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.