1 00:00:09,860 --> 00:00:10,620 Okay. 2 00:00:10,620 --> 00:00:14,290 This is lecture five in linear algebra. 3 00:00:14,290 --> 00:00:19,015 And, it will complete this chapter of the book. 4 00:00:21,620 --> 00:00:24,030 So the last section of this chapter 5 00:00:24,030 --> 00:00:30,060 is two point seven that talks about permutations, which 6 00:00:30,060 --> 00:00:34,240 finished the previous lecture, and transposes, 7 00:00:34,240 --> 00:00:36,650 which also came in the previous lecture. 8 00:00:36,650 --> 00:00:40,750 There's a little more to do with those guys, permutations 9 00:00:40,750 --> 00:00:42,860 and transposes. 10 00:00:42,860 --> 00:00:48,180 But then the heart of the lecture will be the beginning 11 00:00:48,180 --> 00:00:51,910 of what you could say is the beginning of linear algebra, 12 00:00:51,910 --> 00:00:57,470 the beginning of real linear algebra which is seeing 13 00:00:57,470 --> 00:01:01,590 a bigger picture with vector spaces -- not just vectors, 14 00:01:01,590 --> 00:01:06,650 but spaces of vectors and sub-spaces of those spaces. 15 00:01:06,650 --> 00:01:11,110 So we're a little ahead of the syllabus, which 16 00:01:11,110 --> 00:01:13,150 is good, because we're coming to the place 17 00:01:13,150 --> 00:01:15,780 where, there's a lot to 18 00:01:15,780 --> 00:01:16,620 do. 19 00:01:16,620 --> 00:01:17,800 Okay. 20 00:01:17,800 --> 00:01:22,400 So, to begin with permutations. 21 00:01:22,400 --> 00:01:26,790 Can I just -- 22 00:01:26,790 --> 00:01:36,850 so these permutations, those are matrices P and they execute 23 00:01:36,850 --> 00:01:38,400 row exchanges. 24 00:01:42,160 --> 00:01:44,910 And we may need them. 25 00:01:44,910 --> 00:01:47,120 We may have a perfectly good matrix, 26 00:01:47,120 --> 00:01:52,310 a perfect matrix A that's invertible that we can solve A 27 00:01:52,310 --> 00:01:55,580 x=b, but to do it -- 28 00:01:55,580 --> 00:02:00,050 I've got to allow myself that extra freedom 29 00:02:00,050 --> 00:02:05,070 that if a zero shows up in the pivot position I move it away. 30 00:02:05,070 --> 00:02:06,630 I get a non-zero. 31 00:02:06,630 --> 00:02:12,460 I get a proper pivot there by exchanging from a row below. 32 00:02:12,460 --> 00:02:15,000 And you've seen that already, and I just 33 00:02:15,000 --> 00:02:18,200 want to collect the ideas together. 34 00:02:18,200 --> 00:02:21,310 And principle, I could even have to do 35 00:02:21,310 --> 00:02:24,750 that two times, or more times. 36 00:02:24,750 --> 00:02:27,290 So I have to allow -- 37 00:02:27,290 --> 00:02:29,330 to complete the -- 38 00:02:29,330 --> 00:02:34,370 the theory, the possibility that I take my matrix A, 39 00:02:34,370 --> 00:02:38,540 I start elimination, I find out that I need row exchanges 40 00:02:38,540 --> 00:02:41,620 and I do it and continue and I finish. 41 00:02:41,620 --> 00:02:42,750 Okay. 42 00:02:42,750 --> 00:02:49,150 Then all I want to do is say -- and I won't make a big project 43 00:02:49,150 --> 00:02:51,160 out of this -- 44 00:02:51,160 --> 00:02:54,300 what happens to A equal L U? 45 00:02:54,300 --> 00:02:59,570 So A equal L U -- 46 00:02:59,570 --> 00:03:05,130 this was a matrix L with ones on the diagonal and zeroes 47 00:03:05,130 --> 00:03:09,450 above and multipliers below, and this U 48 00:03:09,450 --> 00:03:13,425 we know, with zeroes down here. 49 00:03:16,940 --> 00:03:19,080 That's only possible. 50 00:03:19,080 --> 00:03:21,770 That description of elimination assumes 51 00:03:21,770 --> 00:03:26,480 that we don't have a P, that we don't have any row exchanges. 52 00:03:26,480 --> 00:03:29,570 And now I just want to say, okay, how 53 00:03:29,570 --> 00:03:31,220 do I account for row exchanges? 54 00:03:31,220 --> 00:03:33,990 Because that doesn't. 55 00:03:33,990 --> 00:03:40,030 The P in this factorization is the identity matrix. 56 00:03:40,030 --> 00:03:44,140 The rows were in a good order, we left them there. 57 00:03:44,140 --> 00:03:47,830 Maybe I'll just add a little moment of reality, 58 00:03:47,830 --> 00:03:55,190 too, about how Matlab actually does elimination. 59 00:03:55,190 --> 00:03:59,520 Matlab not only checks whether that pivot is not zero, 60 00:03:59,520 --> 00:04:02,770 as every human would do. 61 00:04:02,770 --> 00:04:05,480 It checks for is that pivot big enough, 62 00:04:05,480 --> 00:04:09,380 because it doesn't like very, very small pivots. 63 00:04:09,380 --> 00:04:13,550 Pivots close to zero are numerically bad. 64 00:04:13,550 --> 00:04:16,660 So actually if we ask Matlab to solve a system, 65 00:04:16,660 --> 00:04:21,420 it will do some elimination some row exchanges, which 66 00:04:21,420 --> 00:04:23,180 we don't think are necessary. 67 00:04:23,180 --> 00:04:27,930 Algebra doesn't say they're necessary, but accuracy -- 68 00:04:27,930 --> 00:04:31,680 numerical accuracy says they are. 69 00:04:31,680 --> 00:04:35,930 Well, we're doing algebra, so here we 70 00:04:35,930 --> 00:04:39,380 will say, well, what do row exchanges do, 71 00:04:39,380 --> 00:04:42,230 but we won't do them unless we have to. 72 00:04:42,230 --> 00:04:45,340 But we may have to. 73 00:04:45,340 --> 00:04:50,380 And then, the result is -- 74 00:04:50,380 --> 00:04:51,850 it's hiding here. 75 00:04:51,850 --> 00:04:56,040 It's the main fact. 76 00:04:56,040 --> 00:05:00,100 This is the description of elimination with row exchanges. 77 00:05:00,100 --> 00:05:12,280 So A equal L U becomes P A equal L U. 78 00:05:12,280 --> 00:05:15,210 So this P is the matrix that does the row exchanges, 79 00:05:15,210 --> 00:05:18,080 and actually it does them -- 80 00:05:18,080 --> 00:05:20,780 it gets the rows into the right order, 81 00:05:20,780 --> 00:05:23,810 into the good order where pivots will not -- 82 00:05:23,810 --> 00:05:27,120 where zeroes won't appear in the pivot position, 83 00:05:27,120 --> 00:05:32,700 where L and U will come out right as up here. 84 00:05:32,700 --> 00:05:36,430 So, that's the point. 85 00:05:36,430 --> 00:05:39,650 Actually, I don't want to labor that point, 86 00:05:39,650 --> 00:05:43,110 that a permutation matrix -- 87 00:05:43,110 --> 00:05:47,260 and you remember what those were. 88 00:05:47,260 --> 00:05:51,280 I'll remind you from last time of what the main points about 89 00:05:51,280 --> 00:05:57,820 permutation matrices were -- 90 00:05:57,820 --> 00:06:00,340 and then just leave this factorization 91 00:06:00,340 --> 00:06:03,820 as the general case. 92 00:06:03,820 --> 00:06:17,220 This is -- any invertible A we get this. 93 00:06:17,220 --> 00:06:20,090 For almost every one, we don't need a P. 94 00:06:20,090 --> 00:06:24,220 But there's that handful that do need row exchanges, 95 00:06:24,220 --> 00:06:26,480 and if we do need them, there they are. 96 00:06:26,480 --> 00:06:30,410 Okay, finally, just to remember what P was. 97 00:06:30,410 --> 00:06:46,440 So permutations, P is the identity matrix 98 00:06:46,440 --> 00:06:51,480 with reordered rows. 99 00:06:56,710 --> 00:06:59,707 I include in reordering the possibility that you just 100 00:06:59,707 --> 00:07:00,540 leave them the same. 101 00:07:00,540 --> 00:07:02,880 So the identity matrix is -- okay. 102 00:07:02,880 --> 00:07:08,490 That's, like, your basic permutation matrix -- 103 00:07:08,490 --> 00:07:12,840 your do-nothing permutation matrix is the identity. 104 00:07:12,840 --> 00:07:15,730 And then there are the ones that exchange two rows and then 105 00:07:15,730 --> 00:07:18,940 the ones that exchange three rows and then then ones that 106 00:07:18,940 --> 00:07:21,040 exchange four -- 107 00:07:21,040 --> 00:07:24,640 well, it gets a little -- 108 00:07:24,640 --> 00:07:27,310 it gets more interesting algebraically 109 00:07:27,310 --> 00:07:30,630 if you've got four rows, you might exchange them 110 00:07:30,630 --> 00:07:32,190 all in one big cycle. 111 00:07:32,190 --> 00:07:35,550 One to two, two to three, three to four, four to one. 112 00:07:35,550 --> 00:07:41,070 Or you might have -- exchange one and two and three and four. 113 00:07:41,070 --> 00:07:42,467 Lots of possibilities there. 114 00:07:42,467 --> 00:07:43,800 In fact, how many possibilities? 115 00:07:46,570 --> 00:07:49,310 The answer was (n)factorial. 116 00:07:49,310 --> 00:07:50,984 This is n(n-1)(n-2)... 117 00:07:56,920 --> 00:07:58,230 (3)(2)(1). 118 00:07:58,230 --> 00:08:05,100 That's the number of -- this counts the reorderings, 119 00:08:05,100 --> 00:08:07,990 the possible reorderings. 120 00:08:07,990 --> 00:08:15,730 So it counts all the n by n permutations. 121 00:08:23,170 --> 00:08:26,200 And all those matrices have these -- 122 00:08:26,200 --> 00:08:32,380 have this nice property that they're all invertible, 123 00:08:32,380 --> 00:08:39,140 because we can bring those rows back into the normal order. 124 00:08:39,140 --> 00:08:42,679 And the matrix that does that is just P -- 125 00:08:42,679 --> 00:08:46,590 is just the same as the transpose. 126 00:08:46,590 --> 00:08:49,180 You might take a permutation matrix, 127 00:08:49,180 --> 00:08:51,820 multiply by its transpose and you will see how -- 128 00:08:51,820 --> 00:08:57,220 that the ones hit the ones and give the ones in the identity 129 00:08:57,220 --> 00:08:58,010 matrix. 130 00:08:58,010 --> 00:08:59,980 So this is a -- 131 00:08:59,980 --> 00:09:02,590 we'll be highly interested in matrices 132 00:09:02,590 --> 00:09:04,980 that have nice properties. 133 00:09:04,980 --> 00:09:09,580 And one property that -- maybe I could rewrite that as P 134 00:09:09,580 --> 00:09:13,880 transpose P is the identity. 135 00:09:13,880 --> 00:09:16,950 That tells me in other words that this 136 00:09:16,950 --> 00:09:19,300 is the inverse of that. 137 00:09:19,300 --> 00:09:20,210 Okay. 138 00:09:20,210 --> 00:09:25,940 We'll be interested in matrices that have P transpose 139 00:09:25,940 --> 00:09:27,940 P equal the identity. 140 00:09:27,940 --> 00:09:30,470 There are more of them than just permutations, 141 00:09:30,470 --> 00:09:35,000 but my point right now is that permutations are like a little 142 00:09:35,000 --> 00:09:36,870 group in the middle -- 143 00:09:36,870 --> 00:09:42,180 in the center of these special matrices. 144 00:09:42,180 --> 00:09:43,270 Okay. 145 00:09:43,270 --> 00:09:47,590 So now we know how many there are. 146 00:09:47,590 --> 00:09:50,750 Twenty four in the case of -- there are twenty four four 147 00:09:50,750 --> 00:09:55,960 by four permutations, there are five factorial which is 148 00:09:55,960 --> 00:10:00,340 a hundred and twenty, five times twenty four would bump us up 149 00:10:00,340 --> 00:10:03,710 to a hundred and twenty -- so listing all the five by five 150 00:10:03,710 --> 00:10:09,990 permutations would be not so much fun. 151 00:10:09,990 --> 00:10:12,380 Okay. 152 00:10:12,380 --> 00:10:15,320 So that's permutations. 153 00:10:15,320 --> 00:10:20,960 Now also in section two seven is some discussion of transposes. 154 00:10:20,960 --> 00:10:22,895 And can I just complete that discussion. 155 00:10:25,610 --> 00:10:30,910 First of all, I haven't even transposed a matrix 156 00:10:30,910 --> 00:10:32,110 on the board here, have I? 157 00:10:32,110 --> 00:10:33,710 So I'd better do it. 158 00:10:33,710 --> 00:10:39,975 So suppose I take a matrix like (1 2 4; 3 3 1). 159 00:10:43,660 --> 00:10:47,640 It's a rectangular matrix, three by two. 160 00:10:47,640 --> 00:10:51,030 And I want to transpose it. 161 00:10:51,030 --> 00:10:53,380 So what's -- 162 00:10:53,380 --> 00:10:59,040 I'll use a T, also Matlab would use a prime. 163 00:10:59,040 --> 00:11:02,050 And the result will be -- 164 00:11:02,050 --> 00:11:07,770 I'll right it here, because this was three rows and two columns, 165 00:11:07,770 --> 00:11:10,060 this was a three by two matrix. 166 00:11:10,060 --> 00:11:13,080 The transpose will be two rows and three columns, 167 00:11:13,080 --> 00:11:15,380 two by three. 168 00:11:15,380 --> 00:11:20,180 So it's short and wider. 169 00:11:20,180 --> 00:11:25,730 And, of course, that row -- that column becomes a row -- 170 00:11:25,730 --> 00:11:27,510 that column becomes the other row. 171 00:11:30,060 --> 00:11:36,200 And at the same time, that row became a column. 172 00:11:36,200 --> 00:11:37,970 This row became a column. 173 00:11:37,970 --> 00:11:41,610 Oh, what's the general formula for the transpose? 174 00:11:41,610 --> 00:11:48,920 So the transpose -- 175 00:11:48,920 --> 00:11:50,740 you see it in numbers. 176 00:11:50,740 --> 00:11:54,570 What I'm going to write is the same thing in symbols. 177 00:11:54,570 --> 00:11:57,920 The numbers are the clearest, of course. 178 00:11:57,920 --> 00:12:04,170 But in symbols, if I take A transpose 179 00:12:04,170 --> 00:12:10,135 and I ask what number is in row I and column J of A transpose? 180 00:12:14,600 --> 00:12:16,320 Well, it came out of A. 181 00:12:16,320 --> 00:12:22,080 It came out A by this flip across the main diagonal. 182 00:12:22,080 --> 00:12:28,030 And, actually, it was the number in A 183 00:12:28,030 --> 00:12:34,540 which was in row J, column I. 184 00:12:34,540 --> 00:12:36,920 So the row and column -- 185 00:12:36,920 --> 00:12:39,880 the row and column numbers just get reversed. 186 00:12:39,880 --> 00:12:42,030 The row number becomes the column number, 187 00:12:42,030 --> 00:12:44,450 the column number becomes the row number. 188 00:12:44,450 --> 00:12:45,610 No problem. 189 00:12:45,610 --> 00:12:46,260 Okay. 190 00:12:46,260 --> 00:12:49,280 Now, a special -- 191 00:12:49,280 --> 00:12:52,880 the best matrices, we could say. 192 00:12:52,880 --> 00:12:58,590 In a lot of applications, symmetric matrices show up. 193 00:12:58,590 --> 00:13:02,145 So can I just call attention to symmetric matrices? 194 00:13:07,960 --> 00:13:08,840 What does that mean? 195 00:13:08,840 --> 00:13:11,110 What does that word symmetric mean? 196 00:13:11,110 --> 00:13:16,000 It means that this transposing doesn't change the matrix. 197 00:13:16,000 --> 00:13:17,690 A transpose equals A. 198 00:13:17,690 --> 00:13:21,255 And an example. 199 00:13:25,740 --> 00:13:29,900 So, let's take a matrix that's symmetric, 200 00:13:29,900 --> 00:13:33,750 so whatever is sitting on the diagonal -- 201 00:13:33,750 --> 00:13:36,800 but now what's above the diagonal, like a one, 202 00:13:36,800 --> 00:13:41,630 had better be there, a seven had better be here, 203 00:13:41,630 --> 00:13:43,410 a nine had better be there. 204 00:13:43,410 --> 00:13:47,460 There's a symmetric matrix. 205 00:13:47,460 --> 00:13:52,420 I happened to use all positive numbers as its entries. 206 00:13:52,420 --> 00:13:54,030 That's not the point. 207 00:13:54,030 --> 00:13:57,160 The point is that if I transpose that matrix, 208 00:13:57,160 --> 00:13:58,880 I get it back again. 209 00:13:58,880 --> 00:14:03,370 So symmetric matrices have this property A transpose equals A. 210 00:14:03,370 --> 00:14:07,150 I guess at this point -- 211 00:14:07,150 --> 00:14:14,130 I'm just asking you to notice this family of matrices that 212 00:14:14,130 --> 00:14:17,830 are unchanged by transposing. 213 00:14:17,830 --> 00:14:22,950 And they're easy to identify, of course. 214 00:14:22,950 --> 00:14:28,060 You know, it's not maybe so easy before we had a case where 215 00:14:28,060 --> 00:14:31,830 the transpose gave the inverse. 216 00:14:31,830 --> 00:14:36,230 That's highly important, but not so simple to see. 217 00:14:36,230 --> 00:14:39,200 This is the case where the transpose gives the same matrix 218 00:14:39,200 --> 00:14:39,720 back again. 219 00:14:39,720 --> 00:14:42,510 That's totally simple to see. 220 00:14:42,510 --> 00:14:43,380 Okay. 221 00:14:43,380 --> 00:14:49,410 Could actually -- maybe I could even say when would we get such 222 00:14:49,410 --> 00:14:51,150 a matrix? 223 00:14:51,150 --> 00:14:55,740 For example, this -- that matrix is absolutely far from 224 00:14:55,740 --> 00:14:57,850 symmetric, right? 225 00:14:57,850 --> 00:15:01,150 The transpose isn't even the same shape -- 226 00:15:01,150 --> 00:15:03,700 because it's rectangular, it turns the -- 227 00:15:03,700 --> 00:15:06,580 lies down on its side. 228 00:15:06,580 --> 00:15:10,970 But let me tell you a way to get a symmetric matrix out of 229 00:15:10,970 --> 00:15:11,880 this. 230 00:15:11,880 --> 00:15:14,640 Multiply those together. 231 00:15:14,640 --> 00:15:17,210 If I multiply this rectangular, shall I 232 00:15:17,210 --> 00:15:18,910 call it R for rectangular? 233 00:15:18,910 --> 00:15:22,890 So let that be R for rectangular matrix 234 00:15:22,890 --> 00:15:27,580 and let that be R transpose, which it is. 235 00:15:27,580 --> 00:15:32,000 Then I think that if I multiply those together, 236 00:15:32,000 --> 00:15:34,390 I get a symmetric matrix. 237 00:15:34,390 --> 00:15:36,780 Can I just do it with the numbers 238 00:15:36,780 --> 00:15:44,700 and then ask you why, how did I know it would be symmetric? 239 00:15:44,700 --> 00:15:52,415 So my point is that R transpose R is always symmetric. 240 00:15:52,415 --> 00:15:52,914 Okay? 241 00:15:57,560 --> 00:16:01,470 And I'm going to do it for that particular R transpose R which 242 00:16:01,470 --> 00:16:02,750 was -- 243 00:16:02,750 --> 00:16:05,815 let's see, the column was one two four three three one. 244 00:16:09,700 --> 00:16:11,700 I called that one R transpose, didn't I, 245 00:16:11,700 --> 00:16:16,030 and I called this guy one two four three three one. 246 00:16:16,030 --> 00:16:17,550 I called that R. 247 00:16:17,550 --> 00:16:19,970 Shall we just do that multiplication? 248 00:16:19,970 --> 00:16:22,960 Okay, so up here I'm getting a ten. 249 00:16:22,960 --> 00:16:27,310 Next to it I'm getting two, a nine, I'm getting an eleven. 250 00:16:27,310 --> 00:16:30,390 Next to that I'm getting four and three, a seven. 251 00:16:30,390 --> 00:16:32,650 Now what do I get there? 252 00:16:32,650 --> 00:16:37,650 This eleven came from one three times two three, right? 253 00:16:37,650 --> 00:16:39,720 Row one, column two. 254 00:16:39,720 --> 00:16:41,530 What goes here? 255 00:16:41,530 --> 00:16:43,360 Row two, column one. 256 00:16:43,360 --> 00:16:46,050 But no difference. 257 00:16:46,050 --> 00:16:50,220 One three two three or two three one three, same thing. 258 00:16:50,220 --> 00:16:51,750 It's going to be an eleven. 259 00:16:51,750 --> 00:16:54,310 That's the symmetry. 260 00:16:54,310 --> 00:16:56,230 I can continue to fill it out. 261 00:16:56,230 --> 00:16:58,100 What -- oh, let's get that seven. 262 00:16:58,100 --> 00:17:00,520 That seven will show up down here, too, 263 00:17:00,520 --> 00:17:02,650 and then four more numbers. 264 00:17:02,650 --> 00:17:06,619 That seven will show up here because one three times four 265 00:17:06,619 --> 00:17:10,630 one gave the seven, but also four one times one three 266 00:17:10,630 --> 00:17:11,599 will give that seven. 267 00:17:11,599 --> 00:17:16,030 Do you see that it works? 268 00:17:16,030 --> 00:17:24,630 Actually, do you want to see it work also in matrix language? 269 00:17:24,630 --> 00:17:27,089 I mean, that's quite convincing, right? 270 00:17:27,089 --> 00:17:29,965 That seven is no accident. 271 00:17:32,630 --> 00:17:35,550 The eleven is no accident. 272 00:17:35,550 --> 00:17:40,680 But just tell me how do I know if I transpose this guy -- 273 00:17:40,680 --> 00:17:42,820 How do I know it's symmetric? 274 00:17:42,820 --> 00:17:45,620 Well, I'm going to transpose it. 275 00:17:45,620 --> 00:17:49,610 And when I transpose it, I'm hoping 276 00:17:49,610 --> 00:17:52,210 I get the matrix back again. 277 00:17:52,210 --> 00:17:54,630 So can I transpose R transpose R? 278 00:17:54,630 --> 00:17:56,157 So just -- so, why? 279 00:17:59,880 --> 00:18:02,945 Well, my suggestion is take the transpose. 280 00:18:09,800 --> 00:18:11,670 That's the only way to show it's symmetric. 281 00:18:11,670 --> 00:18:14,340 Take the transpose and see that it didn't change. 282 00:18:14,340 --> 00:18:19,970 Okay, so I take the transpose of R transpose R. 283 00:18:19,970 --> 00:18:20,470 Okay. 284 00:18:20,470 --> 00:18:23,190 How do I do that? 285 00:18:23,190 --> 00:18:27,680 This is our little practice on the rules for transposes. 286 00:18:27,680 --> 00:18:33,780 So the rule for transposes is the order gets reversed. 287 00:18:33,780 --> 00:18:38,300 Just like inverses, which we did prove, 288 00:18:38,300 --> 00:18:44,390 same rule for transposes and -- which we'll now use. 289 00:18:44,390 --> 00:18:46,110 So the order gets reversed. 290 00:18:46,110 --> 00:18:51,600 It's the transpose of that that comes first, 291 00:18:51,600 --> 00:18:56,880 and the transpose of this that comes -- no. 292 00:18:56,880 --> 00:18:58,010 Is that -- yeah. 293 00:18:58,010 --> 00:19:01,130 That's what I have to write, right? 294 00:19:01,130 --> 00:19:03,500 This is a product of two matrices and I want its 295 00:19:03,500 --> 00:19:04,350 transpose. 296 00:19:04,350 --> 00:19:06,880 So I put the matrices in the opposite order 297 00:19:06,880 --> 00:19:08,480 and I transpose them. 298 00:19:08,480 --> 00:19:09,780 But what have I got here? 299 00:19:09,780 --> 00:19:12,320 What is R transpose transpose? 300 00:19:15,300 --> 00:19:18,670 Well, don't all speak at once. 301 00:19:18,670 --> 00:19:22,080 R transpose transpose, I flipped over the diagonal, 302 00:19:22,080 --> 00:19:28,540 I flipped over the diagonal again, so I've got R. 303 00:19:28,540 --> 00:19:32,770 And that's just my point, that if I started with this matrix, 304 00:19:32,770 --> 00:19:34,740 I transposed it, I got it back again. 305 00:19:37,850 --> 00:19:46,810 So that's the check, without using numbers, but with -- 306 00:19:46,810 --> 00:19:52,510 it checked in two lines that I always get symmetric matrices 307 00:19:52,510 --> 00:19:53,480 this way. 308 00:19:53,480 --> 00:19:55,680 And actually, that's where they come 309 00:19:55,680 --> 00:20:00,220 from in so many practical applications. 310 00:20:00,220 --> 00:20:02,730 Okay. 311 00:20:02,730 --> 00:20:07,580 So now I've said something today about permutations and about 312 00:20:07,580 --> 00:20:14,540 transposes and about symmetry and I'm ready 313 00:20:14,540 --> 00:20:17,290 for chapter three. 314 00:20:17,290 --> 00:20:21,550 Can we take a breath -- 315 00:20:21,550 --> 00:20:25,250 the tape won't take a breath, but the lecturer will, 316 00:20:25,250 --> 00:20:31,980 because to tell you about vector spaces is -- 317 00:20:31,980 --> 00:20:38,220 we really have to start now and think, okay, listen up. 318 00:20:38,220 --> 00:20:39,500 What are vector spaces? 319 00:20:47,400 --> 00:20:48,400 And what are sub-spaces? 320 00:20:51,160 --> 00:20:51,890 Okay. 321 00:20:51,890 --> 00:21:01,430 So, the point is, The main operations that we do -- 322 00:21:01,430 --> 00:21:04,450 what do we do with vectors? 323 00:21:04,450 --> 00:21:05,820 We add them. 324 00:21:05,820 --> 00:21:07,270 We know how to add two vectors. 325 00:21:09,880 --> 00:21:13,930 We multiply them by numbers, usually called scalers. 326 00:21:13,930 --> 00:21:17,350 If we have a vector, we know what three V is. 327 00:21:17,350 --> 00:21:24,030 If we have a vector V and W, we know what V plus W is. 328 00:21:24,030 --> 00:21:26,670 Those are the two operations that we've 329 00:21:26,670 --> 00:21:29,090 got to be able to do. 330 00:21:29,090 --> 00:21:33,220 To legitimately talk about a space of vectors, 331 00:21:33,220 --> 00:21:35,870 the requirement is that we should 332 00:21:35,870 --> 00:21:39,990 be able to add the things and multiply by numbers 333 00:21:39,990 --> 00:21:44,380 and that there should be some decent rules satisfied. 334 00:21:44,380 --> 00:21:45,370 Okay. 335 00:21:45,370 --> 00:21:48,590 So let me start with examples. 336 00:21:48,590 --> 00:21:50,755 So I'm talking now about vector spaces. 337 00:21:56,810 --> 00:21:58,685 And I'm going to start with examples. 338 00:22:06,690 --> 00:22:09,480 Let me say again what this word space is meaning. 339 00:22:09,480 --> 00:22:14,010 When I say that word space, that means to me 340 00:22:14,010 --> 00:22:19,440 that I've got a bunch of vectors, a space of vectors. 341 00:22:19,440 --> 00:22:21,010 But not just any bunch of vectors. 342 00:22:24,010 --> 00:22:28,720 It has to be a space of vectors -- 343 00:22:28,720 --> 00:22:31,300 has to allow me to do the operations that vectors 344 00:22:31,300 --> 00:22:32,740 are for. 345 00:22:32,740 --> 00:22:37,110 I have to be able to add vectors and multiply by numbers. 346 00:22:37,110 --> 00:22:39,560 I have to be able to take linear combinations. 347 00:22:39,560 --> 00:22:43,120 Well, where did we meet linear combinations? 348 00:22:43,120 --> 00:22:48,960 We met them back in, say in R^2. 349 00:22:48,960 --> 00:22:51,370 So there's a vector space. 350 00:22:51,370 --> 00:22:54,010 What's that vector space? 351 00:22:54,010 --> 00:22:59,270 So R two is telling me I'm talking about real numbers 352 00:22:59,270 --> 00:23:01,370 and I'm talking about two real numbers. 353 00:23:01,370 --> 00:23:11,470 So this is all two dimensional vectors -- 354 00:23:11,470 --> 00:23:16,580 real, such as -- 355 00:23:16,580 --> 00:23:18,750 well, I'm not going to be able to list them all. 356 00:23:18,750 --> 00:23:20,210 But let me put a few down. 357 00:23:20,210 --> 00:23:30,420 |3; 2|, |0;0|, |pi; e|. 358 00:23:30,420 --> 00:23:30,920 So on. 359 00:23:35,450 --> 00:23:39,890 And it's natural -- okay. 360 00:23:39,890 --> 00:23:44,270 Let's see, I guess I should do algebra first. 361 00:23:44,270 --> 00:23:46,980 Algebra means what can I do to these vectors? 362 00:23:46,980 --> 00:23:48,130 I can add them. 363 00:23:48,130 --> 00:23:50,520 I can add that to that. 364 00:23:50,520 --> 00:23:51,660 And how do I do it? 365 00:23:51,660 --> 00:23:54,460 A component at a time, of course. 366 00:23:54,460 --> 00:23:58,240 Three two added to zero zero gives me, three two. 367 00:23:58,240 --> 00:24:00,190 Sorry about that. 368 00:24:00,190 --> 00:24:05,780 Three two added to pi e gives me three plus pi, two plus e. 369 00:24:05,780 --> 00:24:07,260 Oh, you know what it does. 370 00:24:07,260 --> 00:24:11,240 And you know the picture that goes with it. 371 00:24:11,240 --> 00:24:14,830 There's the vector three two. 372 00:24:14,830 --> 00:24:19,520 And often, the picture has an arrow. 373 00:24:19,520 --> 00:24:22,550 The vector zero zero, which is a highly important vector -- 374 00:24:22,550 --> 00:24:24,610 it's got, like, the most important here 375 00:24:24,610 --> 00:24:25,930 -- is there. 376 00:24:25,930 --> 00:24:29,840 And of course there's not much of an arrow. 377 00:24:29,840 --> 00:24:35,110 Pi -- I'll have to remember -- pi is about three and a little 378 00:24:35,110 --> 00:24:37,570 more, e is about two and a little more. 379 00:24:37,570 --> 00:24:41,090 So maybe there's pi e. 380 00:24:41,090 --> 00:24:44,690 I never drew pi e before. 381 00:24:44,690 --> 00:24:47,030 It's just natural to -- 382 00:24:47,030 --> 00:24:55,560 this is the first component on the horizontal 383 00:24:55,560 --> 00:24:59,470 and this is the second component, 384 00:24:59,470 --> 00:25:02,010 going up the vertical. 385 00:25:02,010 --> 00:25:02,910 Okay. 386 00:25:02,910 --> 00:25:07,570 And the whole plane is R two. 387 00:25:07,570 --> 00:25:14,980 So R two is, we could say, the plane. 388 00:25:14,980 --> 00:25:17,710 The xy plane. 389 00:25:17,710 --> 00:25:18,920 That's what everybody thinks. 390 00:25:24,770 --> 00:25:32,800 But the point is it's a vector space because all those vectors 391 00:25:32,800 --> 00:25:34,140 are in there. 392 00:25:34,140 --> 00:25:37,380 If I removed one of them -- 393 00:25:37,380 --> 00:25:39,420 Suppose I removed zero zero. 394 00:25:39,420 --> 00:25:43,480 Suppose I tried to take the -- considered the X Y plane with 395 00:25:43,480 --> 00:25:46,360 a puncture, with a point removed. 396 00:25:46,360 --> 00:25:47,200 Like the origin. 397 00:25:47,200 --> 00:25:50,470 That would be, like, awful to take the origin away. 398 00:25:50,470 --> 00:25:52,570 Why is that? 399 00:25:52,570 --> 00:25:54,550 Why do I need the origin there? 400 00:25:54,550 --> 00:25:59,570 Because I have to be allowed -- if I had these other vectors, 401 00:25:59,570 --> 00:26:03,330 I have to be allowed to multiply three two -- 402 00:26:03,330 --> 00:26:05,610 this was three two -- 403 00:26:05,610 --> 00:26:09,820 by anything, by any scaler, including zero. 404 00:26:09,820 --> 00:26:12,020 I've got to be allowed to multiply by zero 405 00:26:12,020 --> 00:26:15,010 and the result's got to be there. 406 00:26:15,010 --> 00:26:18,110 I can't do without that point. 407 00:26:18,110 --> 00:26:23,670 And I have to be able to add three two to the opposite guy, 408 00:26:23,670 --> 00:26:26,800 minus three minus two. 409 00:26:26,800 --> 00:26:29,230 And if I add those I'm back to the origin again. 410 00:26:29,230 --> 00:26:31,360 No way I can do without the origin. 411 00:26:31,360 --> 00:26:36,280 Every vector space has got that zero vector in it. 412 00:26:36,280 --> 00:26:38,650 Okay, that's an easy vector space, 413 00:26:38,650 --> 00:26:42,520 because we have a natural picture of it. 414 00:26:42,520 --> 00:26:43,840 Okay. 415 00:26:43,840 --> 00:26:46,260 Similarly easy is R^3. 416 00:26:50,340 --> 00:26:54,630 This would be all -- let me go up a little here. 417 00:26:54,630 --> 00:26:57,820 This would be -- 418 00:26:57,820 --> 00:27:02,670 R three would be all three dimensional vectors -- 419 00:27:02,670 --> 00:27:09,645 or shall I say vectors with three real components. 420 00:27:14,320 --> 00:27:15,000 Okay. 421 00:27:15,000 --> 00:27:21,030 Let me just to be sure we're together, 422 00:27:21,030 --> 00:27:23,661 let me take the vector three two zero. 423 00:27:29,410 --> 00:27:33,390 Is that a vector in R^2 or R^3? 424 00:27:33,390 --> 00:27:38,490 Definitely it's in R^3. 425 00:27:38,490 --> 00:27:40,150 It's got three components. 426 00:27:40,150 --> 00:27:43,040 One of them happens to be zero, but that's a perfectly okay 427 00:27:43,040 --> 00:27:43,850 number. 428 00:27:43,850 --> 00:27:48,290 So that's a vector in R^3. 429 00:27:48,290 --> 00:27:51,590 We don't want to mix up the -- 430 00:27:51,590 --> 00:27:55,090 I mean, keep these vectors straight and keep R^n straight. 431 00:27:55,090 --> 00:27:57,630 So what's R^n? 432 00:27:57,630 --> 00:27:59,150 R^n. 433 00:27:59,150 --> 00:28:05,990 So this is our big example, is all vectors with n components. 434 00:28:05,990 --> 00:28:11,170 And I'm making these darn things column vectors. 435 00:28:11,170 --> 00:28:14,050 Can I try to follow that convention, 436 00:28:14,050 --> 00:28:17,530 that they'll be column vectors, and their components should 437 00:28:17,530 --> 00:28:20,690 be real numbers. 438 00:28:20,690 --> 00:28:24,830 Later we'll need complex numbers and complex vectors, 439 00:28:24,830 --> 00:28:26,721 but much later. 440 00:28:26,721 --> 00:28:27,220 Okay. 441 00:28:27,220 --> 00:28:28,780 So that's a vector space. 442 00:28:31,420 --> 00:28:33,670 Now, let's see. 443 00:28:33,670 --> 00:28:35,910 What do I have to tell you about vector spaces? 444 00:28:35,910 --> 00:28:44,090 I said the most important thing, which is that we can add any 445 00:28:44,090 --> 00:28:46,760 two of these and we -- still in R^2. 446 00:28:46,760 --> 00:28:50,220 We can multiply by any number and we're still in R^2. 447 00:28:50,220 --> 00:28:53,380 We can take any combination and we're still in R^2. 448 00:28:53,380 --> 00:28:55,290 And same goes for R^n. 449 00:28:55,290 --> 00:29:02,240 It's -- honesty requires me to mention that these operations 450 00:29:02,240 --> 00:29:08,300 of adding and multiplying have to obey a few rules. 451 00:29:08,300 --> 00:29:12,790 Like, we can't just arbitrarily say, okay, the sum of three two 452 00:29:12,790 --> 00:29:15,610 and pi e is zero zero. 453 00:29:15,610 --> 00:29:18,410 It's not. 454 00:29:18,410 --> 00:29:22,650 The sum of three two and minus three two is zero zero. 455 00:29:22,650 --> 00:29:27,030 So -- oh, I'm not going to -- the book, actually, 456 00:29:27,030 --> 00:29:32,420 lists the eight rules that the addition and multiplication 457 00:29:32,420 --> 00:29:34,680 have to satisfy, but they do. 458 00:29:34,680 --> 00:29:38,810 They certainly satisfy it in R^n and usually it's not those 459 00:29:38,810 --> 00:29:42,170 eight rules that are in doubt. 460 00:29:42,170 --> 00:29:50,070 What's -- the question is, can we do those additions and do we 461 00:29:50,070 --> 00:29:51,250 stay in the space? 462 00:29:51,250 --> 00:29:55,580 Let me show you a case where you can't. 463 00:29:55,580 --> 00:29:59,810 So suppose this is going to be not a vector space. 464 00:30:05,490 --> 00:30:08,780 Suppose I take the xy plane -- so there's R^2. 465 00:30:08,780 --> 00:30:11,240 That is a vector space. 466 00:30:11,240 --> 00:30:15,940 Now suppose I just take part of it. 467 00:30:15,940 --> 00:30:17,670 Just this. 468 00:30:17,670 --> 00:30:22,270 Just this one -- this is one quarter of the vector space. 469 00:30:24,910 --> 00:30:29,965 All the vectors with positive or at least not negative 470 00:30:29,965 --> 00:30:30,465 components. 471 00:30:33,070 --> 00:30:37,540 Can I add those safely? 472 00:30:37,540 --> 00:30:38,410 Yes. 473 00:30:38,410 --> 00:30:41,690 If I add a vector with, like, two -- 474 00:30:41,690 --> 00:30:45,030 three two to another vector like five six, 475 00:30:45,030 --> 00:30:48,950 I'm still up in this quarter, no problem with adding. 476 00:30:48,950 --> 00:30:54,860 But there's a heck of a problem with multiplying by scalers, 477 00:30:54,860 --> 00:30:58,690 because there's a lot of scalers that will take me out 478 00:30:58,690 --> 00:31:02,280 of this quarter plane, like negative ones. 479 00:31:02,280 --> 00:31:05,820 If I took three two and I multiplied by minus five, 480 00:31:05,820 --> 00:31:08,240 I'm way down here. 481 00:31:08,240 --> 00:31:12,220 So that's not a vector space, because it's not -- 482 00:31:12,220 --> 00:31:14,250 closed is the right word. 483 00:31:14,250 --> 00:31:17,870 It's not closed under multiplication 484 00:31:17,870 --> 00:31:19,850 by all real numbers. 485 00:31:22,500 --> 00:31:27,150 So a vector space has to be closed under multiplication 486 00:31:27,150 --> 00:31:29,010 and addition of vectors. 487 00:31:29,010 --> 00:31:31,680 In other words, linear combinations. 488 00:31:31,680 --> 00:31:37,560 It -- so, it means that if I give you a few vectors -- 489 00:31:37,560 --> 00:31:39,980 yeah look, here's an important -- here -- 490 00:31:39,980 --> 00:31:42,420 now we're getting to some really important vector spaces. 491 00:31:42,420 --> 00:31:47,460 Well, R^n -- like, they are the most important. 492 00:31:47,460 --> 00:31:52,520 But we will be interested in so- in vector spaces that are 493 00:31:52,520 --> 00:31:55,700 inside R^n. 494 00:31:55,700 --> 00:32:01,790 Vector spaces that follow the rules, but they -- 495 00:32:01,790 --> 00:32:10,140 we don't need all of -- see, there we started with R^2 here, 496 00:32:10,140 --> 00:32:15,060 and took part of it and messed it up. 497 00:32:15,060 --> 00:32:17,420 What we got was not a vector space. 498 00:32:17,420 --> 00:32:25,670 Now tell me a vector space that is part of R^2 and is still 499 00:32:25,670 --> 00:32:31,480 safely -- we can multiply, we can add and we stay in this 500 00:32:31,480 --> 00:32:32,880 smaller vector space. 501 00:32:32,880 --> 00:32:35,680 So it's going to be called a subspace. 502 00:32:35,680 --> 00:32:40,990 So I'm going to change this bad example to a good one. 503 00:32:40,990 --> 00:32:42,440 Okay. 504 00:32:42,440 --> 00:32:45,620 So I'm going to start again with R^2, 505 00:32:45,620 --> 00:32:50,120 but I'm going to take an example -- it is a vector space, 506 00:32:50,120 --> 00:32:53,805 so it'll be a vector space inside R^2. 507 00:32:56,970 --> 00:33:03,560 And we'll call that a subspace of R^2. 508 00:33:06,450 --> 00:33:07,040 Okay. 509 00:33:07,040 --> 00:33:09,010 What can I do? 510 00:33:09,010 --> 00:33:11,960 It's got something in it. 511 00:33:11,960 --> 00:33:14,730 Suppose it's got this vector in it. 512 00:33:14,730 --> 00:33:17,070 Okay. 513 00:33:17,070 --> 00:33:19,740 If that vector's in my little subspace 514 00:33:19,740 --> 00:33:23,500 and it's a true subspace, then there's 515 00:33:23,500 --> 00:33:24,990 got to be some more in it, 516 00:33:24,990 --> 00:33:25,640 right? 517 00:33:25,640 --> 00:33:28,900 I have to be able to multiply that by two, 518 00:33:28,900 --> 00:33:33,660 and that double vector has to be included. 519 00:33:33,660 --> 00:33:36,610 Have to be able to multiply by zero, that vector, 520 00:33:36,610 --> 00:33:39,420 or by half, or by three quarters. 521 00:33:39,420 --> 00:33:40,310 All these vectors. 522 00:33:40,310 --> 00:33:44,470 Or by minus a half, or by minus one. 523 00:33:44,470 --> 00:33:48,730 I have to be able to multiply by any number. 524 00:33:48,730 --> 00:33:52,250 So that is going to say that I have to have that whole line. 525 00:33:52,250 --> 00:33:56,410 Do you see that? 526 00:33:56,410 --> 00:33:58,440 Once I get a vector in there -- 527 00:33:58,440 --> 00:34:03,070 I've got the whole line of all multiples of that vector. 528 00:34:03,070 --> 00:34:09,320 I can't have a vector space without extending to get 529 00:34:09,320 --> 00:34:10,770 those multiples in there. 530 00:34:10,770 --> 00:34:12,399 Now I still have to check addition. 531 00:34:15,000 --> 00:34:16,179 But that comes out okay. 532 00:34:16,179 --> 00:34:20,560 This line is going to work, because I could add something 533 00:34:20,560 --> 00:34:23,219 on the line to something else on the line 534 00:34:23,219 --> 00:34:26,540 and I'm still on the line. 535 00:34:26,540 --> 00:34:28,469 So, example. 536 00:34:28,469 --> 00:34:33,340 So this is all examples of a subspace -- 537 00:34:33,340 --> 00:34:45,199 our example is a line in R^2 actually -- not just any line. 538 00:34:45,199 --> 00:34:50,239 If I took this line, would that -- 539 00:34:50,239 --> 00:34:51,960 so all the vectors on that line. 540 00:34:51,960 --> 00:34:56,929 So that vector and that vector and this vector and this vector 541 00:34:56,929 --> 00:34:58,380 -- 542 00:34:58,380 --> 00:35:05,450 in lighter type, I'm drawing something that doesn't work. 543 00:35:05,450 --> 00:35:07,510 It's not a subspace. 544 00:35:07,510 --> 00:35:09,890 The line in R^2 -- to be a subspace, 545 00:35:09,890 --> 00:35:15,220 the line in R^2 must go through the zero vector. 546 00:35:19,400 --> 00:35:21,700 Because -- why is this line no good? 547 00:35:21,700 --> 00:35:23,140 Let me do a dashed line. 548 00:35:27,500 --> 00:35:31,290 Because if I multiplied that vector on the dashed line 549 00:35:31,290 --> 00:35:34,490 by zero, then I'm down here, I'm not on the dashed line. 550 00:35:34,490 --> 00:35:36,620 Z- zero's got to be. 551 00:35:36,620 --> 00:35:39,920 Every subspace has got to contain zero -- 552 00:35:39,920 --> 00:35:43,230 because I must be allowed to multiply by zero and that will 553 00:35:43,230 --> 00:35:46,300 always give me the zero vector. 554 00:35:46,300 --> 00:35:48,020 Okay. 555 00:35:48,020 --> 00:35:51,410 Now, I was going to make -- 556 00:35:51,410 --> 00:35:54,460 create some subspaces. 557 00:35:54,460 --> 00:35:59,610 Oh, while I'm in R^2, why don't we think of all 558 00:35:59,610 --> 00:36:01,110 the possibilities. 559 00:36:01,110 --> 00:36:03,760 R two, there can't be that many. 560 00:36:03,760 --> 00:36:07,510 So what are the possible subspaces of R^2? 561 00:36:07,510 --> 00:36:08,720 Let me list them. 562 00:36:11,480 --> 00:36:16,220 So I'm listing now the subspaces of R^2. 563 00:36:19,660 --> 00:36:23,750 And one possibility that we always allow 564 00:36:23,750 --> 00:36:29,760 is all of R two, the whole thing, the whole space. 565 00:36:29,760 --> 00:36:34,010 That counts as a subspace of itself. 566 00:36:34,010 --> 00:36:35,830 You always want to allow that. 567 00:36:35,830 --> 00:36:39,750 Then the others are lines -- 568 00:36:39,750 --> 00:36:45,690 any line, meaning infinitely far in both directions 569 00:36:45,690 --> 00:36:49,810 through the zero. 570 00:36:55,110 --> 00:36:57,790 So that's like the whole space -- 571 00:36:57,790 --> 00:37:00,550 that's like whole two D space. 572 00:37:00,550 --> 00:37:02,860 This is like one dimension. 573 00:37:02,860 --> 00:37:05,320 Is this line the same as R^1 ? 574 00:37:05,320 --> 00:37:07,470 No. 575 00:37:07,470 --> 00:37:11,200 You could say it looks a lot like R^1. 576 00:37:11,200 --> 00:37:14,380 R^1 was just a line and this is a line. 577 00:37:14,380 --> 00:37:17,460 But this is a line inside R^2. 578 00:37:17,460 --> 00:37:20,440 The vectors here have two components. 579 00:37:20,440 --> 00:37:23,600 So that's not the same as R^1, because there the vectors only 580 00:37:23,600 --> 00:37:25,570 have one component. 581 00:37:25,570 --> 00:37:29,590 Very close, you could say, but not the same. 582 00:37:29,590 --> 00:37:30,320 Okay. 583 00:37:30,320 --> 00:37:32,250 And now there's a third possibility. 584 00:37:36,550 --> 00:37:40,940 There's a third subspace that's -- 585 00:37:40,940 --> 00:37:47,970 of R^2 that's not the whole thing, and it's not a line. 586 00:37:47,970 --> 00:37:50,170 It's even less. 587 00:37:50,170 --> 00:37:52,840 It's just the zero vector alone. 588 00:37:52,840 --> 00:37:55,170 The zero vector alone, only. 589 00:38:01,250 --> 00:38:05,550 I'll often call this subspace Z, just for zero. 590 00:38:05,550 --> 00:38:07,700 Here's a line, L. 591 00:38:07,700 --> 00:38:10,010 Here's a plane, all of R^2. 592 00:38:10,010 --> 00:38:14,680 So, do you see that the zero vector's okay? 593 00:38:14,680 --> 00:38:16,970 You would just -- to understand subspaces, 594 00:38:16,970 --> 00:38:20,820 we have to know the rules -- and knowing the rules means that we 595 00:38:20,820 --> 00:38:25,040 have to see that yes, the zero vector by itself, 596 00:38:25,040 --> 00:38:27,990 just this guy alone satisfies the rules. 597 00:38:27,990 --> 00:38:28,690 Why's that? 598 00:38:28,690 --> 00:38:31,320 Oh, it's too dumb to tell you. 599 00:38:31,320 --> 00:38:36,430 If I took that and added it to itself, I'm still there. 600 00:38:36,430 --> 00:38:40,320 If I took that and multiplied by seventeen, I'm still there. 601 00:38:40,320 --> 00:38:44,070 So I've done the operations, adding and multiplying 602 00:38:44,070 --> 00:38:47,010 by numbers, that are required, and I didn't go 603 00:38:47,010 --> 00:38:50,300 outside this one point space. 604 00:38:53,570 --> 00:38:57,170 So that's always -- that's the littlest subspace. 605 00:38:57,170 --> 00:39:00,930 And the largest subspace is the whole thing and in-between come 606 00:39:00,930 --> 00:39:02,370 all -- 607 00:39:02,370 --> 00:39:04,080 whatever's in between. 608 00:39:04,080 --> 00:39:04,580 Okay. 609 00:39:04,580 --> 00:39:07,610 So for example, what's in between for R^3? 610 00:39:07,610 --> 00:39:12,100 So if I'm in ordinary three dimensions, the subspace is R, 611 00:39:12,100 --> 00:39:18,250 all of R^3 at one extreme, the zero vector at the bottom. 612 00:39:18,250 --> 00:39:23,430 And then a plane, a plane through the origin. 613 00:39:23,430 --> 00:39:26,510 Or a line, a line through the origin. 614 00:39:26,510 --> 00:39:32,970 So with R^3, the subspaces were R^3, plane through the origin, 615 00:39:32,970 --> 00:39:37,560 line through the origin and a zero vector by itself, 616 00:39:37,560 --> 00:39:43,030 zero zero zero, just that single vector. 617 00:39:43,030 --> 00:39:44,360 Okay, you've got the idea. 618 00:39:47,470 --> 00:39:51,080 But, now comes -- 619 00:39:51,080 --> 00:39:53,350 the reality is -- 620 00:39:53,350 --> 00:39:57,530 what are these -- where do these subspaces come -- 621 00:39:57,530 --> 00:40:00,950 how do they come out of matrices? 622 00:40:00,950 --> 00:40:06,080 And I want to take this matrix -- 623 00:40:06,080 --> 00:40:08,350 oh, let me take that matrix. 624 00:40:08,350 --> 00:40:17,430 So I want to create some subspaces out of that matrix. 625 00:40:17,430 --> 00:40:26,980 Well, one subspace is from the columns. 626 00:40:26,980 --> 00:40:29,760 Okay. 627 00:40:29,760 --> 00:40:34,050 So this is the important subspace, 628 00:40:34,050 --> 00:40:38,190 the first important subspace that comes from that matrix -- 629 00:40:38,190 --> 00:40:40,750 I'm going to -- let me call it A again. 630 00:40:40,750 --> 00:40:44,370 Back to -- okay. 631 00:40:44,370 --> 00:40:48,150 I'm looking at the columns of A. 632 00:40:48,150 --> 00:40:50,530 Those are vectors in R^3. 633 00:40:50,530 --> 00:40:52,380 So the columns are in R^3. 634 00:40:52,380 --> 00:40:58,100 The columns are in R^3. 635 00:41:02,280 --> 00:41:04,585 So I want those columns to be in my subspace. 636 00:41:08,970 --> 00:41:11,960 Now I can't just put two columns in my subspace 637 00:41:11,960 --> 00:41:14,512 and call it a subspace. 638 00:41:14,512 --> 00:41:16,970 What do I have to throw in -- if I'm going to put those two 639 00:41:16,970 --> 00:41:21,460 columns in, what else has got to be there to have a subspace? 640 00:41:21,460 --> 00:41:25,050 I must be able to add those things. 641 00:41:25,050 --> 00:41:28,460 So the sum of those columns -- 642 00:41:28,460 --> 00:41:34,970 so these columns are in R^3, and I have to be able -- 643 00:41:34,970 --> 00:41:37,330 I'm, you know, I want that to be in my subspace, 644 00:41:37,330 --> 00:41:39,080 I want that to be in my subspace, 645 00:41:39,080 --> 00:41:42,880 but therefore I have to be able to multiply them by anything. 646 00:41:42,880 --> 00:41:45,910 Zero zero zero has got to be in my subspace. 647 00:41:45,910 --> 00:41:48,630 I have to be able to add them so that four five five 648 00:41:48,630 --> 00:41:50,150 is in the subspace. 649 00:41:50,150 --> 00:41:53,054 I've got to be able to add one of these plus three of these. 650 00:41:53,054 --> 00:41:54,470 That'll give me some other vector. 651 00:41:57,100 --> 00:42:02,180 I have to be able to take all the linear combinations. 652 00:42:02,180 --> 00:42:14,200 So these are columns in R^3 and all there linear combinations 653 00:42:14,200 --> 00:42:16,920 form a subspace. 654 00:42:21,260 --> 00:42:23,400 What do I mean by linear combinations? 655 00:42:23,400 --> 00:42:26,060 I mean multiply that by something, 656 00:42:26,060 --> 00:42:28,290 multiply that by something and add. 657 00:42:28,290 --> 00:42:33,350 The two operations of linear algebra, multiplying by numbers 658 00:42:33,350 --> 00:42:36,060 and adding vectors. 659 00:42:36,060 --> 00:42:38,930 And, if I include all the results, 660 00:42:38,930 --> 00:42:40,875 then I'm guaranteed to have a subspace. 661 00:42:43,570 --> 00:42:46,860 I've done the job. 662 00:42:46,860 --> 00:42:49,210 And we'll give it a name -- 663 00:42:49,210 --> 00:42:49,960 the column space. 664 00:42:53,740 --> 00:42:54,380 Column space. 665 00:43:01,220 --> 00:43:05,920 And maybe I'll call it C of A. 666 00:43:05,920 --> 00:43:07,120 C for column space. 667 00:43:11,580 --> 00:43:15,020 There's an idea there that -- 668 00:43:15,020 --> 00:43:22,750 Like, the central idea for today's lecture is -- 669 00:43:22,750 --> 00:43:25,220 got a few vectors. 670 00:43:25,220 --> 00:43:27,130 Not satisfied with a few vectors, 671 00:43:27,130 --> 00:43:29,800 we want a space of vectors. 672 00:43:29,800 --> 00:43:33,160 The vectors, they're in -- these vectors in -- are in R^3 , 673 00:43:33,160 --> 00:43:37,050 so our space of vectors will be vectors in R^3. 674 00:43:37,050 --> 00:43:40,940 The key idea's -- we have to be able to take 675 00:43:40,940 --> 00:43:42,400 their combinations. 676 00:43:42,400 --> 00:43:47,300 So tell me, geometrically, if I drew all these things -- 677 00:43:47,300 --> 00:43:50,060 like if I drew one two four, that would be somewhere maybe 678 00:43:50,060 --> 00:43:50,930 there. 679 00:43:50,930 --> 00:43:54,740 If I drew three three one, who knows, might be -- 680 00:43:54,740 --> 00:43:57,140 I don't know, I'll say there. 681 00:43:57,140 --> 00:44:01,690 There's column one, there's column two. 682 00:44:01,690 --> 00:44:06,700 What else -- what's in the whole column space? 683 00:44:06,700 --> 00:44:11,160 How do I draw the whole column space now? 684 00:44:11,160 --> 00:44:13,430 I take all combinations of those two vectors. 685 00:44:15,970 --> 00:44:18,220 Do I get -- well, I guess I actually listed 686 00:44:18,220 --> 00:44:19,160 the possibilities. 687 00:44:19,160 --> 00:44:21,940 Do I get the whole space? 688 00:44:21,940 --> 00:44:24,190 Do I get a plane? 689 00:44:24,190 --> 00:44:26,984 I get more than a line, that's for sure. 690 00:44:26,984 --> 00:44:28,900 And I certainly get more than the zero vector, 691 00:44:28,900 --> 00:44:31,610 but I do get the zero vector included. 692 00:44:31,610 --> 00:44:34,160 What do I get if I combine -- 693 00:44:34,160 --> 00:44:39,115 take all the combinations of two vectors in R^3 ? 694 00:44:44,040 --> 00:44:46,450 So I've got all this stuff on -- 695 00:44:46,450 --> 00:44:49,040 that whole line gets filled out, that whole line gets filled 696 00:44:49,040 --> 00:44:51,190 out, but all in-between gets filled out -- 697 00:44:51,190 --> 00:44:52,900 between the two lines because I -- 698 00:44:52,900 --> 00:44:56,610 I allowed to add something from one line, something 699 00:44:56,610 --> 00:44:57,850 from the other. 700 00:44:57,850 --> 00:44:58,810 You see what's coming? 701 00:44:58,810 --> 00:44:59,643 I'm getting a plane. 702 00:45:05,060 --> 00:45:06,790 That's my -- and it's through the origin. 703 00:45:10,210 --> 00:45:17,950 Those two vectors, namely one two four and three three one, 704 00:45:17,950 --> 00:45:20,590 when I take all their combinations, 705 00:45:20,590 --> 00:45:21,770 I fill out a whole plane. 706 00:45:21,770 --> 00:45:25,240 Please think about that. 707 00:45:25,240 --> 00:45:28,280 That's the picture you have to see. 708 00:45:28,280 --> 00:45:31,940 You sure have to see it in R^3 , because we're going to do it 709 00:45:31,940 --> 00:45:36,880 in R^10, and we may take a combination of five vectors 710 00:45:36,880 --> 00:45:40,740 in R^10, and what will we have? 711 00:45:40,740 --> 00:45:41,630 God knows. 712 00:45:41,630 --> 00:45:44,910 It's some subspace. 713 00:45:44,910 --> 00:45:46,880 We'll have five vectors. 714 00:45:46,880 --> 00:45:49,010 They'll all have ten components. 715 00:45:49,010 --> 00:45:52,320 We take their combinations. 716 00:45:52,320 --> 00:45:58,240 We don't have R^5 , because our vectors have ten components. 717 00:45:58,240 --> 00:46:05,020 And we possibly have, like, some five dimensional flat thing 718 00:46:05,020 --> 00:46:06,680 going through the origin for sure. 719 00:46:09,220 --> 00:46:12,110 Well, of course, if those five vectors were all on the line, 720 00:46:12,110 --> 00:46:13,710 then we would only get that line. 721 00:46:13,710 --> 00:46:16,840 So, you see, there are, like, other possibilities here. 722 00:46:16,840 --> 00:46:21,690 It depends what -- it depends on those five vectors. 723 00:46:21,690 --> 00:46:25,440 Just like if our two columns had been on the same line, 724 00:46:25,440 --> 00:46:28,640 then the column space would have been only a line. 725 00:46:28,640 --> 00:46:31,440 Here it was a plane. 726 00:46:31,440 --> 00:46:31,940 Okay. 727 00:46:35,700 --> 00:46:37,610 I'm going to stop at that point. 728 00:46:37,610 --> 00:46:44,220 That's the central idea of -- the great example of how 729 00:46:44,220 --> 00:46:48,960 to create a subspace from a matrix. 730 00:46:48,960 --> 00:46:51,990 Take its columns, take their combinations, 731 00:46:51,990 --> 00:46:57,360 all their linear combinations and you get the column space. 732 00:46:57,360 --> 00:47:01,060 And that's the central sort of -- 733 00:47:01,060 --> 00:47:04,320 we're looking at linear algebra at a higher level. 734 00:47:04,320 --> 00:47:07,600 When I look at A -- now, I want to look at Ax=b. 735 00:47:07,600 --> 00:47:10,090 That'll be the first thing in the next lecture. 736 00:47:13,650 --> 00:47:17,300 How do I understand Ax=b in this language -- 737 00:47:17,300 --> 00:47:22,580 in this new language of vector spaces and column spaces. 738 00:47:22,580 --> 00:47:24,830 And what are other subspaces? 739 00:47:24,830 --> 00:47:30,230 So the column space is a big one, there are others to come. 740 00:47:30,230 --> 00:47:32,270 Okay, thanks.