1
00:00:00.3 --> 00:00:00
OK.
2
00:00:00 --> 00:00:03
This is lecture twelve.
3
00:00:03 --> 00:00:07
We've reached twelve lectures.
4
00:00:07 --> 00:00:14
And this one is more than the
others about applications of
5
00:00:14 --> 00:00:16
linear algebra.
6
00:00:16 --> 00:00:19
And I'll confess.
7
00:00:19 --> 00:00:26.46
When I'm giving you examples of
the null space and the row
8
00:00:26.46 --> 00:00:31.9
space, I create a little matrix.
9
00:00:31.9 --> 00:00:37
You probably see that I just
invent that matrix as I'm going.
10
00:00:37 --> 00:00:42
And I feel a little guilty
about it, because the truth is
11
00:00:42 --> 00:00:47
that real linear algebra uses
matrices that come from
12
00:00:47 --> 00:00:48.34
somewhere.
13
00:00:48.34 --> 00:00:52
They're not just,
like, randomly invented by the
14
00:00:52 --> 00:00:53
instructor.
15
00:00:53 --> 00:00:57.3
They come from applications.
16
00:00:57.3 --> 00:01:00
They have a definite structure.
17
00:01:00 --> 00:01:06
And anybody who works with them
gets, uses that structure.
18
00:01:06 --> 00:01:10.65
I'll just report,
like, this weekend I was at an
19
00:01:10.65 --> 00:01:13
event with chemistry professors.
20
00:01:13 --> 00:01:19
OK, those guys are row reducing
matrices, and what matrices are
21
00:01:19 --> 00:01:22
they working with?
22
00:01:22 --> 00:01:26
Well, their little matrices
tell them how much of each
23
00:01:26 --> 00:01:31
element goes into the -- or each
molecule, how many molecules of
24
00:01:31 --> 00:01:34
each go into a reaction and what
comes out.
25
00:01:34 --> 00:01:38
And by row reduction they get a
clearer picture of a complicated
26
00:01:38 --> 00:01:39
reaction.
27
00:01:39 --> 00:01:43
And this weekend I'm going to
-- to a sort of birthday party
28
00:01:43 --> 00:01:45.7
at Mathworks.
29
00:01:45.7 --> 00:01:49
So Mathworks is out Route 9 in
Natick.
30
00:01:49 --> 00:01:53
That's where Matlab is created.
31
00:01:53 --> 00:01:56
It's a very,
very successful,
32
00:01:56 --> 00:02:00
software, tremendously
successful.
33
00:02:00 --> 00:02:06
And the conference will be
about how linear algebra is
34
00:02:06 --> 00:02:07.9
used.
35
00:02:07.9 --> 00:02:13
And so I feel better today to
talk about what I think is the
36
00:02:13 --> 00:02:17
most important model in applied
math.
37
00:02:17 --> 00:02:20.95
And the discrete version is a
graph.
38
00:02:20.95 --> 00:02:23
So can I draw a graph?
39
00:02:23 --> 00:02:27
Write down the matrix that's
associated with it,
40
00:02:27 --> 00:02:31
and that's a great source of
matrices.
41
00:02:31 --> 00:02:33.9
You'll see.
42
00:02:33.9 --> 00:02:39
So a graph is just,
so a graph -- to repeat -- has
43
00:02:39 --> 00:02:40
nodes and edges.
44
00:02:40 --> 00:02:41
OK.
45
00:02:41 --> 00:02:46
And I'm going to write down the
graph, a graph,
46
00:02:46 --> 00:02:50
so I'm just creating a small
graph here.
47
00:02:50 --> 00:02:56
As I mentioned last time,
we would be very interested in
48
00:02:56 --> 00:03:00
the graph of all,
websites.
49
00:03:00 --> 00:03:03
Or the graph of all telephones.
50
00:03:03 --> 00:03:08
I mean -- or the graph of all
people in the world.
51
00:03:08 --> 00:03:14
Here let me take just,
maybe nodes one two three --
52
00:03:14 --> 00:03:20
well, I better put in an -- I'll
put in that edge and maybe an
53
00:03:20 --> 00:03:27.3
edge to, to a node four,
and another edge to node four.
54
00:03:27.3 --> 00:03:28
How's that?
55
00:03:28 --> 00:03:32
So there's a graph with four
nodes.
56
00:03:32 --> 00:03:37
So n will be four in my -- n
equal four nodes.
57
00:03:37 --> 00:03:43
And the matrix will have m
equal the number -- there'll be
58
00:03:43 --> 00:03:48
a row for every edge,
so I've got one two three four
59
00:03:48 --> 00:03:50
five edges.
60
00:03:50 --> 00:03:55
So that will be the number of
rows.
61
00:03:55 --> 00:03:59.17
And I have to to write down the
matrix that I want to,
62
00:03:59.17 --> 00:04:02.54
I want to study,
I need to give a direction to
63
00:04:02.54 --> 00:04:06
every edge, so I know a plus and
a minus direction.
64
00:04:06 --> 00:04:08
So I'll just do that with an
arrow.
65
00:04:08 --> 00:04:11
Say from one to two,
one to three,
66
00:04:11 --> 00:04:13
two to three,
one to four,
67
00:04:13 --> 00:04:15
three to four.
68
00:04:15 --> 00:04:19
That just tells me,
if I have current flowing on
69
00:04:19 --> 00:04:25
these edges then I know whether
it's -- to count it as positive
70
00:04:25 --> 00:04:31
or negative according as whether
it's with the arrow or against
71
00:04:31 --> 00:04:32
the arrow.
72
00:04:32 --> 00:04:36
But I just drew those arrows
arbitrarily.
73
00:04:36 --> 00:04:36
OK.
74
00:04:36 --> 00:04:41
Because I --
my example is going to come --
75
00:04:41 --> 00:04:47
the example I'll -- the words
that I will use will be words
76
00:04:47 --> 00:04:50.33
like potential,
potential difference,
77
00:04:50.33 --> 00:04:51
currents.
78
00:04:51 --> 00:04:55
In other words,
I'm thinking of an electrical
79
00:04:55 --> 00:04:55
network.
80
00:04:55 --> 00:04:59
But that's just one
possibility.
81
00:04:59 --> 00:05:03
My applied math class builds on
this example.
82
00:05:03 --> 00:05:09
It could be a hydraulic
network, so we could be doing,
83
00:05:09 --> 00:05:11
flow of water,
flow of oil.
84
00:05:11 --> 00:05:15
Other examples,
this could be a structure.
85
00:05:15 --> 00:05:20.39
Like the -- a design for a
bridge or a design for a
86
00:05:20.39 --> 00:05:23
Buckminster Fuller dome.
87
00:05:23 --> 00:05:27
Or many other possibilities,
so many.
88
00:05:27 --> 00:05:31
So l- but let's take potentials
and currents as,
89
00:05:31 --> 00:05:36
as a basic example,
and let me create the matrix
90
00:05:36 --> 00:05:40
that tells you exactly what the
graph tells you.
91
00:05:40 --> 00:05:44
So now I'll call it the
incidence matrix,
92
00:05:44 --> 00:05:47.1
incidence matrix.
93
00:05:47.1 --> 00:05:47
OK.
94
00:05:47 --> 00:05:51
So let me write it down,
and you'll see,
95
00:05:51 --> 00:05:54
what its properties are.
96
00:05:54 --> 00:05:58
So every row corresponds to an
edge.
97
00:05:58 --> 00:06:04
I have five rows from five
edges, and let me write down
98
00:06:04 --> 00:06:07
again what this graph looks
like.
99
00:06:07 --> 00:06:13
OK, the first edge,
edge one, goes from node one to
100
00:06:13 --> 00:06:15.25
two.
101
00:06:15.25 --> 00:06:21
So I'm going to put in a minus
one and a plus one in th- this
102
00:06:21 --> 00:06:25.51
corresponds to node one two
three and four,
103
00:06:25.51 --> 00:06:27.22
the four columns.
104
00:06:27.22 --> 00:06:33
The five rows correspond -- the
first row corresponds to edge
105
00:06:33 --> 00:06:33
one.
106
00:06:33 --> 00:06:38
Edge one leaves node one and
goes into node two,
107
00:06:38 --> 00:06:42
and that --
and it doesn't touch three and
108
00:06:42 --> 00:06:43
four.
109
00:06:43 --> 00:06:47
Edge two, edge two goes -- oh,
I haven't numbered these edges.
110
00:06:47 --> 00:06:50
I just figured that was
probably edge one,
111
00:06:50 --> 00:06:52
but I didn't say so.
112
00:06:52 --> 00:06:54
Let me take that to be edge
one.
113
00:06:54 --> 00:06:57
Let me take this to be edge
two.
114
00:06:57 --> 00:06:59
Let me take this to be edge
three.
115
00:06:59 --> 00:07:02.1
This is edge four.
116
00:07:02.1 --> 00:07:05
Ho, I'm discovering -- no,
wait a minute.
117
00:07:05 --> 00:07:06
Did I number that twice?
118
00:07:06 --> 00:07:08.12
Here's edge four.
119
00:07:08.12 --> 00:07:09.68
And here's edge five.
120
00:07:09.68 --> 00:07:09
OK?
121
00:07:09 --> 00:07:10
All right.
122
00:07:10 --> 00:07:13
So, so edge one,
as I said, goes from node one
123
00:07:13 --> 00:07:14
to two.
124
00:07:14 --> 00:07:18
Edge two goes from two to
three, node two to three,
125
00:07:18 --> 00:07:23
so minus one and one in the
second and third columns.
126
00:07:23 --> 00:07:27
Edge three goes from one to
three.
127
00:07:27 --> 00:07:34
I'm, I'm tempted to stop for a
moment with those three edges.
128
00:07:34 --> 00:07:39
Edges one two three,
those form what would we,
129
00:07:39 --> 00:07:44
what do you call the,
the little, the little,
130
00:07:44 --> 00:07:51
the subgraph formed by edges
one, two, and three?
131
00:07:51 --> 00:07:52
That's a loop.
132
00:07:52 --> 00:07:58.32
And the number of loops and the
position of the loops will be
133
00:07:58.32 --> 00:07:59
crucial.
134
00:07:59 --> 00:07:59
OK.
135
00:07:59 --> 00:08:03
Actually, here's a interesting
point about loops.
136
00:08:03 --> 00:08:08
If I look at those rows,
corresponding to edges one two
137
00:08:08 --> 00:08:12
three, and these guys made a
loop.
138
00:08:12 --> 00:08:18
You want to tell me -- if I
just looked at that much of the
139
00:08:18 --> 00:08:22
matrix it would be natural for
me to ask, are those rows
140
00:08:22 --> 00:08:23
independent?
141
00:08:23 --> 00:08:26
Are the rows independent?
142
00:08:26 --> 00:08:31
And can you tell from looking
at that if they are or are not
143
00:08:31 --> 00:08:32.49
independent?
144
00:08:32.49 --> 00:08:36
Do you see a,
a relation between those three
145
00:08:36 --> 00:08:37
rows?
146
00:08:37 --> 00:08:38
Yes.
147
00:08:38 --> 00:08:43
If I add that row to that row,
I get this row.
148
00:08:43 --> 00:08:49
So, so that's like a hint here
that loops correspond to
149
00:08:49 --> 00:08:56
dependent, linearly dependent
column -- linearly dependent
150
00:08:56 --> 00:08:56
rows.
151
00:08:56 --> 00:09:02
OK, let me complete the
incidence matrix.
152
00:09:02 --> 00:09:06
Number four,
edge four is going from node
153
00:09:06 --> 00:09:08
one to node four.
154
00:09:08 --> 00:09:14
And the fifth edge is going
from node three to node four.
155
00:09:14 --> 00:09:14
OK.
156
00:09:14 --> 00:09:16
There's my matrix.
157
00:09:16 --> 00:09:21
It came from the five edges and
the four nodes.
158
00:09:21 --> 00:09:27
And if I had a big graph,
I'd have a big matrix.
159
00:09:27 --> 00:09:31
And what questions do I ask
about matrices?
160
00:09:31 --> 00:09:35
Can I ask -- here's the review
now.
161
00:09:35 --> 00:09:40
There's a matrix that comes
from somewhere.
162
00:09:40 --> 00:09:45
If, if it was a big graph,
it would be a large matrix,
163
00:09:45 --> 00:09:49
but a lot of zeros,
right?
164
00:09:49 --> 00:09:52
Because every row only has two
non-zeros.
165
00:09:52 --> 00:09:56
So the number of -- it's a very
sparse matrix.
166
00:09:56 --> 00:10:00.15
The number of non-zeros is
exactly two times five,
167
00:10:00.15 --> 00:10:01
it's two m.
168
00:10:01 --> 00:10:03
Every row only has two
non-zeros.
169
00:10:03 --> 00:10:06
And that's with a lot of
structure.
170
00:10:06 --> 00:10:09
And -- that was the point I
wanted to begin with,
171
00:10:09 --> 00:10:14
that graphs,
that real graphs from real --
172
00:10:14 --> 00:10:19
real matrices from genuine
problems have structure.
173
00:10:19 --> 00:10:19
OK.
174
00:10:19 --> 00:10:25
We can ask, and because of the
structure, we can answer,
175
00:10:25 --> 00:10:29
the, the main questions about
matrices.
176
00:10:29 --> 00:10:35
So first question,
what about the null space?
177
00:10:35 --> 00:10:40
So what I asking if I ask you
for the null space of that
178
00:10:40 --> 00:10:40
matrix?
179
00:10:40 --> 00:10:46
I'm asking you if I'm looking
at the columns of the matrix,
180
00:10:46 --> 00:10:49
four columns,
and I'm asking you,
181
00:10:49 --> 00:10:51
are those columns independent?
182
00:10:51 --> 00:10:58.6
If the columns are independent,
then what's in the null space?
183
00:10:58.6 --> 00:11:01
Only the zero vector,
right?
184
00:11:01 --> 00:11:07
The null space contains --
tells us what combinations of
185
00:11:07 --> 00:11:13
the columns -- it tells us how
to combine columns to get zero.
186
00:11:13 --> 00:11:20
Can -- and is there anything in
the null space of this matrix
187
00:11:20 --> 00:11:24
other than just the zero vector?
188
00:11:24 --> 00:11:28
In other words,
are those four columns
189
00:11:28 --> 00:11:30.96
independent or dependent?
190
00:11:30.96 --> 00:11:31
OK.
191
00:11:31 --> 00:11:33
That's our question.
192
00:11:33 --> 00:11:37
Let me, I don't know if you see
the answer.
193
00:11:37 --> 00:11:40
Whether there's -- so let's
see.
194
00:11:40 --> 00:11:44.6
I guess we could do it
properly.
195
00:11:44.6 --> 00:11:47
We could solve Ax=0.
196
00:11:47 --> 00:11:53
So let me solve Ax=0 to find
the null space.
197
00:11:53 --> 00:11:53
OK.
198
00:11:53 --> 00:11:55
What's Ax?
199
00:11:55 --> 00:12:00
Can I put x in here in,
in little letters?
200
00:12:00 --> 00:12:05
x1, x2, x3, x4,
that's -- it's got four
201
00:12:05 --> 00:12:06
columns.
202
00:12:06 --> 00:12:12.6
Ax now is that matrix times x.
203
00:12:12.6 --> 00:12:15
And what do I get for Ax?
204
00:12:15 --> 00:12:21
If the camera can keep that
matrix multiplication there,
205
00:12:21 --> 00:12:23
I'll put the answer here.
206
00:12:23 --> 00:12:28.47
Ax equal -- what's the first
component of Ax?
207
00:12:28.47 --> 00:12:34
Can you take that first row,
minus one one zero zero,
208
00:12:34 --> 00:12:40
and multiply by the x,
and of course you get x2-x1.
209
00:12:40 --> 00:12:44
The second row,
I get x3-x2.
210
00:12:44 --> 00:12:49
From the third row,
I get x3-x1.
211
00:12:49 --> 00:12:54
From the fourth row,
I get x4-x1.
212
00:12:54 --> 00:13:00
And from the fifth row,
I get x4-x3.
213
00:13:00 --> 00:13:08
And I want to know when is the
thing zero.
214
00:13:08 --> 00:13:10
This is my equation,
Ax=0.
215
00:13:10 --> 00:13:16
Notice what that matrix A is
doing, what we've created a
216
00:13:16 --> 00:13:21
matrix that computes the
differences across every edge,
217
00:13:21 --> 00:13:24
the differences in potential.
218
00:13:24 --> 00:13:30
Let me even begin to give this
interpretation.
219
00:13:30 --> 00:13:38.12
I'm going to think of this
vector x, which is x1 x2 x3 x4,
220
00:13:38.12 --> 00:13:42
as the potentials at the nodes.
221
00:13:42 --> 00:13:49
So I'm introducing a word,
potentials at the nodes.
222
00:13:49 --> 00:13:57
And now if I multiply by A,
I get these -- I get these five
223
00:13:57 --> 00:14:01
components, x2-x1,
et cetera.
224
00:14:01 --> 00:14:05
And what are they?
225
00:14:05 --> 00:14:08
They're potential differences.
226
00:14:08 --> 00:14:11
That's what A computes.
227
00:14:11 --> 00:14:16
If I have potentials at the
nodes and I multiply by A,
228
00:14:16 --> 00:14:21
it gives me the potential
differences, the differences in
229
00:14:21 --> 00:14:24
potential, across the edges.
230
00:14:24 --> 00:14:24
OK.
231
00:14:24 --> 00:14:29.1
When are those differences all
zero?
232
00:14:29.1 --> 00:14:31
So I'm looking for the null
space.
233
00:14:31 --> 00:14:36
Of course, if all the (x)s are
zero then I get zero.
234
00:14:36 --> 00:14:40
That, that just tells me,
of course, the zero vector is
235
00:14:40 --> 00:14:42
in the null space.
236
00:14:42 --> 00:14:45
But w- there's more in the null
space.
237
00:14:45 --> 00:14:50
Those columns are -- of A are
dependent, right --
238
00:14:50 --> 00:14:54
because I can find solutions to
that equation.
239
00:14:54 --> 00:14:57
Tell me -- the null space.
240
00:14:57 --> 00:15:03.04
Tell me one vector in the null
space, so tell me an x,
241
00:15:03.04 --> 00:15:08
it's got four components,
and it makes that thing zero.
242
00:15:08 --> 00:15:11
So what's a good x to do that?
243
00:15:11 --> 00:15:16
One one one one,
constant potential.
244
00:15:16 --> 00:13:21
If the potentials are constant,
then all the potential
245
00:13:21 --> 00:11:41
differences are zero,
and that x is in the null
246
00:11:41 --> 00:11:28
space.
247
00:11:28 --> 00:10:21
What else is in the null space?
248
00:10:21 --> 00:08:52
If it -- yeah,
let me ask you just always,
249
00:08:52 --> 00:07:38
give me a basis for the null
space.
250
00:07:38 --> 00:05:37
A basis for the null space will
be just that.1
251
00:05:37 --> 00:06:21
That's --, that's it.
252
00:06:21 --> 00:07:31
That's a basis for the null
space.
253
00:07:31 --> 00:09:38
The null space is actually one
dimensional, and it's the line
254
00:09:38 --> 00:10:46
of all vectors through that one.
255
00:10:46 --> 00:12:36
So there's a basis for it,
and here is the whole null
256
00:12:36 --> 00:12:48
space.
257
00:12:48 --> 00:14:53
Any multiple of one one one
one, it's the whole line in four
258
00:14:53 --> 00:15:57
dimensional space.
259
00:15:57 --> 00:16:01
Do you see that that's the null
space?
260
00:16:01 --> 00:16:07
So the, so the dimension of the
null space of A is one.
261
00:16:07 --> 00:16:13
And there's a basis for it,
and there's everything that's
262
00:16:13 --> 00:16:14
in it.
263
00:16:14 --> 00:16:14
Good.
264
00:16:14 --> 00:16:19.6
And what does that mean
physically?
265
00:16:19.6 --> 00:16:24
I mean, what does that mean in
the application?
266
00:16:24 --> 00:16:27
That guy in the null space.
267
00:16:27 --> 00:16:34
It means that the potentials
can only be determined up to a
268
00:16:34 --> 00:16:35
constant.
269
00:16:35 --> 00:16:41
Potential differences are what
make current flow.
270
00:16:41 --> 00:16:45
That's what makes things
happen.
271
00:16:45 --> 00:16:48
It's these potential
differences that will make
272
00:16:48 --> 00:16:50
something move in the,
in our network,
273
00:16:50 --> 00:16:52
between x2- between node two
and node one.
274
00:16:52 --> 00:16:55
Nothing will move if all
potentials are the same.
275
00:16:55 --> 00:16:57
If all potentials are c,
c, c, and c,
276
00:16:57 --> 00:16:59.19
then nothing will move.
277
00:16:59.19 --> 00:17:02
So we're, we have this one
parameter, this arbitrary
278
00:17:02 --> 00:17:05
constant that raises or drops
all the potentials.
279
00:17:05 --> 00:17:09
It's like ranking football
teams, whatever.
280
00:17:09 --> 00:17:14
We have a, there's a,
there's a constant -- or
281
00:17:14 --> 00:17:19
looking at temperatures,
you know, there's a flow of
282
00:17:19 --> 00:17:24.9
heat from higher temperature to
lower temperature.
283
00:17:24.9 --> 00:17:28
If temperatures are equal
there's no flow,
284
00:17:28 --> 00:17:34
and therefore we can measure --
we can measure temperatures by,
285
00:17:34 --> 00:17:38
Celsius or we can start at
absolute zero.
286
00:17:38 --> 00:17:44
And that arbitrary -- it's the
same arbitrary constant that,
287
00:17:44 --> 00:17:47
that was there in calculus.
288
00:17:47 --> 00:17:50
In calculus,
right, when you took the
289
00:17:50 --> 00:17:55
integral, the indefinite
integral, there was a plus c,
290
00:17:55 --> 00:18:00
and you had to set a starting
point to know what that c was.
291
00:18:00 --> 00:18:06.18
So here what often happens is
we fix one of the potentials,
292
00:18:06.18 --> 00:18:08
like the last one.
293
00:18:08 --> 00:18:13
So a typical thing would be to
ground that node.
294
00:18:13 --> 00:18:15
To set its potential at zero.
295
00:18:15 --> 00:18:19.96
And if we do that,
if we fix that potential so
296
00:18:19.96 --> 00:18:25
it's not unknown anymore,
then that column disappears and
297
00:18:25 --> 00:18:29
we have three columns,
and those three columns are
298
00:18:29 --> 00:18:31
independent.
299
00:18:31 --> 00:18:36
So I'll leave the column in
there, but we'll remember that
300
00:18:36 --> 00:18:39
grounding a node is the way to
get it out.
301
00:18:39 --> 00:18:43
And grounding a node is the way
to -- setting a node -- setting
302
00:18:43 --> 00:18:47
a potential to zero tells us
the, the base for all
303
00:18:47 --> 00:18:48
potentials.
304
00:18:48 --> 00:18:50.46
Then we can compute the others.
305
00:18:50.46 --> 00:18:50
OK.
306
00:18:50 --> 00:18:55
But what's the --
now I've talked enough to ask
307
00:18:55 --> 00:18:58
what the rank of the matrix is?
308
00:18:58 --> 00:18:59
What's the rank then?
309
00:18:59 --> 00:19:01
The rank of the matrix.
310
00:19:01 --> 00:19:04
So we have a five by four
matrix.
311
00:19:04 --> 00:19:08
We've located its null space,
one dimensional.
312
00:19:08 --> 00:19:12
How many independent columns do
we have?
313
00:19:12 --> 00:19:14.4
What's the rank?
314
00:19:14.4 --> 00:19:15
It's three.
315
00:19:15 --> 00:19:21
And the first three columns,
or actually any three columns,
316
00:19:21 --> 00:19:23
will be independent.
317
00:19:23 --> 00:19:28
Any three potentials are
independent, good variables.
318
00:19:28 --> 00:19:33
The fourth potential is not,
we need to set,
319
00:19:33 --> 00:19:36
and typically we ground that
node.
320
00:19:36 --> 00:19:38
OK.
321
00:19:38 --> 00:19:39.32
Rank is three.
322
00:19:39.32 --> 00:19:40
Rank equals three.
323
00:19:40 --> 00:19:41.15
OK.
324
00:19:41.15 --> 00:19:45
Let's see, do I want to ask you
about the column space?
325
00:19:45 --> 00:19:50
The column space is all
combinations of those columns.
326
00:19:50 --> 00:19:54.4
I could say more about it and I
will.
327
00:19:54.4 --> 00:20:00
Let me go to the null space of
A transpose, because the
328
00:20:00 --> 00:20:06
equation A transpose y equals
zero is probably the most
329
00:20:06 --> 00:20:11
fundamental equation of applied
mathematics.
330
00:20:11 --> 00:20:15
All right, let's talk about
that.
331
00:20:15 --> 00:20:18.6
That deserves our attention.
332
00:20:18.6 --> 00:20:21
A transpose y equals zero.
333
00:20:21 --> 00:20:26.4
Let's -- let me put it on here.
334
00:20:26.4 --> 00:20:26
OK.
335
00:20:26 --> 00:20:29
So A transpose y equals zero.
336
00:20:29 --> 00:20:34
So now I'm finding the null
space of A transpose.
337
00:20:34 --> 00:20:39
Oh, and if I ask you its
dimension, you could tell me
338
00:20:39 --> 00:20:40
what it is.
339
00:20:40 --> 00:20:45
What's the dimension of the
null space of A transpose?
340
00:20:45 --> 00:20:51
We now know enough to answer
that question.
341
00:20:51 --> 00:20:57
What's the general formula for
the dimension of the null space
342
00:20:57 --> 00:20:58
of A transpose?
343
00:20:58 --> 00:21:02
A transpose,
let me even write out A
344
00:21:02 --> 00:21:03
transpose.
345
00:21:03 --> 00:21:07.26
This A transpose will be n by
m, right?
346
00:21:07.26 --> 00:21:07
n by m.
347
00:21:07 --> 00:21:11
In this case,
it'll be four by five.
348
00:21:11 --> 00:21:15.9
Those columns will become rows.
349
00:21:15.9 --> 00:21:23
Minus one zero minus one minus
one zero is now the first row.
350
00:21:23 --> 00:21:30
The second row of the matrix,
one minus one and three zeros.
351
00:21:30 --> 00:21:35
The third column now becomes
the third row,
352
00:21:35 --> 00:21:38.4
zero one one zero minus one.
353
00:21:38.4 --> 00:21:43
And the fourth column becomes
the fourth row.
354
00:21:43 --> 00:21:45.9
OK, good.
355
00:21:45.9 --> 00:21:48
There's A transpose.
356
00:21:48 --> 00:21:52
That multiplies y,
y1 y2 y3 y4 and y5.
357
00:21:52 --> 00:21:52
OK.
358
00:21:52 --> 00:21:57
Now you've had time to think
about this question.
359
00:21:57 --> 00:22:04
What's the dimension of the
null space, if I set all those
360
00:22:04 --> 00:22:04
-- wow.
361
00:22:04 --> 00:22:11
Usually -- sometime during this
semester, I'll drop one of these
362
00:22:11 --> 00:22:15
erasers behind there.
363
00:22:15 --> 00:22:18
That's a great moment.
364
00:22:18 --> 00:22:22
There's no recovery.
365
00:22:22 --> 00:22:29
There's -- centuries of erasers
are back there.
366
00:22:29 --> 00:22:30.38
OK.
367
00:22:30.38 --> 00:22:38.5
OK, what's the dimension of the
null space?
368
00:22:38.5 --> 00:22:45
Give me the general formula
first in terms of r and m and n.
369
00:22:45 --> 00:22:51
This is like crucial,
you -- we struggled to,
370
00:22:51 --> 00:22:59
to decide what dimension meant,
and then we figured out what it
371
00:22:59 --> 00:23:06
equaled for an m by n matrix of
rank r, and the answer was m-r,
372
00:23:06 --> 00:23:08.75
right?
373
00:23:08.75 --> 00:23:14
There are m=5 components,
m=5 columns of A transpose.
374
00:23:14 --> 00:23:19
And r of those columns are
pivot columns,
375
00:23:19 --> 00:23:22
because it'll have r pivots.
376
00:23:22 --> 00:23:24
It has rank r.
377
00:23:24 --> 00:23:29
And m-r are the free ones now
for A transpose,
378
00:23:29 --> 00:23:35.9
so that's five minus three,
so that's two.
379
00:23:35.9 --> 00:23:39
And I would like to find this
null space.
380
00:23:39 --> 00:23:41
I know its dimension.
381
00:23:41 --> 00:23:44
Now I want to find out a basis
for it.
382
00:23:44 --> 00:23:48
And I want to understand what
this equation is.
383
00:23:48 --> 00:23:52
So let me say what A transpose
y actually represents,
384
00:23:52 --> 00:23:56
why I'm interested in that
equation.
385
00:23:56 --> 00:24:02
I'll put it down with those old
erasers and continue this.
386
00:24:02 --> 00:24:07
Here's the great picture of
applied mathematics.
387
00:24:07 --> 00:24:09
So let me complete that.
388
00:24:09 --> 00:24:15
There's a matrix that I'll call
C that connects potential
389
00:24:15 --> 00:24:17
differences to currents.
390
00:24:17 --> 00:24:23.72
So I'll call these --
these are currents on the
391
00:24:23.72 --> 00:24:27.07
edges, y1 y2 y3 y4 and y5.
392
00:24:27.07 --> 00:24:31
Those are currents on the
edges.
393
00:24:31 --> 00:24:38
And this relation between
current and potential difference
394
00:24:38 --> 00:24:40
is Ohm's Law.
395
00:24:40 --> 00:24:43
This here is Ohm's Law.
396
00:24:43 --> 00:24:51
Ohm's Law says that the current
on an edge is some number times
397
00:24:51 --> 00:24:54.9
the potential drop.
398
00:24:54.9 --> 00:24:59
That's -- and that number is
the conductance of the edge,
399
00:24:59 --> 00:25:01
one over the resistance.
400
00:25:01 --> 00:25:06
This is the old current is,
is, the relation of current,
401
00:25:06 --> 00:25:09
resistance, and change in
potential.
402
00:25:09 --> 00:25:14
So it's a change in potential
that makes some current happen,
403
00:25:14 --> 00:25:19
and it's Ohm's Law that says
how much current happens.
404
00:25:19 --> 00:25:19
OK.
405
00:25:19 --> 00:25:28.06
And then the final step of this
framework is the equation A
406
00:25:28.06 --> 00:25:31
transpose y equals zero.
407
00:25:31 --> 00:25:36
And that's -- what is that
saying?
408
00:25:36 --> 00:25:39
It has a famous name.
409
00:25:39 --> 00:25:47
It's Kirchoff's Current Law,
KCL, Kirchoff's Current Law,
410
00:25:47 --> 00:25:52.2
A transpose y equals zero.
411
00:25:52.2 --> 00:25:56
So that when I'm solving,
and when I go back up with this
412
00:25:56 --> 00:26:00
blackboard and solve A transpose
y equals zero,
413
00:26:00 --> 00:26:04
it's this pattern of -- that I
want you to see.
414
00:26:04 --> 00:26:08
That we had rectangular
matrices, but -- and real
415
00:26:08 --> 00:26:12
applications,
but in those real applications
416
00:26:12 --> 00:26:15
comes A and A transpose.
417
00:26:15 --> 00:26:21
So our four subspaces are
exactly the right things to know
418
00:26:21 --> 00:26:22
about.
419
00:26:22 --> 00:26:23
All right.
420
00:26:23 --> 00:26:28
Let's know about that null
space of A transpose.
421
00:26:28 --> 00:26:31
Wait a minute,
where'd it go?
422
00:26:31 --> 00:26:32
There it is.
423
00:26:32 --> 00:26:33
OK.
424
00:26:33 --> 00:26:33
OK.
425
00:26:33 --> 00:26:37
Null space of A transpose.
426
00:26:37 --> 00:26:40
We know what its dimension
should be.
427
00:26:40 --> 00:26:44
Let's find out -- tell me a
vector in it.
428
00:26:44 --> 00:26:47
Tell me -- now,
so what I asking you?
429
00:26:47 --> 00:26:53
I'm asking you for five
currents that satisfy Kirchoff's
430
00:26:53 --> 00:26:54
Current Law.
431
00:26:54 --> 00:26:59
So we better understand what
that law says.
432
00:26:59 --> 00:27:03
That, that law,
A transpose y equals zero,
433
00:27:03 --> 00:27:08
what does that say,
say in the first row of A
434
00:27:08 --> 00:27:09
transpose?
435
00:27:09 --> 00:27:15
That says -- the so the first
row of A transpose says minus y1
436
00:27:15 --> 00:27:18
minus y3 minus y4 is zero.
437
00:27:18 --> 00:27:22
Where did that equation come
from?
438
00:27:22 --> 00:27:26.7
Let me -- I'll redraw the
graph.
439
00:27:26.7 --> 00:27:31
Can I redraw the graph here,
so that we -- maybe here,
440
00:27:31 --> 00:27:35
so that we see again -- there
was node one,
441
00:27:35 --> 00:27:40.02
node two, node three,
node four was off here.
442
00:27:40.02 --> 00:27:42
That was, that was our graph.
443
00:27:42 --> 00:27:45
We had currents on those.
444
00:27:45 --> 00:27:48
We had a current y1 going
there.
445
00:27:48 --> 00:27:53
We had a current y --
what were the other,
446
00:27:53 --> 00:27:58
what are those edge numbers?
y4 here and y3 here.
447
00:27:58 --> 00:28:00
And then a y2 and a y5.
448
00:28:00 --> 00:28:06
I'm, I'm just copying what was
on the other board so it's ea-
449
00:28:06 --> 00:28:08
convenient to see it.
450
00:28:08 --> 00:28:14
What is this equation telling
me, this first equation of
451
00:28:14 --> 00:28:17
Kirchoff's Current Law?
452
00:28:17 --> 00:28:20
What does that mean for that
graph?
453
00:28:20 --> 00:28:24
Well, I see y1,
y3, and y4 as the currents
454
00:28:24 --> 00:28:25.57
leaving node one.
455
00:28:25.57 --> 00:28:29
So sure enough,
the first equation refers to
456
00:28:29 --> 00:28:31
node one, and what does it say?
457
00:28:31 --> 00:28:34
It says that the net flow is
zero.
458
00:28:34 --> 00:28:39.42
That, that equation A transpose
y, Kirchoff's Current Law,
459
00:28:39.42 --> 00:28:44
is a balance equation,
a conservation law.
460
00:28:44 --> 00:28:48
Physicists, be overjoyed,
right, by this stuff.
461
00:28:48 --> 00:28:52
It, it says that in equals out.
462
00:28:52 --> 00:28:56
And in this case,
the three arrows are all going
463
00:28:56 --> 00:29:01
out, so it says y1,
y3, and y4 add to zero.
464
00:29:01 --> 00:29:03
Let's take the next one.
465
00:29:03 --> 00:29:09
The second row is y1-y2,
and that's all that's in that
466
00:29:09 --> 00:29:10.95
row.
467
00:29:10.95 --> 00:29:16
And that must have something to
do with node two.
468
00:29:16 --> 00:29:20
And sure enough,
it says y1=y2,
469
00:29:20 --> 00:29:24.31
current in equals current out.
470
00:29:24.31 --> 00:29:29
The third one,
y2 plus y3 minus y5 equals
471
00:29:29 --> 00:29:29
zero.
472
00:29:29 --> 00:29:37
That certainly will be what's
up at the third node.
473
00:29:37 --> 00:29:40
y2 coming in,
y3 coming in,
474
00:29:40 --> 00:29:43
y5 going out has to balance.
475
00:29:43 --> 00:29:47
And finally,
y4 plus y5 equals zero says
476
00:29:47 --> 00:29:53.08
that at this node,
y4 plus y5, the total flow,
477
00:29:53.08 --> 00:29:53
is zero.
478
00:29:53 --> 00:29:59
We don't -- you know,
charge doesn't accumulate at
479
00:29:59 --> 00:30:02
the nodes.
480
00:30:02 --> 00:30:04
It travels around.
481
00:30:04 --> 00:30:04
OK.
482
00:30:04 --> 00:30:11
Now give me -- I come back now
to the linear algebra question.
483
00:30:11 --> 00:30:16
What's a vector y that solves
these equations?
484
00:30:16 --> 00:30:22
Can I figure out what the null
space is for this matrix,
485
00:30:22 --> 00:30:27.7
A transpose,
by looking at the graph?
486
00:30:27.7 --> 00:30:31.42
I'm happy if I don't have to do
elimination.
487
00:30:31.42 --> 00:30:34
I can do elimination,
we know how to do,
488
00:30:34 --> 00:30:38
we know how to find the null
space basis.
489
00:30:38 --> 00:30:43
We can do elimination on this
matrix, and we'll get it into a
490
00:30:43 --> 00:30:48
good reduced row echelon form,
and the special solutions will
491
00:30:48 --> 00:30:50
pop right out.
492
00:30:50 --> 00:30:55
But I would like to -- even to
do it without that.
493
00:30:55 --> 00:31:00
Let me just ask you first,
if I did elimination on that,
494
00:31:00 --> 00:31:04
on that, matrix,
what would the last row become?
495
00:31:04 --> 00:31:10
What would the last row -- if I
do elimination on that matrix,
496
00:31:10 --> 00:31:15
the last row of R will be all
zeros, right?
497
00:31:15 --> 00:31:15
Why?
498
00:31:15 --> 00:31:18
Because the rank is three.
499
00:31:18 --> 00:31:20
We only going to have three
pivots.
500
00:31:20 --> 00:31:25
And the fourth row will be all
zeros when we eliminate.
501
00:31:25 --> 00:31:29
So elimination will tell us
what, what we spotted earlier,
502
00:31:29 --> 00:31:34
what's the null space -- all
the, all the information,
503
00:31:34 --> 00:31:37.4
what are the dependencies.
504
00:31:37.4 --> 00:31:42
We'll find those by
elimination, but here in a real
505
00:31:42 --> 00:31:46
example, we can find them by
thinking.
506
00:31:46 --> 00:31:46
OK.
507
00:31:46 --> 00:31:51
Again, my question is,
what is a solution y?
508
00:31:51 --> 00:31:57
How could current travel around
this network without collecting
509
00:31:57 --> 00:32:01.5
any charge at the nodes?
510
00:32:01.5 --> 00:32:02
Tell me a y.
511
00:32:02 --> 00:32:03
OK.
512
00:32:03 --> 00:32:08
So a basis for the null space
of A transpose.
513
00:32:08 --> 00:32:11
How many vectors I looking for?
514
00:32:11 --> 00:32:12
Two.
515
00:32:12 --> 00:32:15.48
It's a two dimensional space.
516
00:32:15.48 --> 00:32:19
My basis should have two
vectors in it.
517
00:32:19 --> 00:32:22
Give me one.
518
00:32:22 --> 00:32:24
One set of currents.
519
00:32:24 --> 00:32:27
Suppose, let me start it.
520
00:32:27 --> 00:32:31
Let me start with y1 as one.
521
00:32:31 --> 00:32:31
OK.
522
00:32:31 --> 00:32:38
So one unit of -- one amp
travels on edge one with the
523
00:32:38 --> 00:32:39
arrow.
524
00:32:39 --> 00:32:41
OK, then what?
525
00:32:41 --> 00:32:42
What is y2?
526
00:32:42 --> 00:32:46
It's one also,
right?
527
00:32:46 --> 00:32:51
And of course what you did was
solve Kirchoff's Current Law
528
00:32:51 --> 00:32:53
quickly in the second equation.
529
00:32:53 --> 00:32:54
OK.
530
00:32:54 --> 00:32:59
Now we've got one amp leaving
node one, coming around to node
531
00:32:59 --> 00:32:59
three.
532
00:32:59 --> 00:33:01
What shall we do now?
533
00:33:01 --> 00:33:06
Well, what shall I take for y3
in other words?
534
00:33:06 --> 00:33:10
Oh, I've got a choice,
but why not make it what you
535
00:33:10 --> 00:33:12
said, negative one.
536
00:33:12 --> 00:33:17
So I have just sent current,
one amp, around that loop.
537
00:33:17 --> 00:33:20
What shall y4 and y5 be in this
case?
538
00:33:20 --> 00:33:23
We could take them to be zero.
539
00:33:23 --> 00:33:26.68
This satisfies Kirchoff's
Current Law.
540
00:33:26.68 --> 00:33:31
We could check it patiently,
that minus y1 minus y3 gives
541
00:33:31 --> 00:33:33
zero.
542
00:33:33 --> 00:33:35
We know y1 is y2.
543
00:33:35 --> 00:33:39
The others, y4 plus y5 is
certainly zero.
544
00:33:39 --> 00:33:45
Any current around a loop
satisfies -- satisfies the
545
00:33:45 --> 00:33:46.39
Current Law.
546
00:33:46.39 --> 00:33:46
OK.
547
00:33:46 --> 00:33:50
Now you know how to get another
one.
548
00:33:50 --> 00:33:54.9
Take current around this loop.
549
00:33:54.9 --> 00:34:00
So now let y3 be one,
y5 be one, and y4 be minus one.
550
00:34:00 --> 00:34:06
And so, so we have the first
basis vector sent current around
551
00:34:06 --> 00:34:12
that loop, the second basis
vector sends current around that
552
00:34:12 --> 00:34:12
loop.
553
00:34:12 --> 00:34:18
And I've -- and those are
independent, and I've got two
554
00:34:18 --> 00:34:23
solutions --
two vectors in the null space
555
00:34:23 --> 00:34:27
of A transpose,
two solutions to Kirchoff's
556
00:34:27 --> 00:34:28
Current Law.
557
00:34:28 --> 00:34:34
Of course you would say what
about sending current around the
558
00:34:34 --> 00:34:35
big loop.
559
00:34:35 --> 00:34:37
What about that vector?
560
00:34:37 --> 00:34:41
One for y1, one for y2,
nothing f- on y3,
561
00:34:41 --> 00:34:46.3
one for y5, and minus one for
y4.
562
00:34:46.3 --> 00:34:47
What about that?
563
00:34:47 --> 00:34:52
Is that, is that in the null
space of A transpose?
564
00:34:52 --> 00:34:52
Sure.
565
00:34:52 --> 00:34:57
So why don't we now have a
third vector in the basis?
566
00:34:57 --> 00:35:01
Because it's not independent,
right?
567
00:35:01 --> 00:35:03
It's not independent.
568
00:35:03 --> 00:35:06
This vector is the sum of those
two.
569
00:35:06 --> 00:35:12.4
If I send current around that
and around that --
570
00:35:12.4 --> 00:35:17
then on this edge y3 it's going
to cancel out and I'll have
571
00:35:17 --> 00:35:22
altogether current around the
whole, the outside loop.
572
00:35:22 --> 00:35:27
That's what this one is,
but it's a combination of those
573
00:35:27 --> 00:35:28
two.
574
00:35:28 --> 00:35:33
Do you see that I've now,
I've identified the null space
575
00:35:33 --> 00:35:37
of A transpose --
but more than that,
576
00:35:37 --> 00:35:41
we've solved Kirchoff's Current
Law.
577
00:35:41 --> 00:35:45
And understood it in terms of
the network.
578
00:35:45 --> 00:35:45
OK.
579
00:35:45 --> 00:35:48
So that's the null space of A
transpose.
580
00:35:48 --> 00:35:55.5
I guess I -- there's always one
more space to ask you about.
581
00:35:55.5 --> 00:36:01
Let's see, I guess I need the
row space of A,
582
00:36:01 --> 00:36:05
the column space of A
transpose.
583
00:36:05 --> 00:36:10
So what's N,
what's its dimension?
584
00:36:10 --> 00:36:10.96
Yup?
585
00:36:10.96 --> 00:36:18
What's the dimension of the row
space of A?
586
00:36:18 --> 00:36:21
If I look at the original A,
it had five rows.
587
00:36:21 --> 00:36:23
How many were independent?
588
00:36:23 --> 00:36:27
Oh, I guess I'm asking you the
rank again, right?
589
00:36:27 --> 00:36:29
And the answer is three,
right?
590
00:36:29 --> 00:36:31.36
Three independent rows.
591
00:36:31.36 --> 00:36:34
When I transpose it,
there's three independent
592
00:36:34 --> 00:36:35
columns.
593
00:36:35 --> 00:36:39.75
Are those columns independent,
those three?
594
00:36:39.75 --> 00:36:45
The first three columns,
are they the pivot columns of
595
00:36:45 --> 00:36:46
the matrix?
596
00:36:46 --> 00:36:46
No.
597
00:36:46 --> 00:36:50
Those three columns are not
independent.
598
00:36:50 --> 00:36:54
There's a in fact,
this tells me a relation
599
00:36:54 --> 00:36:57.25
between them.
600
00:36:57.25 --> 00:37:01
There's a vector in the null
space that says the first column
601
00:37:01 --> 00:37:04
plus the second column equals
the third column.
602
00:37:04 --> 00:37:08.78
They're not independent because
they come from a loop.
603
00:37:08.78 --> 00:37:12
So the pivot columns,
the pivot columns of this
604
00:37:12 --> 00:37:15
matrix will be the first,
the second, not the third,
605
00:37:15 --> 00:37:17
but the fourth.
606
00:37:17 --> 00:37:22
One, columns one,
two, and four are OK.
607
00:37:22 --> 00:37:29
Where are they -- those are the
columns of A transpose,
608
00:37:29 --> 00:37:32
those correspond to edges.
609
00:37:32 --> 00:37:37
So there's edge one,
there's edge two,
610
00:37:37 --> 00:37:40
and there's edge four.
611
00:37:40 --> 00:37:45
So there's a --
that's like -- is a,
612
00:37:45 --> 00:37:46
smaller graph.
613
00:37:46 --> 00:37:51
If I just look at the part of
the graph that I've,
614
00:37:51 --> 00:37:57.66
that I've, thick -- used with
thick edges, it has the same
615
00:37:57.66 --> 00:37:58
four nodes.
616
00:37:58 --> 00:38:01
It only has three edges.
617
00:38:01 --> 00:38:07.8
And the, those edges correspond
to the independent guys.
618
00:38:07.8 --> 00:38:12
And in the graph there -- those
three edges have no loop,
619
00:38:12 --> 00:38:13
right?
620
00:38:13 --> 00:38:18
The independent ones are the
ones that don't have a loop.
621
00:38:18 --> 00:38:21
All the -- dependencies came
from loops.
622
00:38:21 --> 00:38:27.42
They were the things in the
null space of A transpose.
623
00:38:27.42 --> 00:38:31
If I take three pivot columns,
there are no dependencies among
624
00:38:31 --> 00:38:34
them, and they form a graph
without a loop,
625
00:38:34 --> 00:38:38
and I just want to ask you
what's the name for a graph
626
00:38:38 --> 00:38:39
without a loop?
627
00:38:39 --> 00:38:43
So a graph without a loop is --
has got not very many edges,
628
00:38:43 --> 00:38:44
right?
629
00:38:44 --> 00:38:47
I've got four nodes and it only
has three edges,
630
00:38:47 --> 00:38:51.9
and if I put another edge in,
I would have a loop.
631
00:38:51.9 --> 00:38:58
So it's this graph with no
loops, and it's the one where
632
00:38:58 --> 00:39:02
the rows of A are independent.
633
00:39:02 --> 00:39:08
And what's a graph called that
has no loops?
634
00:39:08 --> 00:39:10
It's called a tree.
635
00:39:10 --> 00:39:16.72
So a tree is the name for a
graph with no loops.
636
00:39:16.72 --> 00:39:21
And just to take one last step
here.
637
00:39:21 --> 00:39:26
Using our formula for
dimension.
638
00:39:26 --> 00:39:36
Using our formula for
dimension, let's look -- once at
639
00:39:36 --> 00:39:38
this formula.
640
00:39:38 --> 00:39:47
The dimension of the null space
of A transpose is m-r.
641
00:39:47 --> 00:39:48.52
OK.
642
00:39:48.52 --> 00:39:58
This is the number of loops,
number of independent loops.
643
00:39:58 --> 00:40:03
m is the number of edges.
644
00:40:03 --> 00:40:07.3
And what is r?
645
00:40:07.3 --> 00:40:11
What is r for our -- we'll have
to remember way back.
646
00:40:11 --> 00:40:17
The rank came -- from looking
at the columns of our matrix.
647
00:40:17 --> 00:40:18.88
So what's the rank?
648
00:40:18.88 --> 00:40:20
Let's just remember.
649
00:40:20 --> 00:40:25
Rank was -- you remember there
was one -- we had a one
650
00:40:25 --> 00:40:29
dimensional --
rank was n minus one,
651
00:40:29 --> 00:40:32
that's what I'm struggling to
say.
652
00:40:32 --> 00:40:38
Because there were n columns
coming from the n nodes,
653
00:40:38 --> 00:40:42
so it's minus,
the number of nodes minus one,
654
00:40:42 --> 00:40:47
because of that C,
that one one one one vector in
655
00:40:47 --> 00:40:49
the null space.
656
00:40:49 --> 00:40:53
The columns were not
independent.
657
00:40:53 --> 00:40:59
There was one dependency,
so we needed n minus one.
658
00:40:59 --> 00:41:02
This is a great formula.
659
00:41:02 --> 00:41:09
This is like the first shall I,
-- write it slightly
660
00:41:09 --> 00:41:10
differently?
661
00:41:10 --> 00:41:17.4
The number of edges -- let me
put things -- have I got it
662
00:41:17.4 --> 00:41:19.25
right?
663
00:41:19.25 --> 00:41:24.36
Number of edges is m,
the number -- r- is m-r,
664
00:41:24.36 --> 00:41:24.7
OK.
665
00:41:24.7 --> 00:41:31
So, so I'm getting -- let me
put the number of nodes on the
666
00:41:31 --> 00:41:32
other side.
667
00:41:32 --> 00:41:39
So I -- the number of nodes --
I'll move that to the other side
668
00:41:39 --> 00:41:46.39
-- minus the number of edges
plus the number of loops is -- I
669
00:41:46.39 --> 00:41:50
have minus, minus one is one.
670
00:41:50 --> 00:41:55
The number of nodes minus the
number of edges plus the number
671
00:41:55 --> 00:41:56
of loops is one.
672
00:41:56 --> 00:41:58
These are like zero dimensional
guys.
673
00:41:58 --> 00:42:00
They're the points on the
graph.
674
00:42:00 --> 00:42:03
The edges are like one
dimensional things,
675
00:42:03 --> 00:42:05.77
they're, they connect nodes.
676
00:42:05.77 --> 00:42:08.64
The loops are like two
dimensional things.
677
00:42:08.64 --> 00:42:11.3
They have, like,
an area.
678
00:42:11.3 --> 00:42:15
And this count works for every
graph.
679
00:42:15 --> 00:42:19
And it's known as Euler's
Formula.
680
00:42:19 --> 00:42:24
We see Euler again,
that guy never stopped.
681
00:42:24 --> 00:42:24
OK.
682
00:42:24 --> 00:42:29
And can we just check -- so
what I saying?
683
00:42:29 --> 00:42:36.5
I'm saying that linear algebra
proves Euler's Formula.
684
00:42:36.5 --> 00:42:41
Euler's Formula is this great
topology fact about any graph.
685
00:42:41 --> 00:42:46.73
I'll draw, let me draw another
graph, let me draw a graph with
686
00:42:46.73 --> 00:42:48.52
more edges and loops.
687
00:42:48.52 --> 00:42:50
Let me put in lots of -- OK.
688
00:42:50 --> 00:42:53
I just drew a graph there.
689
00:42:53 --> 00:42:57
So what are the,
what are the quantities in that
690
00:42:57 --> 00:42:58
formula?
691
00:42:58 --> 00:43:01
How many nodes have I got?
692
00:43:01 --> 00:43:02
Looks like five.
693
00:43:02 --> 00:43:05
How many edges have I got?
694
00:43:05 --> 00:43:08
One two three four five six
seven.
695
00:43:08 --> 00:43:10
How many loops have I got?
696
00:43:10 --> 00:43:12
One two three.
697
00:43:12 --> 00:43:15
And Euler's right,
I always get one.
698
00:43:15 --> 00:43:19
That, this formula,
is extremely useful in
699
00:43:19 --> 00:43:25
understanding the relation of
these quantities --
700
00:43:25 --> 00:43:30
the number of nodes,
the number of edges,
701
00:43:30 --> 00:43:34
and the number of loops.
702
00:43:34 --> 00:43:34
OK.
703
00:43:34 --> 00:43:42
Just complete this lecture by
completing this picture,
704
00:43:42 --> 00:43:44
this cycle.
705
00:43:44 --> 00:43:52
So let me come to the -- so
this expresses the equations of
706
00:43:52 --> 00:43:55.8
applied math.
707
00:43:55.8 --> 00:44:00
This, let me call these
potential differences,
708
00:44:00 --> 00:44:00
say, E.
709
00:44:00 --> 00:44:01
So E is A x.
710
00:44:01 --> 00:44:05
That's the equation for this
step.
711
00:44:05 --> 00:44:09.59
The currents come from the
potential differences.
712
00:44:09.59 --> 00:44:10
y is C E.
713
00:44:10 --> 00:44:17
The potential -- the currents
satisfy Kirchoff's Current Law.
714
00:44:17 --> 00:44:21
Those are the equations of --
with no source terms.
715
00:44:21 --> 00:44:26
Those are the equations of
electrical circuits of many --
716
00:44:26 --> 00:44:30
those are like the,
the most basic three equations.
717
00:44:30 --> 00:44:33
Applied math comes in this
structure.
718
00:44:33 --> 00:44:38
The only thing I haven't got
yet in the picture is an outside
719
00:44:38 --> 00:44:42
source to make something happen.
720
00:44:42 --> 00:44:46
I could add a current source
here, I could,
721
00:44:46 --> 00:44:52
I could add external currents
going in and out of nodes.
722
00:44:52 --> 00:44:55
I could add batteries in the
edges.
723
00:44:55 --> 00:44:57
Those are two ways.
724
00:44:57 --> 00:45:02
If I add batteries in the
edges, they, they come into
725
00:45:02 --> 00:45:03
here.
726
00:45:03 --> 00:45:07
Let me add current sources.
727
00:45:07 --> 00:45:11
If I add current sources,
those come in here.
728
00:45:11 --> 00:45:15
So there's a,
there's where current sources
729
00:45:15 --> 00:45:20
go, because the F is a like a
current coming from outside.
730
00:45:20 --> 00:45:24
So we have our edges,
we have our graph,
731
00:45:24 --> 00:45:29
and then I send one amp into
this node and out of this node
732
00:45:29 --> 00:45:32
--
and that gives me,
733
00:45:32 --> 00:45:35
a right-hand side in Kirchoff's
Current Law.
734
00:45:35 --> 00:45:40
And can I -- to complete the
lecture, I'm just going to put
735
00:45:40 --> 00:45:43
these three equations together.
736
00:45:43 --> 00:45:46
So I start with x,
my unknown.
737
00:45:46 --> 00:45:47
I multiply by A.
738
00:45:47 --> 00:45:51
That gives me the potential
differences.
739
00:45:51 --> 00:45:58
That was our matrix A that the
whole thing started with.
740
00:45:58 --> 00:45:59
I multiply by C.
741
00:45:59 --> 00:46:05
Those are the physical
constants in Ohm's Law.
742
00:46:05 --> 00:46:06
Now I have y.
743
00:46:06 --> 00:46:11
I multiply y by A transpose,
and now I have F.
744
00:46:11 --> 00:46:16
So there is the whole thing.
745
00:46:16 --> 00:46:19
There's the basic equation of
applied math.
746
00:46:19 --> 00:46:24
Coming from these three steps,
in which the last step is this
747
00:46:24 --> 00:46:25
balance equation.
748
00:46:25 --> 00:46:29
There's always a balance
equation to look for.
749
00:46:29 --> 00:46:33
These are the -- when I say the
most basic equations of applied
750
00:46:33 --> 00:46:36
mathematics --
I should say,
751
00:46:36 --> 00:46:37
in equilibrium.
752
00:46:37 --> 00:46:40.07
Time isn't in this problem.
753
00:46:40.07 --> 00:46:43
I'm not -- and Newton's Law
isn't acting here.
754
00:46:43 --> 00:46:47
I'm, I'm looking at the --
equations when everything has
755
00:46:47 --> 00:46:51
settled down,
how do the currents distribute
756
00:46:51 --> 00:46:53
in the network.
757
00:46:53 --> 00:46:58
And of course there are big
codes to solve the -- this is
758
00:46:58 --> 00:47:04
the basic problem of numerical
linear algebra for systems of
759
00:47:04 --> 00:47:08.85
equations, because that's how
they come.
760
00:47:08.85 --> 00:47:11
And my final question.
761
00:47:11 --> 00:47:17.5
What can you tell me about this
matrix A transpose C A?
762
00:47:17.5 --> 00:47:19
Or even A transpose A?
763
00:47:19 --> 00:47:23
I'll just close with that
question.
764
00:47:23 --> 00:47:28
What do you know about the
matrix A transpose A?
765
00:47:28 --> 00:47:31
It is always symmetric,
right.
766
00:47:31 --> 00:47:32
OK, thank.
767
00:47:32 --> 00:47:38
So I'll see you Wednesday for a
full review of these chapters,
768
00:47:38 --> 00:47:41
and Friday you get to tell me.
769
00:47:41 --> 00:47:44
Thanks.