1 00:00:09,510 --> 00:00:10,100 OK. 2 00:00:10,100 --> 00:00:16,500 Uh this is the review lecture for the first part 3 00:00:16,500 --> 00:00:21,880 of the course, the Ax=b part of the course. 4 00:00:21,880 --> 00:00:28,710 And the exam will emphasize chapter three. 5 00:00:28,710 --> 00:00:31,890 Because those are the --0 chapter 6 00:00:31,890 --> 00:00:34,940 three was about the rectangular matrices 7 00:00:34,940 --> 00:00:39,750 where we had null spaces and null spaces of A transpose, 8 00:00:39,750 --> 00:00:43,450 and ranks, and all the things that 9 00:00:43,450 --> 00:00:49,890 are so clear when the matrix is square and invertible, 10 00:00:49,890 --> 00:00:57,080 they became things to think about for rectangular matrices. 11 00:00:57,080 --> 00:01:01,590 So, and vector spaces and subspaces and above all 12 00:01:01,590 --> 00:01:04,400 those four subspaces. 13 00:01:04,400 --> 00:01:07,940 OK, I'm thinking to start at least I'll 14 00:01:07,940 --> 00:01:13,210 just look at old exams, read out questions, write 15 00:01:13,210 --> 00:01:15,320 on the board what I need to and we 16 00:01:15,320 --> 00:01:17,980 can see what the answers are. 17 00:01:17,980 --> 00:01:22,420 The first one I see is one I can just read out. 18 00:01:22,420 --> 00:01:23,930 Well, I'll write a little. 19 00:01:23,930 --> 00:01:32,145 Suppose u, v and w are nonzero vectors in R^7. 20 00:01:39,230 --> 00:01:44,190 What are the possible -- they span a -- a vector space. 21 00:01:44,190 --> 00:01:47,510 They span a subspace of R^7, and what are the possible 22 00:01:47,510 --> 00:01:49,130 dimensions? 23 00:01:49,130 --> 00:01:50,840 So that's a straightforward question, 24 00:01:50,840 --> 00:01:55,610 what are the possible dimensions of the subspace spanned 25 00:01:55,610 --> 00:01:58,040 by u, v and w? 26 00:01:58,040 --> 00:02:03,430 OK, one, two, or three, right. 27 00:02:03,430 --> 00:02:05,320 One, two or three. 28 00:02:05,320 --> 00:02:08,930 Couldn't be more because we've only got three vectors, 29 00:02:08,930 --> 00:02:12,430 and couldn't be zero because -- 30 00:02:15,580 --> 00:02:19,030 because I told you the vectors were nonzero. 31 00:02:19,030 --> 00:02:23,020 Otherwise if I allowed the possibility that those were all 32 00:02:23,020 --> 00:02:27,319 the zero vector -- then the zero-dimensional subspace would 33 00:02:27,319 --> 00:02:28,110 have been in there. 34 00:02:28,110 --> 00:02:29,250 OK. 35 00:02:29,250 --> 00:02:35,970 Now can I jump to a more serious question? 36 00:02:35,970 --> 00:02:37,600 OK. 37 00:02:37,600 --> 00:02:41,850 We have a five by three matrix. 38 00:02:41,850 --> 00:02:44,860 And I'm calling it U. 39 00:02:44,860 --> 00:02:47,010 I'm saying it's in echelon form. 40 00:02:47,010 --> 00:02:50,600 And it has three pivots, r=3. 41 00:02:50,600 --> 00:02:51,810 Three pivots. 42 00:02:57,560 --> 00:02:58,140 Ok. 43 00:02:58,140 --> 00:03:01,820 First question what's the null space? 44 00:03:01,820 --> 00:03:03,865 What's the null space of this matrix 45 00:03:03,865 --> 00:03:11,910 U, so this matrix is five by three, and I find it helpful to 46 00:03:11,910 --> 00:03:13,860 just see 47 00:03:13,860 --> 00:03:17,750 visually what five by three means, what that shape is. 48 00:03:17,750 --> 00:03:19,420 Three columns. 49 00:03:19,420 --> 00:03:20,810 Three columns in U, 50 00:03:20,810 --> 00:03:23,040 then five rows, 51 00:03:23,040 --> 00:03:24,430 three pivots, 52 00:03:24,430 --> 00:03:26,650 and what's the null space? 53 00:03:26,650 --> 00:03:29,360 The null space of U is -- 54 00:03:29,360 --> 00:03:33,000 and it asks for a spec-of course I'm 55 00:03:33,000 --> 00:03:35,390 looking for an answer that isn't just 56 00:03:35,390 --> 00:03:37,340 the definition of the null space, 57 00:03:37,340 --> 00:03:42,030 but is the null space of this matrix, with this information. 58 00:03:42,030 --> 00:03:44,220 And what is it? 59 00:03:44,220 --> 00:03:46,110 It's only the zero vector. 60 00:03:46,110 --> 00:03:48,810 Because we're told that the rank is three, 61 00:03:48,810 --> 00:03:53,840 so those three columns must be independent, no combination -- 62 00:03:53,840 --> 00:03:58,890 of those columns is the zero vector except -- 63 00:03:58,890 --> 00:04:01,671 so the only thing in this null space is the zero vector, 64 00:04:01,671 --> 00:04:02,170 and I -- 65 00:04:02,170 --> 00:04:05,030 I could even say what that vector is, zero, columns also? 66 00:04:05,030 --> 00:04:06,200 zero, zero. 67 00:04:06,200 --> 00:04:07,870 That's OK. 68 00:04:07,870 --> 00:04:10,020 So that's what's in the null space. 69 00:04:10,020 --> 00:04:15,015 All right? let me go on with -- this question has multiple 70 00:04:15,015 --> 00:04:15,515 parts. 71 00:04:18,089 --> 00:04:24,950 What's the -- oh now it asks you about a ten by three matrix, B, 72 00:04:24,950 --> 00:04:31,530 which is the matrix U and two U. 73 00:04:31,530 --> 00:04:37,340 It actually -- I would probably be writing R -- 74 00:04:37,340 --> 00:04:40,630 and maybe I should be writing R here now. 75 00:04:40,630 --> 00:04:47,800 This exam goes back a few years when I emphasized U more 76 00:04:47,800 --> 00:04:49,980 than R. 77 00:04:49,980 --> 00:04:54,630 Now, what's the echelon form for that matrix? 78 00:04:54,630 --> 00:04:57,150 So the echelon form, what's the rank, 79 00:04:57,150 --> 00:04:59,125 Yes. and what's the echelon form? 80 00:05:04,160 --> 00:05:06,270 Let's suppose this is in reduced echelon form, 81 00:05:06,270 --> 00:05:06,360 so that I could be using the letter R. 82 00:05:06,360 --> 00:05:06,480 So I'll ask for the reduced row echelon form so imagine that 83 00:05:06,480 --> 00:05:06,610 So I'm seeing the same length in b, three, and also in x. 84 00:05:06,610 --> 00:05:07,110 these are -- 85 00:05:07,110 --> 00:05:09,880 U is in reduced row echelon form but now 86 00:05:09,880 --> 00:05:15,500 I've doubled the height of the matrix, 87 00:05:15,500 --> 00:05:22,720 So the (x)-s that it multiplies have three components, 88 00:05:22,720 --> 00:05:29,150 what will happen when we do row reduction? 89 00:05:29,150 --> 00:05:32,950 What row reduction will take us to what matrix here? 90 00:05:39,170 --> 00:05:42,270 So you start doing elimination. 91 00:05:42,270 --> 00:05:44,590 You're doing elimination on single rows. 92 00:05:44,590 --> 00:05:48,050 But of course we're allowed to think of blocks. 93 00:05:48,050 --> 00:05:51,820 So what, well, what's the answer look like? 94 00:05:51,820 --> 00:05:53,921 U and z- or R -- 95 00:05:53,921 --> 00:05:56,170 let's -- I'll stay with this letter U but I'm what was 96 00:05:56,170 --> 00:05:56,660 the name of that --2 winning a million dollars or really 97 00:05:56,660 --> 00:06:00,030 thinking it's in reduced form, and zero. 98 00:06:00,030 --> 00:06:00,710 OK. 99 00:06:00,710 --> 00:06:01,810 Fine. 100 00:06:01,810 --> 00:06:10,390 Then it asks oh, further, it asks about this matrix. 101 00:06:10,390 --> 00:06:15,720 U, U, U, and zero. 102 00:06:15,720 --> 00:06:18,190 OK, what's the echelon form of this? 103 00:06:18,190 --> 00:06:20,050 So it's just like practice in thinking 104 00:06:20,050 --> 00:06:22,560 through what would row elimination, 105 00:06:22,560 --> 00:06:25,790 what would row reduction do. 106 00:06:25,790 --> 00:06:29,450 Have I thought this through, so what -- what are we -- 107 00:06:29,450 --> 00:06:31,920 if we start doing elimination, basically we're going 108 00:06:31,920 --> 00:06:34,030 to subtract these rows from these -- 109 00:06:34,030 --> 00:06:36,080 In the column space, because I do 110 00:06:36,080 --> 00:06:40,190 know that it can so it's going to take us to U, U, zero, 111 00:06:40,190 --> 00:06:44,090 and minus U, For example, I'm going to ask you for a I guess, 112 00:06:44,090 --> 00:06:44,590 right? 113 00:06:44,590 --> 00:06:47,240 But is it a- is it three by three? 114 00:06:47,240 --> 00:06:53,690 Take the thing all the way to R -- let's suppose U is really R. 115 00:06:53,690 --> 00:06:56,660 Suppose that we're really going for the reduced row echelon 116 00:06:56,660 --> 00:06:56,930 doing linear algebra of course -- 117 00:06:56,930 --> 00:06:58,390 but I didn't have to do the form. 118 00:06:58,390 --> 00:07:01,090 Then would we stop there? 119 00:07:01,090 --> 00:07:03,260 No. 120 00:07:03,260 --> 00:07:07,740 We would clean out, we would -- we could use this to -- 121 00:07:07,740 --> 00:07:10,770 is that right, can I so I took this row -- 122 00:07:10,770 --> 00:07:13,540 these rows away from these to get there. 123 00:07:13,540 --> 00:07:16,885 Now I take these rows away from these, so that gives me zero. 124 00:07:19,460 --> 00:07:25,030 There. there?1 And now what more would I do if I'm really 125 00:07:25,030 --> 00:07:29,460 shooting for R, the reduced row echelon form? 126 00:07:29,460 --> 00:07:33,200 I would -- then I want plus ones in the pivot, 127 00:07:33,200 --> 00:07:37,750 so I would multiply through by minus one to get plus there. 128 00:07:37,750 --> 00:07:42,170 So essentially I'm seeing reduced row echelon form 129 00:07:42,170 --> 00:07:46,180 there and there, and there's just one little twist 130 00:07:46,180 --> 00:07:47,040 still to go. 131 00:07:47,040 --> 00:07:51,324 Do you see what that final twist might be? 132 00:07:51,324 --> 00:07:52,490 It certainly has three rows. 133 00:07:52,490 --> 00:07:57,790 To have if -- if U is in reduced row echelon form and now 134 00:07:57,790 --> 00:08:02,490 I'm looking at U, U, there's one little step to take, 135 00:08:02,490 --> 00:08:05,090 this isn't like a big deal at all, but -- 136 00:08:05,090 --> 00:08:08,880 but if I really want this to be in reduced form, 137 00:08:08,880 --> 00:08:11,050 what would I still -- 138 00:08:11,050 --> 00:08:14,740 might I still have to do? 139 00:08:14,740 --> 00:08:17,750 I might have some zero rows here, 140 00:08:17,750 --> 00:08:20,720 I might have some zero rows here that strictly 141 00:08:20,720 --> 00:08:22,720 should move to the bottom. 142 00:08:22,720 --> 00:08:26,079 Well, I'm not going to make a project out of that. 143 00:08:26,079 --> 00:08:26,620 first of all? 144 00:08:26,620 --> 00:08:27,760 OK. 145 00:08:27,760 --> 00:08:29,445 What's the rank of that matrix? 146 00:08:32,510 --> 00:08:37,250 What's the rank of this matrix C? 147 00:08:37,250 --> 00:08:40,854 Given that I know that the original U has rank three, 148 00:08:40,854 --> 00:08:42,020 what's the rank of this guy? 149 00:08:42,020 --> 00:08:42,080 Six, right. 150 00:08:42,080 --> 00:08:42,289 That has rank six, I can tell. 151 00:08:42,289 --> 00:08:43,030 What was -- what's the rank of this B, while -- 152 00:08:43,030 --> 00:08:43,159 while we're at on TV for a few weeks, did you see that, 153 00:08:43,159 --> 00:08:43,659 it? 154 00:08:43,659 --> 00:08:56,690 The rank of B, is that six or three? 155 00:08:56,690 --> 00:08:58,200 So A is a three by three matrix. 156 00:08:58,200 --> 00:08:59,300 Three is right. 157 00:08:59,300 --> 00:08:59,940 Three is right. 158 00:08:59,940 --> 00:09:02,520 Because we actually got it to where we could just see three 159 00:09:02,520 --> 00:09:03,330 pivots. 160 00:09:03,330 --> 00:09:08,230 OK, and oh, now finally this easy one, 161 00:09:08,230 --> 00:09:15,020 what's the dimension be solved exactly when B is in the column 162 00:09:15,020 --> 00:09:18,410 space, of the null space -- 163 00:09:18,410 --> 00:09:22,370 of the null space of C transpose? 164 00:09:22,370 --> 00:09:22,980 Oh, boy. 165 00:09:22,980 --> 00:09:27,260 OK, so what do I -- if I want the dimension of a null space, 166 00:09:27,260 --> 00:09:29,240 I want to know the size of the matrix -- 167 00:09:29,240 --> 00:09:32,450 so what's the size of the matrix C? 168 00:09:32,450 --> 00:09:38,200 It looks like it's ten by six, is it? 169 00:09:38,200 --> 00:09:49,280 Ten by six, so C is ten by six, so m is ten, so C has ten rows, 170 00:09:49,280 --> 00:09:54,190 C transpose has ten columns, so there are ten columns there. 171 00:09:54,190 --> 00:09:57,950 So how many free variables have I got, once I -- 172 00:09:57,950 --> 00:10:00,950 There's quite a few trues, shall I take a poll, 173 00:10:00,950 --> 00:10:03,950 if I start with the ten columns in C transpose, 174 00:10:03,950 --> 00:10:06,050 that's the m for the original C. 175 00:10:06,050 --> 00:10:07,640 And what do I subtract off? 176 00:10:07,640 --> 00:10:08,140 Six. 177 00:10:08,140 --> 00:10:10,240 Because we said that was the rank. 178 00:10:10,240 --> 00:10:11,540 So I'm left with four. 179 00:10:11,540 --> 00:10:12,040 Thanks. 180 00:10:12,040 --> 00:10:12,540 OK. 181 00:10:12,540 --> 00:10:16,840 If b is in the -- and what would -- what does the exam say 182 00:10:16,840 --> 00:10:19,080 So I think that's the right answer -- 183 00:10:19,080 --> 00:10:23,580 the dimension of the null space of C transpose would be four. 184 00:10:23,580 --> 00:10:24,870 Right. 185 00:10:24,870 --> 00:10:25,870 OK. 186 00:10:25,870 --> 00:10:27,570 Yeah. 187 00:10:27,570 --> 00:10:28,650 OK. 188 00:10:28,650 --> 00:10:34,710 So that's one question, at least it brought in some -- 189 00:10:34,710 --> 00:10:37,130 some of the dimension counts. 190 00:10:37,130 --> 00:10:37,630 OK. 191 00:10:37,630 --> 00:10:40,088 Here's another type of question. think about, what's the -- 192 00:10:40,088 --> 00:10:44,520 what's the shape of the matrix, 193 00:10:44,520 --> 00:10:52,870 I give you an equation, Ax equals two four two. 194 00:10:52,870 --> 00:10:54,590 And I give you the complete solution. 195 00:10:54,590 --> 00:10:57,992 And what's its rank? 196 00:10:57,992 --> 00:10:59,325 But I don't give you the matrix. 197 00:11:05,390 --> 00:11:10,810 And another -- there's another vector, zero, zero, one. 198 00:11:10,810 --> 00:11:11,310 OK. 199 00:11:15,951 --> 00:11:16,450 All right. 200 00:11:16,450 --> 00:11:22,660 My first question is what's the dimension of the row space? 201 00:11:22,660 --> 00:11:24,160 Of the matrix A? 202 00:11:24,160 --> 00:11:31,890 So the main thing that you want to get from this question 203 00:11:31,890 --> 00:11:36,130 is that a question could start this way. 204 00:11:36,130 --> 00:11:38,770 exam, don't just tell me. 205 00:11:38,770 --> 00:11:40,890 Sort of backward way. 206 00:11:40,890 --> 00:11:45,130 so I guess I'm asking you what is 207 00:11:45,130 --> 00:11:50,420 the column space for this basis for the null space. 208 00:11:50,420 --> 00:11:56,770 By giving you the answer and not telling you what the problem 209 00:11:56,770 --> 00:12:02,590 homework, so I was watching it, so there were three -- 210 00:12:02,590 --> 00:12:03,650 the is. 211 00:12:03,650 --> 00:12:07,890 But we can get a lot of information, 212 00:12:07,890 --> 00:12:11,590 and sometimes we can get complete information 213 00:12:11,590 --> 00:12:13,710 about that matrix A. 214 00:12:13,710 --> 00:12:14,240 OK. 215 00:12:14,240 --> 00:12:19,530 So what's the dimension of the row space of A? 216 00:12:19,530 --> 00:12:21,120 What's the rank? 217 00:12:21,120 --> 00:12:27,470 Tell me about what's the size of the matrix, yeah, just -- 218 00:12:27,470 --> 00:12:32,230 These are the things we want to matrix b 219 00:12:32,230 --> 00:12:36,470 and I'll answer them without multiplying it out.2 220 00:12:36,470 --> 00:12:41,760 Its rank -- tell me something about its null space, 221 00:12:41,760 --> 00:12:46,000 I heard the right answer for the rank, 222 00:12:46,000 --> 00:12:49,700 the rank is one in this case. 223 00:12:49,700 --> 00:12:50,230 Why? 224 00:12:50,230 --> 00:12:53,940 Because the dimension of the null space, 225 00:12:53,940 --> 00:12:59,760 so the dimension of this instant?3 the null space of A 226 00:12:59,760 --> 00:13:05,050 is from knowing that that's the complete solution, it's two. 227 00:13:05,050 --> 00:13:08,760 I'm seeing two vectors here, and they're 228 00:13:08,760 --> 00:13:12,460 independent in the null space of A, 229 00:13:12,460 --> 00:13:16,170 because they have to be in the 230 00:13:16,170 --> 00:13:20,400 OK, which -- did you watch that quiz, 231 00:13:20,400 --> 00:13:26,750 there was a quiz program null space of A if I'm allowed 232 00:13:26,750 --> 00:13:31,650 to throw into the solution any amount of those vectors, 233 00:13:31,650 --> 00:13:35,620 that tells me that's the null space part then. 234 00:13:35,620 --> 00:13:38,400 So the dimension of the null space is two, and then I -- 235 00:13:38,400 --> 00:13:44,010 of course I know the dimensions of all the -- four subspaces. 236 00:13:44,010 --> 00:13:46,090 Now actually it asks what's the matrix? 237 00:13:46,090 --> 00:13:52,440 Well, what's the matrix in this case? 238 00:13:52,440 --> 00:14:01,210 Do we want to -- shall I try to figure that out? 239 00:14:01,210 --> 00:14:01,790 Sure. 240 00:14:01,790 --> 00:14:03,290 Let's -- you'd like me to do it, OK. 241 00:14:03,290 --> 00:14:04,164 Well, what's the ---- 242 00:14:04,164 --> 00:14:05,590 I actually say in this in the 243 00:14:05,590 --> 00:14:09,630 So what about the matrix, or let me at least start it, 244 00:14:09,630 --> 00:14:10,140 OK. 245 00:14:10,140 --> 00:14:14,720 If A times this x gives two, four, two, 246 00:14:14,720 --> 00:14:19,300 what does that tell me about the matrix A? 247 00:14:19,300 --> 00:14:26,210 If A times that x solves that equation then it tells me that 248 00:14:26,210 --> 00:14:28,900 the first column is -- 249 00:14:28,900 --> 00:14:35,560 the first column of A is -- one, two, one, right. 250 00:14:35,560 --> 00:14:37,580 The first column of A has to be one, two, one, 251 00:14:37,580 --> 00:14:37,650 because if I multiply by x, that's 252 00:14:37,650 --> 00:14:37,730 going to multiply just matrix? the first column, 253 00:14:37,730 --> 00:14:37,790 and give me two, four, two. 254 00:14:37,790 --> 00:14:37,900 And then I've got two more columns to find, 255 00:14:37,900 --> 00:14:38,000 and what information have I got to find them with? 256 00:14:38,000 --> 00:14:38,070 A basis for the null space. 257 00:14:38,070 --> 00:14:38,130 I've got the null space. 258 00:14:38,130 --> 00:14:38,230 So the fact that this is in the null space, 259 00:14:38,230 --> 00:14:38,370 what does that tell So what is if b has the form -- 260 00:14:38,370 --> 00:14:40,411 so I guess I'm asking what's me about the matrix? 261 00:14:40,411 --> 00:14:59,047 A matrix that has zero, zero, one in its null space? 262 00:15:05,510 --> 00:15:10,840 That tells me that the last column of the matrix is zeroes. 263 00:15:10,840 --> 00:15:11,880 so how many think true? 264 00:15:11,880 --> 00:15:13,820 Because this is in the null space, 265 00:15:13,820 --> 00:15:14,897 the last column has to be 266 00:15:14,897 --> 00:15:16,230 When could it be solved? zeroes. 267 00:15:16,230 --> 00:15:19,510 And because this is in the null space, what's 268 00:15:19,510 --> 00:15:23,850 the second column? 269 00:15:23,850 --> 00:15:25,880 Well, this in the null space means 270 00:15:25,880 --> 00:15:27,910 that if I multiply A by that vector 271 00:15:27,910 --> 00:15:32,530 I must be getting zeroes, so I think that better be minus one, 272 00:15:32,530 --> 00:15:44,370 minus two, and minus one. 273 00:15:44,370 --> 00:15:45,300 OK. 274 00:15:45,300 --> 00:15:50,340 That's a type of question that just brings out 275 00:15:50,340 --> 00:15:53,050 the information that's in that complete solution. 276 00:15:53,050 --> 00:15:53,180 And then actually I go on to ask what vectors -- for what 277 00:15:53,180 --> 00:15:53,250 OK. vectors B can Ax=b be solved? 278 00:15:53,250 --> 00:15:53,380 Ax=b can be solved if what -- so I'm looking for a condition 279 00:15:53,380 --> 00:15:53,460 Definitely not. on b, if any. interesting -- 280 00:15:53,460 --> 00:15:53,550 the novel point was there were three ways that 281 00:15:53,550 --> 00:15:53,640 Can it be solved for every right-hand side b? 282 00:15:53,640 --> 00:15:54,140 No. 283 00:15:54,140 --> 00:16:07,315 And the answer is -- 284 00:16:11,410 --> 00:16:25,505 yes or no -- 285 00:16:29,555 --> 00:16:30,054 right. 286 00:16:34,097 --> 00:16:34,596 solvable? 287 00:16:38,700 --> 00:16:44,920 the column space of this matrix, and what is it? 288 00:16:44,920 --> 00:16:50,810 It's so the column space of that matrix is all multiples b 289 00:16:50,810 --> 00:16:54,490 Can you tell me something -- 290 00:16:54,490 --> 00:16:59,170 so I'll ask questions about this -- 291 00:16:59,170 --> 00:17:04,609 b is a multiple of one, two, one. 292 00:17:04,609 --> 00:17:05,290 Right? 293 00:17:05,290 --> 00:17:14,140 I can solve the thing if it's a multiple of one, two, one, 294 00:17:14,140 --> 00:17:18,220 and of course sure enough -- 295 00:17:18,220 --> 00:17:24,140 yeah, that was a multiple of one, two, one, 296 00:17:24,140 --> 00:17:26,829 and so I had a solution. 297 00:17:26,829 --> 00:17:34,640 So this is a case where we've got lots of null space. 298 00:17:34,640 --> 00:17:44,760 Let me just recall rank is big, don't forget those cases, 299 00:17:44,760 --> 00:17:57,390 don't forget the other cases when r is as big as it can be, 300 00:17:57,390 --> 00:18:00,580 OK. r equal m or r equal n. 301 00:18:00,580 --> 00:18:04,970 Those are -- we had a full lecture on that, the full rank, 302 00:18:04,970 --> 00:18:09,480 full lecture, and important -- important case. 303 00:18:09,480 --> 00:18:13,471 I gave you every chance to think about that. 304 00:18:13,471 --> 00:18:13,970 OK. 305 00:18:13,970 --> 00:18:15,760 I'll just move on. 306 00:18:15,760 --> 00:18:19,800 I think this is the best type of review. 307 00:18:19,800 --> 00:18:23,380 So I'm going to solve Bx equals zero. 308 00:18:23,380 --> 00:18:26,080 It's just brings these ideas out. 309 00:18:26,080 --> 00:18:30,511 Apologies to the camera while I recover glasses and exam. 310 00:18:30,511 --> 00:18:31,010 OK. 311 00:18:31,010 --> 00:18:33,700 How about a few true-false ones? 312 00:18:33,700 --> 00:18:37,100 Actually there won't be a true-false on the quiz. 313 00:18:37,100 --> 00:18:40,980 But it gives us a moment of quick review. 314 00:18:40,980 --> 00:18:44,500 True or false, how do you feel about it at 315 00:18:44,500 --> 00:18:45,190 Here's one. 316 00:18:45,190 --> 00:18:46,930 If the null space -- 317 00:18:46,930 --> 00:18:49,400 I have a square matrix. 318 00:18:49,400 --> 00:18:52,740 If its null space is just the zero vector, 319 00:18:52,740 --> 00:18:57,750 what about the null space of A transpose? 320 00:18:57,750 --> 00:19:00,840 If the null space of A is just the zero vector, 321 00:19:00,840 --> 00:19:03,580 and the matrix is square, what do I 322 00:19:03,580 --> 00:19:06,430 know about the null space of A transpose? 323 00:19:06,430 --> 00:19:07,420 Also the zero vector. 324 00:19:07,420 --> 00:19:07,960 Good. 325 00:19:07,960 --> 00:19:10,270 And that's a very very important fact. 326 00:19:10,270 --> 00:19:11,630 OK. 327 00:19:11,630 --> 00:19:15,000 How about this? 328 00:19:15,000 --> 00:19:20,450 These -- look at the space of five by five matrices 329 00:19:20,450 --> 00:19:23,280 as a vector space. 330 00:19:23,280 --> 00:19:25,690 So it's actually a twenty-five-dimensional vector 331 00:19:25,690 --> 00:19:26,190 space. 332 00:19:26,190 --> 00:19:28,990 All five by five matrices. 333 00:19:28,990 --> 00:19:31,770 Look at the invertible matrices. 334 00:19:31,770 --> 00:19:35,430 Do they form a subspace? 335 00:19:35,430 --> 00:19:40,430 So I have this five by -- a space of all five by five 336 00:19:40,430 --> 00:19:41,580 matrices. 337 00:19:41,580 --> 00:19:44,180 I can add them, I can multiply by numbers. 338 00:19:44,180 --> 00:19:48,110 But now I narrow down to the invertible ones. 339 00:19:48,110 --> 00:19:50,700 And I ask are they a subspace? 340 00:19:50,700 --> 00:19:54,600 And you -- your answer is -- 341 00:19:54,600 --> 00:19:57,418 quiet, but nevertheless definite, no. 342 00:20:00,130 --> 00:20:00,800 Right? 343 00:20:00,800 --> 00:20:04,040 Because if I add two invertible matrices I have no idea if the 344 00:20:04,040 --> 00:20:06,570 No. answer is invertible. 345 00:20:06,570 --> 00:20:09,340 If I multiply that invertible -- well, 346 00:20:09,340 --> 00:20:11,600 it doesn't even have the zero matrix in it, 347 00:20:11,600 --> 00:20:13,630 it couldn't be a subspace. 348 00:20:13,630 --> 00:20:16,020 I have to be able to multiply by zero -- 349 00:20:16,020 --> 00:20:21,010 and stay in my subspace, and the invertible ones wouldn't work. 350 00:20:21,010 --> 00:20:24,210 Well, the singular ones wouldn't work either. 351 00:20:24,210 --> 00:20:27,540 They have zero -- the zero matrix is in the singular 352 00:20:27,540 --> 00:20:31,720 matrices, but if I add two singular matrices I don't know 353 00:20:31,720 --> 00:20:33,410 if the answer is singular or not. 354 00:20:33,410 --> 00:20:34,340 OK. 355 00:20:34,340 --> 00:20:36,420 So another true-false. 356 00:20:36,420 --> 00:20:36,510 If b squared equals zero then b equals zero. 357 00:20:36,510 --> 00:20:36,580 columns then the question is does Ax=b, 358 00:20:36,580 --> 00:20:37,904 is it always True or false? 359 00:20:37,904 --> 00:20:40,320 If b squared equals zero, true, false? you could get help, 360 00:20:40,320 --> 00:20:40,820 right -- 361 00:20:40,820 --> 00:20:54,590 b squared equals zero, b has to be a square -- square matrix, 362 00:20:54,590 --> 00:20:57,960 so that I can multiply it by itself, 363 00:20:57,960 --> 00:21:01,910 does that imply that B is zero? 364 00:21:06,450 --> 00:21:10,470 Are there matrices whose square could be the zero matrix? 365 00:21:14,720 --> 00:21:15,630 Yes or no? 366 00:21:15,630 --> 00:21:17,830 Yes there are. 367 00:21:17,830 --> 00:21:20,350 There are matrices whose square is the zero matrix. 368 00:21:20,350 --> 00:21:21,725 So this statement is false. 369 00:21:24,280 --> 00:21:26,980 If b squared is zero, we don't know that b is zero. 370 00:21:26,980 --> 00:21:31,290 For example -- the best example is that matrix. 371 00:21:31,290 --> 00:21:35,510 That matrix is a dangerous matrix. 372 00:21:35,510 --> 00:21:42,580 It will come up in later parts of this course as an example 373 00:21:42,580 --> 00:21:44,890 of what can go wrong. 374 00:21:44,890 --> 00:21:47,720 And here is a real simple -- so this -- 375 00:21:47,720 --> 00:21:52,010 so if I square that without doing the multiplication 376 00:21:52,010 --> 00:21:55,940 and finding the matrix b. matrix, 377 00:21:55,940 --> 00:22:04,770 I do get the zero matrix,and it shows -- 378 00:22:04,770 --> 00:22:05,270 OK. 379 00:22:05,270 --> 00:22:16,220 A system of m equations in m unknowns 380 00:22:16,220 --> 00:22:30,620 is solvable for every right-hand side 381 00:22:30,620 --> 00:22:36,490 if the columns are independent. 382 00:22:36,490 --> 00:22:39,870 So can I say that again, I'm -- 383 00:22:39,870 --> 00:22:41,750 I'll write it down then 384 00:22:41,750 --> 00:22:42,440 OK. 385 00:22:42,440 --> 00:22:43,820 for short. 386 00:22:43,820 --> 00:22:47,270 m by m matrix independent 387 00:22:47,270 --> 00:22:52,110 So give me a basis for -- 388 00:22:52,110 --> 00:22:56,250 for the null space of B. 389 00:22:56,250 --> 00:22:57,630 Let's see. 390 00:22:57,630 --> 00:23:03,150 but you could only use each way once, 391 00:23:03,150 --> 00:23:09,370 so you couldn't like use them all the time. 392 00:23:09,370 --> 00:23:11,440 So remember that? 393 00:23:11,440 --> 00:23:17,650 You could -- so you could poll the audience, 394 00:23:17,650 --> 00:23:21,800 and that was a very -- 395 00:23:21,800 --> 00:23:26,630 that was a hundred percent successful way, 396 00:23:26,630 --> 00:23:31,460 so I'll poll the audience on this. 397 00:23:31,460 --> 00:23:34,920 If the other possibility -- 398 00:23:34,920 --> 00:23:40,440 another possibility you could call your friend, right, 399 00:23:40,440 --> 00:23:48,030 or he's your friend until he gives you the wrong answer, 400 00:23:48,030 --> 00:23:49,410 which -- 401 00:23:49,410 --> 00:23:57,010 that turned out subspaces, then A is some multiple of B. 402 00:23:57,010 --> 00:24:01,840 to be very unreliable, you know, you'd 403 00:24:01,840 --> 00:24:05,990 call up your brother or something 404 00:24:05,990 --> 00:24:12,200 and ask him for the capital of whatever, Bosnia. 405 00:24:12,200 --> 00:24:15,940 What does he know, he makes some guess, 406 00:24:15,940 --> 00:24:29,560 So what -- I would just take extreme cases if it was me, 407 00:24:29,560 --> 00:24:32,602 matrix, independent columns, is Ax=b always solvable? 408 00:24:32,602 --> 00:24:33,810 Maybe just hands up for that? 409 00:24:33,810 --> 00:24:34,310 A few. 410 00:24:39,360 --> 00:24:42,420 And who says no? 411 00:24:42,420 --> 00:24:46,730 Oh, gosh, this audience is not reliable. 412 00:24:46,730 --> 00:24:47,230 Fifty fifty. 413 00:24:47,230 --> 00:24:51,070 I guess I'd say, I'd vote yes. 414 00:24:51,070 --> 00:24:54,120 Because independent columns, that 415 00:24:54,120 --> 00:24:57,190 means that the rank is the full size m, 416 00:24:57,190 --> 00:24:59,890 I have a matrix of rank m. 417 00:24:59,890 --> 00:25:01,530 That means it's -- 418 00:25:01,530 --> 00:25:04,710 I mean it's square, so it's an invertible matrix, 419 00:25:04,710 --> 00:25:07,330 and nothing could go wrong. 420 00:25:07,330 --> 00:25:08,090 Yes. 421 00:25:08,090 --> 00:25:12,070 So that's the good case and we always expect it in chapter 422 00:25:12,070 --> 00:25:18,130 two, but of course chapter three is -- 423 00:25:18,130 --> 00:25:24,990 only one of the possibilities. 424 00:25:24,990 --> 00:25:26,360 OK. 425 00:25:26,360 --> 00:25:41,460 Let me move on to another question from an old quiz. 426 00:25:41,460 --> 00:25:42,840 OK. 427 00:25:42,840 --> 00:25:44,210 OK. 428 00:25:44,210 --> 00:25:46,960 Let's see. 429 00:25:46,960 --> 00:26:07,550 I'm going to give you a matrix, but I'm going to give it to you 430 00:26:07,550 --> 00:26:08,920 OK. 431 00:26:08,920 --> 00:26:19,910 as a product of a couple of matrices, 432 00:26:19,910 --> 00:26:32,260 one, one, zero, zero, one, zero, one, zero, one, 433 00:26:32,260 --> 00:26:39,130 times another matrix, one, zero, 434 00:26:39,130 --> 00:26:52,860 I would like to ask you questions about that matrix 435 00:26:52,860 --> 00:27:07,960 minus one, two, zero, one, one, minus one, and all zeroes. 436 00:27:07,960 --> 00:27:09,330 OK. 437 00:27:09,330 --> 00:27:11,070 Let's see, what dimension I in -- 438 00:27:11,070 --> 00:27:17,210 the null space of B is a subspace of R. 439 00:27:17,210 --> 00:27:20,110 What size vectors I looking for here? 440 00:27:20,110 --> 00:27:24,124 Because if we don't know the size, 441 00:27:24,124 --> 00:27:26,040 we aren't going to find it, right? the null -- 442 00:27:26,040 --> 00:27:33,530 this matrix is three by four obviously. 443 00:27:33,530 --> 00:27:37,950 So if we're looking for the null space we're looking for those 444 00:27:37,950 --> 00:27:39,870 vectors x in R^4. 445 00:27:39,870 --> 00:27:40,830 OK. 446 00:27:40,830 --> 00:27:45,490 So the null space of B is certainly a subspace of R^4. 447 00:27:45,490 --> 00:27:48,490 What do you think its dimension is? 448 00:27:48,490 --> 00:27:51,930 Of course once we find the basis we 449 00:27:51,930 --> 00:27:55,010 would know the dimension immediately, but let's 450 00:27:55,010 --> 00:28:00,590 stop first, what's the rank of this matrix B? 451 00:28:00,590 --> 00:28:05,010 Let's see, what -- is that matrix invertible, 452 00:28:05,010 --> 00:28:06,306 that square one there? 453 00:28:10,890 --> 00:28:15,110 Let's say sure, I think it is, yes, 454 00:28:15,110 --> 00:28:17,330 that matrix B looks invertible. 455 00:28:17,330 --> 00:28:21,230 Is that pretty clear? 456 00:28:21,230 --> 00:28:23,090 Yeah. 457 00:28:23,090 --> 00:28:24,180 Yeah. 458 00:28:24,180 --> 00:28:28,450 So I've gone wrong in this course already, 459 00:28:28,450 --> 00:28:33,210 but I'll still hope that that matrix is invertible. 460 00:28:33,210 --> 00:28:36,650 Yeah, yeah, because if I look for a combination of those 461 00:28:36,650 --> 00:28:41,600 three columns -- well, I couldn't use this middle column 462 00:28:41,600 --> 00:28:46,890 because it would have a one and in a position that I -- 463 00:28:46,890 --> 00:28:51,370 column is otherwise all zero, so a combination that gives zero 464 00:28:51,370 --> 00:28:56,180 can't give us that problem, and then the other two are clearly 465 00:28:56,180 --> 00:29:00,090 independent sets -- so that matrix is invertible. 466 00:29:00,090 --> 00:29:03,600 Later we could take a determinant or other things. 467 00:29:03,600 --> 00:29:04,100 OK. 468 00:29:04,100 --> 00:29:06,500 What's the setup? 469 00:29:06,500 --> 00:29:11,850 If I have an invertible matrix, a nice invertible square 470 00:29:11,850 --> 00:29:16,620 matrix, times this guy, times this second factor, 471 00:29:16,620 --> 00:29:18,840 and I'm looking for the null space, 472 00:29:18,840 --> 00:29:20,305 does this have any effect? 473 00:29:22,960 --> 00:29:26,490 Is the null -- so what I'm asking is is the null space 474 00:29:26,490 --> 00:29:33,500 of B the same as the null space of just this part? 475 00:29:33,500 --> 00:29:35,950 I think so. 476 00:29:35,950 --> 00:29:36,820 I think so. 477 00:29:36,820 --> 00:29:41,470 If Bx is zero, then multiplying by that guy 478 00:29:41,470 --> 00:29:42,870 I'll still have zero. 479 00:29:42,870 --> 00:29:47,970 But also if this times some x give zero, 480 00:29:47,970 --> 00:29:51,550 I could always multiply on the left by the inverse of that, 481 00:29:51,550 --> 00:29:55,100 because it is invertible, and I would discover 482 00:29:55,100 --> 00:29:59,040 that this kind of Bx is zero. 483 00:29:59,040 --> 00:30:02,700 You want me to write some of that down -- 484 00:30:02,700 --> 00:30:06,730 if I have a product here, C times -- 485 00:30:06,730 --> 00:30:14,040 times D, say, and if C is invertible, 486 00:30:14,040 --> 00:30:18,660 the null space of CD, well, it will the same 487 00:30:18,660 --> 00:30:20,880 as the null space of D. 488 00:30:20,880 --> 00:30:23,770 If C is invertible. 489 00:30:26,600 --> 00:30:29,570 Multiplying by an invertible matrix on the left 490 00:30:29,570 --> 00:30:32,520 can't change the null space. 491 00:30:32,520 --> 00:30:33,050 OK. 492 00:30:33,050 --> 00:30:35,550 So basically I'm asking you for the null space of this. 493 00:30:35,550 --> 00:30:38,920 so when do I know -- well, I would say suppose the matrix 494 00:30:38,920 --> 00:30:40,540 I don't have to do the multiplication 495 00:30:40,540 --> 00:30:43,210 because I have C is invertible. 496 00:30:43,210 --> 00:30:46,220 That first factor C is invertible. 497 00:30:46,220 --> 00:30:48,550 It's not going to change the null space. 498 00:30:48,550 --> 00:30:49,320 OK. 499 00:30:49,320 --> 00:30:55,430 So can we just write down a basis now for the null space? 500 00:30:55,430 --> 00:30:59,060 So what's the basis for the null space of -- of that? 501 00:30:59,060 --> 00:31:06,080 So basis for the null space I'm looking for the two -- 502 00:31:06,080 --> 00:31:08,330 there are two pivot columns obviously. 503 00:31:08,330 --> 00:31:10,200 It clearly has rank two. 504 00:31:10,200 --> 00:31:15,450 If -- so true or false, if A and B have the same four 505 00:31:15,450 --> 00:31:18,070 I'm looking for the two special solutions. 506 00:31:18,070 --> 00:31:21,070 They'll come from the third and the fourth. 507 00:31:21,070 --> 00:31:22,200 The free variables. 508 00:31:22,200 --> 00:31:25,940 OK. so if the third free variable is a one, 509 00:31:25,940 --> 00:31:29,690 then I think probably I need a minus one there 510 00:31:29,690 --> 00:31:32,320 and a one there, it looks like. 511 00:31:32,320 --> 00:31:36,690 Do you agree that if I then do that multiplication I'll get 512 00:31:36,690 --> 00:31:37,190 zero? 513 00:31:37,190 --> 00:31:40,940 And if I have one in the fourth variable, then 514 00:31:40,940 --> 00:31:46,190 maybe I need a one in the second variable and maybe a minus two 515 00:31:46,190 --> 00:31:47,970 in the third. 516 00:31:47,970 --> 00:31:51,310 So I just reasoned that through and then if I look back I see 517 00:31:51,310 --> 00:31:55,450 sure enough that the free variable part that I sometimes 518 00:31:55,450 --> 00:31:59,410 call F, that up -- that two by two corner, 519 00:31:59,410 --> 00:32:06,250 is sitting here with all its signs reversed. 520 00:32:06,250 --> 00:32:08,720 So that's -- here I'm seeing minus F, 521 00:32:08,720 --> 00:32:13,110 and here I'm seeing the identity in the null space matrix. 522 00:32:13,110 --> 00:32:15,460 OK, so that's the null space. 523 00:32:15,460 --> 00:32:23,980 Another question is solve Bx equal one, zero, one. 524 00:32:23,980 --> 00:32:25,190 OK. 525 00:32:25,190 --> 00:32:34,890 So that's one question, now solve complete solutions. 526 00:32:34,890 --> 00:32:39,090 To Bx equal one, zero, one. 527 00:32:42,360 --> 00:32:45,200 OK. 528 00:32:45,200 --> 00:32:49,020 Yeah, so I guess I'm seeing if I wanted 529 00:32:49,020 --> 00:32:54,790 to get one, zero, one - What's our particular solution? 530 00:32:54,790 --> 00:32:56,460 So I'm looking for a particular solution 531 00:32:56,460 --> 00:32:57,720 and then the null space 532 00:32:57,720 --> 00:32:59,390 part. 533 00:32:59,390 --> 00:33:00,090 OK. 534 00:33:00,090 --> 00:33:05,360 I-- actually the first column of B, 535 00:33:05,360 --> 00:33:08,840 so what's the first column of our matrix B? 536 00:33:08,840 --> 00:33:10,770 It's the vector one, zero, one. 537 00:33:10,770 --> 00:33:14,140 The first column of our matrix agrees 538 00:33:14,140 --> 00:33:15,430 with the right-hand side. 539 00:33:15,430 --> 00:33:18,670 So I guess I'm thinking x particular 540 00:33:18,670 --> 00:33:22,570 plus x null space will be the particular solution, 541 00:33:22,570 --> 00:33:27,814 since the first column of B is exactly right, that's great. 542 00:33:27,814 --> 00:33:29,980 And then I have C times that first null space vector 543 00:33:29,980 --> 00:33:37,881 and D times the other null space vector. 544 00:33:37,881 --> 00:33:38,380 Right? 545 00:33:38,380 --> 00:33:42,700 The two -- the null space part of the solution, 546 00:33:42,700 --> 00:33:48,060 as always has the arbitrary constants, 547 00:33:48,060 --> 00:33:51,380 the particular solution doesn't have any arbitrary constants, 548 00:33:51,380 --> 00:33:54,840 it's one particular solution, and in this case it'll -- 549 00:33:54,840 --> 00:33:56,160 that one would do. 550 00:33:56,160 --> 00:33:57,070 OK. 551 00:33:57,070 --> 00:33:57,750 Fine. 552 00:33:57,750 --> 00:34:03,030 so those are questions taken from old quizzes, 553 00:34:03,030 --> 00:34:09,781 any questions coming to mind? 554 00:34:09,781 --> 00:34:10,280 Yeah. 555 00:34:10,280 --> 00:34:10,780 Q: value. 556 00:34:10,780 --> 00:34:11,300 OK. 557 00:34:11,300 --> 00:34:15,000 Well, so that particular x particular, 558 00:34:15,000 --> 00:34:22,330 it says that let's see, when I multiply by this guy, 559 00:34:22,330 --> 00:34:26,850 I'm going to get the first column of B. 560 00:34:26,850 --> 00:34:30,050 That -- if that's a solution, I multiply B, 561 00:34:30,050 --> 00:34:34,060 B times this x will be the first column of B, 562 00:34:34,060 --> 00:34:39,480 and so I'm saying that the first column of this B agrees with 563 00:34:39,480 --> 00:34:41,300 the right-hand side. 564 00:34:41,300 --> 00:34:47,340 So I'm saying that look at the first column of that matrix B. 565 00:34:47,340 --> 00:34:50,780 If you do the multiplication, it's -- 566 00:34:50,780 --> 00:34:52,659 so what's the first column of that matrix? 567 00:34:52,659 --> 00:34:54,370 Is that how you do that multiplication? 568 00:34:54,370 --> 00:34:56,203 I multiply that matrix by that first column. 569 00:34:56,203 --> 00:34:59,400 And it picks out one, zero, one. 570 00:34:59,400 --> 00:35:11,020 So the first column of B is exactly that. 571 00:35:11,020 --> 00:35:18,650 And therefore a particular solution will be this guy. 572 00:35:18,650 --> 00:35:19,960 So I'll repeat that question. 573 00:35:19,960 --> 00:35:21,150 Yeah. 574 00:35:21,150 --> 00:35:21,650 OK. 575 00:35:21,650 --> 00:35:22,150 Yes. 576 00:35:22,150 --> 00:35:23,170 Q: particular solution. 577 00:35:23,170 --> 00:35:26,320 Any of the solutions can be the particular one 578 00:35:26,320 --> 00:35:28,180 that we pick out. 579 00:35:28,180 --> 00:35:31,940 So like this plus -- plus this would be another particular 580 00:35:31,940 --> 00:35:32,610 OK. 581 00:35:32,610 --> 00:35:33,110 solution. 582 00:35:43,290 --> 00:35:48,260 It would be another solution. 583 00:35:48,260 --> 00:35:53,040 The particular is just telling us only take one. 584 00:35:53,040 --> 00:35:57,370 But it's not telling us which one we have to take. 585 00:35:57,370 --> 00:35:58,850 We take the most convenient one. 586 00:35:58,850 --> 00:36:02,560 I guess in this -- in this problem that was that one. 587 00:36:02,560 --> 00:36:03,480 Good. 588 00:36:03,480 --> 00:36:05,750 Other questions? 589 00:36:05,750 --> 00:36:06,250 Yes. 590 00:36:06,250 --> 00:36:10,550 And this pattern of particular plus 591 00:36:10,550 --> 00:36:15,370 null space, of course, that's going throughout mathematics 592 00:36:15,370 --> 00:36:17,770 of linear systems. 593 00:36:17,770 --> 00:36:20,970 We're really doing mathematics of linear systems here. 594 00:36:20,970 --> 00:36:24,070 Our systems are discrete and they're finite-dimensional -- 595 00:36:24,070 --> 00:36:31,860 and so it's linear algebra, but this particular plus null space 596 00:36:31,860 --> 00:36:37,850 goes -- that doesn't depend on having finite matrices -- 597 00:36:37,850 --> 00:36:42,990 that spreads much -- that spreads everywhere. 598 00:36:42,990 --> 00:36:46,280 OK, I'm going to just like to encourage 599 00:36:46,280 --> 00:36:48,370 you to take problems out of the book, 600 00:36:48,370 --> 00:36:50,250 let me do the same myself. 601 00:36:50,250 --> 00:36:54,160 OK well here's some easy true or falses. 602 00:36:54,160 --> 00:36:58,440 I don't know why the author put these in here. 603 00:36:58,440 --> 00:36:59,530 OK. 604 00:36:59,530 --> 00:37:03,720 If m=n, then the row space equals the column space. 605 00:37:03,720 --> 00:37:07,390 is invertible -- suppose A is an invertible 606 00:37:07,390 --> 00:37:09,790 So these are true or falses. 607 00:37:09,790 --> 00:37:13,230 If m equals n, so that means the matrix is square, 608 00:37:13,230 --> 00:37:17,410 then the row space equals the column space? 609 00:37:17,410 --> 00:37:18,760 False, good. 610 00:37:18,760 --> 00:37:20,260 Good, what is equal there? 611 00:37:20,260 --> 00:37:25,830 What can I say is equal, if M -- well, 612 00:37:25,830 --> 00:37:27,440 yeah. 613 00:37:27,440 --> 00:37:29,610 Yeah it -- so that's definitely false -- 614 00:37:29,610 --> 00:37:31,160 the row space and the column space, 615 00:37:31,160 --> 00:37:37,550 and this matrix is like always a good example to consider. 616 00:37:37,550 --> 00:37:40,530 So there's a square matrix but it's row 617 00:37:40,530 --> 00:37:44,200 space is the multiples of zero, one, 618 00:37:44,200 --> 00:37:48,310 and its column space is the multiples of one, zero. 619 00:37:48,310 --> 00:37:50,240 Very different. 620 00:37:50,240 --> 00:37:51,680 The row space and the column space 621 00:37:51,680 --> 00:37:54,230 are totally different for that matrix. 622 00:37:54,230 --> 00:37:57,730 Now of course if the matrix was symmetric, 623 00:37:57,730 --> 00:38:01,780 well, then clearly the row space equals the column space. 624 00:38:01,780 --> 00:38:02,280 OK. 625 00:38:02,280 --> 00:38:03,930 How about this question? 626 00:38:03,930 --> 00:38:08,910 The matrices A and minus A share the same four subspaces? 627 00:38:11,490 --> 00:38:15,230 Do the matrices A and minus A have the same column space, 628 00:38:15,230 --> 00:38:17,100 do they have the same null space, 629 00:38:17,100 --> 00:38:18,480 do they have the same row 630 00:38:18,480 --> 00:38:19,050 space? 631 00:38:19,050 --> 00:38:20,960 What's the answer on that? 632 00:38:20,960 --> 00:38:31,120 Yes or no. 633 00:38:31,120 --> 00:38:31,620 Yes. 634 00:38:31,620 --> 00:38:32,310 Good. 635 00:38:32,310 --> 00:38:33,100 OK. 636 00:38:33,100 --> 00:38:33,900 How about this? 637 00:38:33,900 --> 00:38:40,020 If A and B have the same four subspaces, 638 00:38:40,020 --> 00:38:41,470 then A is a multiple of B. 639 00:38:41,470 --> 00:38:43,261 If -- suppose those subspaces are the same. 640 00:38:43,261 --> 00:38:44,680 Then is A a multiple of B? 641 00:38:44,680 --> 00:38:45,510 OK. 642 00:38:45,510 --> 00:38:53,000 How how do you answer a question like that? 643 00:38:53,000 --> 00:38:56,500 Of course if you want to answer it yes, then I would -- 644 00:38:56,500 --> 00:38:59,460 then they'd have to think of a reason why. 645 00:38:59,460 --> 00:39:02,050 If you want to answer no way, then you would -- 646 00:39:02,050 --> 00:39:06,410 and I would sort of like first I would try to think no, 647 00:39:06,410 --> 00:39:10,540 I would say can I come up with an example where it isn't true? 648 00:39:10,540 --> 00:39:17,680 Let me repeat the question. 649 00:39:17,680 --> 00:39:24,810 And then write the answer. 650 00:39:24,810 --> 00:39:30,520 matrix, then what -- 651 00:39:30,520 --> 00:39:40,510 suppose it's six by six invertible matrix, 652 00:39:40,510 --> 00:39:59,070 then what's its row space, and its column space is all of R^6, 653 00:39:59,070 --> 00:40:17,620 and the null space, and the null space of A transpose would be 654 00:40:17,620 --> 00:40:21,900 the zero vector. 655 00:40:21,900 --> 00:40:36,170 So every invertible matrix is going to give that answer. 656 00:40:36,170 --> 00:40:49,020 If I have a six by six invertible matrix, 657 00:40:49,020 --> 00:40:51,602 I know what those subspaces are. 658 00:40:51,602 --> 00:40:53,060 Heck, that was back in chapter two, 659 00:40:53,060 --> 00:40:55,720 when I didn't even know what subspaces were. 660 00:40:55,720 --> 00:41:01,890 The row space and column space are both all six-dimensional 661 00:41:01,890 --> 00:41:05,470 space -- the whole space, and the rank is six, 662 00:41:05,470 --> 00:41:10,280 in other words, and the null spaces have zero dimension. 663 00:41:10,280 --> 00:41:11,870 So do you see now the answer? 664 00:41:11,870 --> 00:41:13,060 So A and B could be. 665 00:41:13,060 --> 00:41:16,970 So A and B could be for example any -- so I'm going to say 666 00:41:16,970 --> 00:41:17,470 false. 667 00:41:17,470 --> 00:41:21,070 Because A and B for example -- 668 00:41:21,070 --> 00:41:34,230 So an example: A and B any invertible six by six, six 669 00:41:34,230 --> 00:41:34,730 by six. 670 00:41:34,730 --> 00:41:39,230 So those would have the same four subspaces 671 00:41:39,230 --> 00:41:40,480 but they wouldn't be the same. 672 00:41:40,480 --> 00:41:44,890 Of course th- there should be something about those matrices 673 00:41:44,890 --> 00:41:45,930 that would be the same. 674 00:41:45,930 --> 00:41:50,070 It's sort of a natural problem, so now actually we're 675 00:41:50,070 --> 00:41:52,120 getting to a math question. 676 00:41:52,120 --> 00:41:54,390 The answer is this is not true. 677 00:41:54,390 --> 00:41:59,300 One matrix doesn't have to be a multiple of the other. 678 00:41:59,300 --> 00:42:02,420 But there must be something that's true. 679 00:42:02,420 --> 00:42:07,770 And that would be sort of like a natural question to ask. 680 00:42:07,770 --> 00:42:15,850 If they have the same subspaces, same four subspaces, 681 00:42:15,850 --> 00:42:29,700 then what -- what could you -- 682 00:42:29,700 --> 00:42:32,480 instinct wasn't necessarily right. 683 00:42:32,480 --> 00:42:37,700 But I hope you now see that the correct answer is false. 684 00:42:37,700 --> 00:42:40,550 And then you might think OK, well, they certainly 685 00:42:40,550 --> 00:42:43,530 do have the same rank. 686 00:42:43,530 --> 00:42:51,860 But do -- obviously if they have the same four subspaces, 687 00:42:51,860 --> 00:42:54,460 they have the same rank. 688 00:42:54,460 --> 00:42:58,620 I might say if they have the well, 689 00:42:58,620 --> 00:43:03,820 I could extend that question and think about other possibilities 690 00:43:03,820 --> 00:43:08,500 and finally come up with something that was true. 691 00:43:08,500 --> 00:43:11,620 But I won't press that one. 692 00:43:11,620 --> 00:43:15,270 Let me keep going with practice questions. 693 00:43:15,270 --> 00:43:19,430 And these practice questions are quite appropriate I 694 00:43:19,430 --> 00:43:21,510 think for the exam. 695 00:43:21,510 --> 00:43:22,030 OK. 696 00:43:22,030 --> 00:43:23,070 let's see. 697 00:43:23,070 --> 00:43:29,310 If I exchange two rows of A which subspaces stay the same? 698 00:43:29,310 --> 00:43:32,950 So I'm trying to take out questions 699 00:43:32,950 --> 00:43:38,670 that we can answer without you know we can answer quickly. 700 00:43:38,670 --> 00:43:45,560 If I have a matrix A, and I exchange two of its rows, which 701 00:43:45,560 --> 00:43:47,570 subspaces stay the same? 702 00:43:47,570 --> 00:43:50,560 The row space does stay the same. 703 00:43:50,560 --> 00:43:53,440 And the null space stays the same. 704 00:43:53,440 --> 00:43:54,270 Good. 705 00:43:54,270 --> 00:43:55,200 Good. 706 00:43:55,200 --> 00:43:56,690 Correct. 707 00:43:56,690 --> 00:43:59,170 Column space would be a wrong answer. 708 00:43:59,170 --> 00:44:00,280 OK. 709 00:44:00,280 --> 00:44:03,140 all right, here's a question. 710 00:44:03,140 --> 00:44:05,730 Oh, this leads into the next chapter. 711 00:44:05,730 --> 00:44:08,990 Why can the vector one, two, three not 712 00:44:08,990 --> 00:44:12,280 be a row and also in the null space? 713 00:44:12,280 --> 00:44:15,040 Fitting we close with this question. 714 00:44:15,040 --> 00:44:19,860 Close is -- so V equal this one, two, 715 00:44:19,860 --> 00:44:33,900 three can't be in the null space of a matrix and the row space. 716 00:44:33,900 --> 00:44:40,690 And my question is why not? 717 00:44:40,690 --> 00:44:41,378 Why not? 718 00:44:45,030 --> 00:44:47,950 So this is a question that we can 719 00:44:47,950 --> 00:44:51,360 because it's sort of asked in a straightforward way, 720 00:44:51,360 --> 00:44:55,410 we can figure out an answer. 721 00:44:55,410 --> 00:44:57,460 Well, actually yeah -- 722 00:44:57,460 --> 00:45:00,550 I'll even pin it down, it can't be in the null space -- 723 00:45:00,550 --> 00:45:04,420 and be a row. 724 00:45:04,420 --> 00:45:07,050 I'll even pin it down further. 725 00:45:07,050 --> 00:45:12,550 Ask it to be a row of A. 726 00:45:12,550 --> 00:45:15,000 Why not? 727 00:45:15,000 --> 00:45:18,820 So I'm -- we know the dimensions of these spaces. 728 00:45:18,820 --> 00:45:26,330 But now I'm asking you sort of like the overlap between -- 729 00:45:26,330 --> 00:45:30,320 so the null space and the row space, 730 00:45:30,320 --> 00:45:33,210 those are in the same n-dimensional space. 731 00:45:33,210 --> 00:45:41,410 Those are -- well, those are both subspaces of n-dimensional 732 00:45:41,410 --> 00:45:45,060 space, and I'm basically saying they can't overlap. 733 00:45:45,060 --> 00:45:49,040 I can't have a vector like this, a typical vector, that's 734 00:45:49,040 --> 00:45:53,110 in the null space and it's also a row of the matrix. 735 00:45:53,110 --> 00:45:54,580 Why is that? 736 00:45:54,580 --> 00:45:56,350 So that's a new sort of idea. 737 00:45:56,350 --> 00:45:58,320 Let's just see what it would mean. 738 00:45:58,320 --> 00:46:04,860 I mean that A times this V, why can this A times this 739 00:46:04,860 --> 00:46:07,770 V it can't be zero. 740 00:46:07,770 --> 00:46:12,410 Oh well, if it's zero, so this is -- 741 00:46:12,410 --> 00:46:16,830 I'm getting it into the null space here. 742 00:46:16,830 --> 00:46:20,750 So this is -- now let's put that vector's in the null space, 743 00:46:20,750 --> 00:46:27,600 why can't the first row of a matrix be one, two, three? 744 00:46:27,600 --> 00:46:35,470 I can fill out the matrix as I like. 745 00:46:35,470 --> 00:46:37,010 Why is that impossible? 746 00:46:37,010 --> 00:46:39,310 Well, you're seeing it's impossible, right? 747 00:46:39,310 --> 00:46:44,690 That if that was a row of the matrix and in the null space, 748 00:46:44,690 --> 00:46:48,421 that number would not be zero, it would be fourteen. 749 00:46:48,421 --> 00:46:48,920 Right. 750 00:46:48,920 --> 00:46:52,960 So now we actually are beginning to get a more complete picture 751 00:46:52,960 --> 00:46:55,170 of these four subspaces. 752 00:46:55,170 --> 00:47:00,220 The two that are over in n-dimensional space, 753 00:47:00,220 --> 00:47:04,450 they actually only share the zero vector. 754 00:47:04,450 --> 00:47:08,110 The intersection of the null space and the row space 755 00:47:08,110 --> 00:47:09,310 is only the zero vector. 756 00:47:09,310 --> 00:47:14,970 And in fact the null space is perpendicular to the row space. 757 00:47:14,970 --> 00:47:18,920 That'll be the first topic let's see, 758 00:47:18,920 --> 00:47:23,940 we have a holiday Monday -- 759 00:47:23,940 --> 00:47:29,800 and I'll see you Wednesday with perpendiculars. 760 00:47:29,800 --> 00:47:33,980 And I'll see you Friday. 761 00:47:33,980 --> 00:47:39,710 So good luck on the quiz.