130
because we're just
getting the idea of rank.
603
00:38:27,130 --> 00:38:29,770
What's the rank of that matrix?
604
00:38:29,770 --> 00:38:32,260
Two, exactly, two.
605
00:38:32,260 --> 00:38:33,770
There will be two pivots.
606
00:38:33,770 --> 00:38:36,770
What will the row
reduced echelon form be?
607
00:38:36,770 --> 00:38:38,850
Anybody know that one?
608
00:38:38,850 --> 00:38:42,230
Actually, tell me not only --
you have to tell me not only
609
00:38:42,230 --> 00:38:45,110
the, there'll be two pivots
but which will be the pivot
610
00:38:45,110 --> 00:38:46,550
columns.
611
00:38:46,550 --> 00:38:50,060
Which columns of this matrix
will be pivot columns?
612
00:38:50,060 --> 00:38:53,140
So the first column
is fine, and then
613
00:38:53,140 --> 00:38:55,720
I go on to the next
column, and what do I get?
614
00:38:55,720 --> 00:38:57,640
Do I get a second
pivot out of --
615
00:38:57,640 --> 00:39:00,410
will I get a second
pivot in this position?
616
00:39:00,410 --> 00:39:01,300
Yes.
617
00:39:01,300 --> 00:39:07,200
So the pivots, when I get all
the way to R, will be there.
618
00:39:07,200 --> 00:39:13,860
And here will be some numbers.
619
00:39:13,860 --> 00:39:18,670
This is the part that
I previously called F.
620
00:39:18,670 --> 00:39:23,720
This is the part that -- the
pivot columns in R will be
621
00:39:23,720 --> 00:39:25,680
the identity matrix.
622
00:39:25,680 --> 00:39:31,300
There are no zero rows, no zero
rows, because the rank is two.
623
00:39:31,300 --> 00:39:34,630
But there'll be stuff over here.
624
00:39:34,630 --> 00:39:42,530
And that will, enter the special
solutions and the null space.
625
00:39:42,530 --> 00:39:43,230
OK.
626
00:39:43,230 --> 00:39:51,840
So this is a typical matrix
with r=m smaller than n.
627
00:39:51,840 --> 00:39:56,135
Now finally I've got a
space here for r=m=n.
628
00:40:01,540 --> 00:40:06,190
I'm off in the corner here with
the most important case of all.
629
00:40:06,190 --> 00:40:08,910
So what's up with this matrix?
630
00:40:08,910 --> 00:40:10,970
So let me give an example.
631
00:40:10,970 --> 00:40:15,065
OK, brilliant example, 1 2 3 1.
632
00:40:19,860 --> 00:40:23,628
Tell me what -- how do I
describe a matrix that has rank
633
00:40:23,628 --> 00:40:24,127
r=m=n?
634
00:40:26,930 --> 00:40:32,560
So the matrix is square,
right, it's a square matrix.
635
00:40:32,560 --> 00:40:36,350
And if I know its rank is
-- it's full rank, now.
636
00:40:36,350 --> 00:40:39,290
I don't have to say full
column rank or full row rank --
637
00:40:39,290 --> 00:40:43,800
I just say full rank, because
the count, column count
638
00:40:43,800 --> 00:40:47,130
and the row count are
the same, and the rank
639
00:40:47,130 --> 00:40:49,040
is as big as it can be.
640
00:40:49,040 --> 00:40:51,090
And what kind of a
matrix have I got?
641
00:40:53,920 --> 00:40:56,670
It's invertible.
642
00:40:56,670 --> 00:41:01,510
So that's exactly the
invertible matrices.
643
00:41:01,510 --> 00:41:06,310
r=m=n means the -- what's
the row echelon form,
644
00:41:06,310 --> 00:41:10,920
the reduced row echelon form,
for an invertible matrix?
645
00:41:10,920 --> 00:41:14,630
For a square, nice,
square, invertible matrix?
646
00:41:14,630 --> 00:41:17,320
It's I.
647
00:41:17,320 --> 00:41:18,880
Right.
648
00:41:18,880 --> 00:41:25,530
So you see that the,
the good matrices
649
00:41:25,530 --> 00:41:31,580
are the ones that kind of
come out trivially in R.
650
00:41:31,580 --> 00:41:34,270
You reduce them all the
way to the identity matrix.
651
00:41:34,270 --> 00:41:37,900
What's the null space for
this, for this matrix?
652
00:41:37,900 --> 00:41:41,170
Can I just hammer
away with questions?
653
00:41:41,170 --> 00:41:43,160
What's the null space
for this matrix?
654
00:41:45,910 --> 00:41:51,570
The null space of that matrix
is the zero vector only.
655
00:41:51,570 --> 00:41:54,660
The zero vector only.
656
00:41:54,660 --> 00:41:58,530
What are the conditions
to solve Ax=b?
657
00:41:58,530 --> 00:42:01,950
Which right-hand sides b are OK?
658
00:42:01,950 --> 00:42:07,560
If I want to solve Ax=b for
this example, so A is this,
659
00:42:07,560 --> 00:42:14,140
b is b1 b2, what are the
conditions on b1 and b2?
660
00:42:14,140 --> 00:42:17,170
None at all, right.
661
00:42:17,170 --> 00:42:21,850
So this is the case, this is
the case where I can solve --
662
00:42:21,850 --> 00:42:25,660
so I've coming back here, I
can -- since the rank equals m,
663
00:42:25,660 --> 00:42:28,460
I can solve for every b.
664
00:42:28,460 --> 00:42:33,840
And since the rank is also
n, there's a unique solution.
665
00:42:33,840 --> 00:42:36,640
Let me summarize the
whole picture here.
666
00:42:39,210 --> 00:42:41,210
Here's the whole picture.
667
00:42:41,210 --> 00:42:44,780
I could have r=m=n.
668
00:42:44,780 --> 00:42:51,190
This is the case where this
is the identity matrix.
669
00:42:51,190 --> 00:42:53,595
And this is the case where
there is one solution.
670
00:42:56,790 --> 00:43:02,300
That's the square
invertible chapter two case.
671
00:43:02,300 --> 00:43:04,420
Now we're into chapter three.
672
00:43:04,420 --> 00:43:08,330
We could have r=m
smaller than n.
673
00:43:11,920 --> 00:43:13,730
Now that's what
we had over there,
674
00:43:13,730 --> 00:43:17,610
and the row echelon form
looked like the identity
675
00:43:17,610 --> 00:43:18,960
with some zero rows.
676
00:43:21,960 --> 00:43:27,565
And that was the case where
there are zero or one solution.
677
00:43:31,420 --> 00:43:34,420
When I say solution
I mean to Ax=b.
678
00:43:37,540 --> 00:43:39,590
So this case,
there's always one.
679
00:43:39,590 --> 00:43:42,130
This case there's zero or one.
680
00:43:42,130 --> 00:43:45,640
And now let me take the
case of full column rank,
681
00:43:45,640 --> 00:43:55,860
but some, extra rows.
682
00:43:55,860 --> 00:44:00,000
So now R has --
683
00:44:00,000 --> 00:44:04,700
well, the identity --
684
00:44:04,700 --> 00:44:08,280
I'm almost tempted to write
the identity matrix and then F,
685
00:44:08,280 --> 00:44:10,130
but that isn't
necessarily right.
686
00:44:15,970 --> 00:44:20,360
I have -- is that right?
687
00:44:20,360 --> 00:44:24,310
Am I getting this correct here?
688
00:44:24,310 --> 00:44:25,430
Oh, I'm not!
689
00:44:25,430 --> 00:44:26,390
My God!
690
00:44:26,390 --> 00:44:33,330
This is the case R equals n,
the columns, the columns are,
691
00:44:33,330 --> 00:44:34,390
are OK.
692
00:44:34,390 --> 00:44:38,770
That's the case that was on that
board, r=n, full column rank.
693
00:44:38,770 --> 00:44:43,240
Now I want the case
where m is smaller than n
694
00:44:43,240 --> 00:44:46,550
and I've got extra columns.
695
00:44:46,550 --> 00:44:47,050
OK.
696
00:44:47,050 --> 00:44:47,810
There we go.
697
00:44:52,030 --> 00:44:57,140
So this is now the
case of full row rank,
698
00:44:57,140 --> 00:45:02,070
and it looks like
I F except that I
699
00:45:02,070 --> 00:45:08,430
can't be sure that the pivot
columns are the first columns.
700
00:45:08,430 --> 00:45:14,690
So the I and the F, could
be partly mixed into the I.
701
00:45:14,690 --> 00:45:18,670
Can I write that
with just like that?
702
00:45:18,670 --> 00:45:23,830
So the F could be sort
of partly into the I
703
00:45:23,830 --> 00:45:28,450
if the first columns
weren't the pivot columns.
704
00:45:28,450 --> 00:45:31,760
Now how many solutions
in this case?
705
00:45:31,760 --> 00:45:34,330
There's always a solution.
706
00:45:34,330 --> 00:45:35,850
This is the existence case.
707
00:45:35,850 --> 00:45:37,020
There's always a solution.
708
00:45:37,020 --> 00:45:39,160
We're not getting any zero rows.
709
00:45:39,160 --> 00:45:41,430
There are no zero rows here.
710
00:45:41,430 --> 00:45:45,635
So there's always either one
or infinitely many solutions.
711
00:45:50,061 --> 00:45:50,560
OK.
712
00:45:53,230 --> 00:45:55,600
Actually, I guess there's
always an infinite number,
713
00:45:55,600 --> 00:46:02,760
because we always have some
null space to deal with.
714
00:46:02,760 --> 00:46:06,150
Then the final case is
where r is smaller than m
715
00:46:06,150 --> 00:46:08,880
and smaller than n.
716
00:46:08,880 --> 00:46:09,380
OK.
717
00:46:09,380 --> 00:46:14,830
Now that's the case
where R is the identity
718
00:46:14,830 --> 00:46:19,920
with some free stuff but
with some zero rows too.
719
00:46:19,920 --> 00:46:23,580
And that's the case where
there's either no solution --
720
00:46:23,580 --> 00:46:29,200
because we didn't get a zero
equals zero for some bs,
721
00:46:29,200 --> 00:46:32,330
or infinitely many solutions.
722
00:46:37,100 --> 00:46:38,010
OK.
723
00:46:38,010 --> 00:46:44,310
Do you -- this board really
summarizes the lecture,
724
00:46:44,310 --> 00:46:47,370
and this sentence
summarizes the lecture.
725
00:46:47,370 --> 00:46:55,070
The rank tells you everything
about the number of solutions.
726
00:46:55,070 --> 00:46:57,010
That number, the
rank r, tells you
727
00:46:57,010 --> 00:47:01,620
all the information except the
exact entries in the solutions.
728
00:47:01,620 --> 00:47:04,140
For that you go to the matrix.
729
00:47:04,140 --> 00:47:05,150
OK, good.
730
00:47:05,150 --> 00:47:08,770
Have a great weekend, and
I'll see you on Monday.