0:00:00.520,0:00:04.360 A couple of videos ago, I made[br]the statement that the rank of 0:00:04.360,0:00:08.280 a matrix A is equal to the[br]rank of its transpose. 0:00:08.280,0:00:09.910 And I made a bit of a[br]hand wavy argument. 0:00:09.910,0:00:12.310 It was at the end of the[br]video, and I was tired. 0:00:12.310,0:00:13.710 It was actually the[br]end of the day. 0:00:13.710,0:00:16.810 And I thought it'd be worthwhile[br]to maybe flush this 0:00:16.810,0:00:17.300 out a little bit. 0:00:17.300,0:00:18.620 Because it's an important[br]take away. 0:00:18.620,0:00:21.550 It'll help us understand[br]everything we've learned a 0:00:21.550,0:00:23.200 little bit better. 0:00:23.200,0:00:25.560 So, let's understand-- I'm[br]actually going to start with 0:00:25.560,0:00:26.830 the rank of A transpose. 0:00:31.630,0:00:37.080 The rank of A transpose is equal[br]to the dimension of the 0:00:37.080,0:00:40.020 column space of A transpose. 0:00:40.020,0:00:42.690 That's the definition[br]of the rank. 0:00:42.690,0:00:46.810 The dimension of the column[br]space of A transpose is the 0:00:46.810,0:00:53.950 number of basis vectors for the 0:00:53.950,0:00:55.330 column space of A transpose. 0:00:55.330,0:00:56.810 That's what dimension is. 0:00:56.810,0:00:59.900 For any subspace, you figure out[br]how many basis vectors you 0:00:59.900,0:01:01.910 need in that subspace,[br]and you count them, 0:01:01.910,0:01:02.830 and that's your dimension. 0:01:02.830,0:01:07.180 So, it's the number of basis[br]vectors for the column space 0:01:07.180,0:01:10.150 of A transpose, which is, of[br]course, the same thing. 0:01:10.150,0:01:12.530 This thing we've seen multiple[br]times, is the same thing as 0:01:12.530,0:01:13.780 the row space of A. 0:01:17.690,0:01:17.950 Right? 0:01:17.950,0:01:20.220 The columns of A transpose[br]are the same thing 0:01:20.220,0:01:21.785 as the rows of A. 0:01:21.785,0:01:24.310 It's because You switch the[br]rows and the columns. 0:01:24.310,0:01:27.480 Now, how can we figure out the[br]number of basis vectors we 0:01:27.480,0:01:30.390 need for the column space[br]of A transpose, or the 0:01:30.390,0:01:32.040 row space of A? 0:01:32.040,0:01:34.160 Let's just think about what[br]the column space of A 0:01:34.160,0:01:36.330 transpose is telling us. 0:01:36.330,0:01:38.290 So, it's equivalent to--[br]so let's say, let 0:01:38.290,0:01:39.690 me draw A like this. 0:01:43.400,0:01:44.420 That's a matrix A. 0:01:44.420,0:01:47.160 Let's say it's an[br]m by n matrix. 0:01:47.160,0:01:49.210 Let me just write it as a[br]bunch of row vectors. 0:01:49.210,0:01:51.040 I could also write it as a bunch[br]of column vectors, but 0:01:51.040,0:01:53.150 right now let's stick[br]to the row vectors. 0:01:53.150,0:01:55.420 So we have row one. 0:01:55.420,0:01:57.420 The transpose of[br]column vectors. 0:01:57.420,0:02:02.460 That's row one, and we're going[br]to have row two, and 0:02:02.460,0:02:05.710 we're going to go all the[br]way down to row m. 0:02:05.710,0:02:06.010 Right? 0:02:06.010,0:02:06.970 It's an m by n matrix. 0:02:06.970,0:02:10.289 Each of these vectors are[br]members of rn, because they're 0:02:10.289,0:02:11.690 going to have n entries in them 0:02:11.690,0:02:13.800 because we have n columns. 0:02:13.800,0:02:15.730 So, that's what A is[br]going to look like. 0:02:15.730,0:02:17.050 A is going look like that. 0:02:17.050,0:02:20.660 And then A transpose, all[br]of these rows are 0:02:20.660,0:02:22.480 going to become columns. 0:02:22.480,0:02:27.670 A transpose is going to look[br]like this. r1, r2, 0:02:27.670,0:02:30.890 all the way to rm. 0:02:30.890,0:02:33.880 And this is of course going[br]to be an n by m matrix. 0:02:33.880,0:02:35.485 You swap these out. 0:02:35.485,0:02:38.670 So, all these rows are[br]going to be columns. 0:02:38.670,0:02:39.650 Right? 0:02:39.650,0:02:41.590 And, obviously the column[br]space-- or maybe not so 0:02:41.590,0:02:47.350 obviously-- the column space of[br]A transpose is equal to the 0:02:47.350,0:02:56.270 span of r1, r2, all[br]the way to rm. 0:02:56.270,0:02:56.900 Right? 0:02:56.900,0:02:58.390 It's equal to the span[br]of these things. 0:02:58.390,0:03:00.780 Or you could equivocally call[br]it, it's equal to the span of 0:03:00.780,0:03:01.470 the rows of A. 0:03:01.470,0:03:03.740 That's why it's also called[br]the row space. 0:03:03.740,0:03:12.560 This is equal to the span[br]of the rows of A. 0:03:12.560,0:03:14.530 These two things[br]are equivalent. 0:03:14.530,0:03:16.100 Now, these are the span. 0:03:16.100,0:03:18.260 That means this is some subspace[br]that's all of the 0:03:18.260,0:03:20.700 linear combinations of these[br]columns, or all of the linear 0:03:20.700,0:03:22.090 combinations of these rows. 0:03:22.090,0:03:26.030 If we want the basis for it, we[br]want to find a minimum set 0:03:26.030,0:03:29.280 of linearly independent vectors[br]that we could use to 0:03:29.280,0:03:30.880 construct any of[br]these columns. 0:03:30.880,0:03:34.830 Or that we could use construct[br]any of these rows, right here. 0:03:34.830,0:03:37.270 Now, what happens when[br]we put A into 0:03:37.270,0:03:40.070 reduced row echelon form? 0:03:40.070,0:03:46.290 We do a bunch of row operations[br]to put it into 0:03:46.290,0:03:49.020 reduced row echelon form. 0:03:49.020,0:03:49.140 Right? 0:03:49.140,0:03:52.150 Do a bunch of row operations[br]and you eventually get 0:03:52.150,0:03:53.050 something like this. 0:03:53.050,0:03:57.410 You'll get the reduced row[br]echelon form of A. 0:03:57.410,0:03:59.610 The reduced row echelon form[br]of A is going to look 0:03:59.610,0:04:00.840 something like this. 0:04:00.840,0:04:04.180 You're going to have some pivot[br]rows, some rows that 0:04:04.180,0:04:05.650 have pivot entries. 0:04:05.650,0:04:08.010 Let's say that's one of them. 0:04:08.010,0:04:08.980 Let's say that's one of them. 0:04:08.980,0:04:11.390 This will all have 0's[br]all the way down. 0:04:11.390,0:04:12.770 This one will have 0's. 0:04:12.770,0:04:14.760 Your pivot entry has to[br]be the only non-zero 0:04:14.760,0:04:16.180 entry in it's column. 0:04:16.180,0:04:18.220 And everything to the left[br]of it all has to be 0. 0:04:18.220,0:04:19.790 Let's say that this one isn't. 0:04:19.790,0:04:21.360 These are some non-zero[br]values. 0:04:21.360,0:04:22.600 These are 0. 0:04:22.600,0:04:24.690 We have another pivot[br]entry over here. 0:04:24.690,0:04:25.340 Everything else is 0. 0:04:25.340,0:04:29.350 Let's say everything else[br]are non-pivot entries. 0:04:29.350,0:04:32.510 So you come here and you have[br]a certain number of pivot 0:04:32.510,0:04:35.350 rows, or a certain number[br]of pivot entries, right? 0:04:35.350,0:04:37.630 And you got there by performing[br]linear row 0:04:37.630,0:04:38.880 operations on these guys. 0:04:38.880,0:04:41.670 So those linear row operations--[br]you know, I take 0:04:41.670,0:04:44.655 3 times row two, and I add it[br]to row one, and that's going 0:04:44.655,0:04:45.790 to become my new row two. 0:04:45.790,0:04:47.840 And you keep doing that and[br]you get these things here. 0:04:47.840,0:04:49.170 So, these things[br]here are linear 0:04:49.170,0:04:50.890 combinations of those guys. 0:04:50.890,0:04:52.830 Or another way to do it, you[br]can reverse those row 0:04:52.830,0:04:53.380 operations. 0:04:53.380,0:04:56.170 I could start with these[br]guys right here. 0:04:56.170,0:04:58.990 And I could just as easily[br]perform the reverse row 0:04:58.990,0:05:00.420 operations. 0:05:00.420,0:05:02.470 Any linear operation, you can[br]perform the reverse of it. 0:05:02.470,0:05:04.040 We've seen that multiple[br]times. 0:05:04.040,0:05:09.690 You could perform row operations[br]with these guys to 0:05:09.690,0:05:11.420 get all of these guys. 0:05:11.420,0:05:15.070 Or another way to view it is,[br]these vectors here, these row 0:05:15.070,0:05:20.400 vectors right here, they span[br]all of these-- or all of these 0:05:20.400,0:05:23.170 row vectors can be represented[br]of linear combinations of your 0:05:23.170,0:05:24.190 pivot rows right here. 0:05:24.190,0:05:29.280 Obviously, your non-pivot rows[br]are going to be all 0's. 0:05:29.280,0:05:31.380 And those are useless. 0:05:31.380,0:05:33.670 But, your pivot rows, if you[br]take linear combinations of 0:05:33.670,0:05:37.870 them, you can clearly do reverse[br]row echelon form and 0:05:37.870,0:05:39.190 get back to your matrix. 0:05:39.190,0:05:41.280 So, all of these guys can[br]be represented as linear 0:05:41.280,0:05:42.730 combinations of them. 0:05:42.730,0:05:47.140 And all of these pivot entries[br]are by definition-- well, 0:05:47.140,0:05:48.590 almost by definition--[br]they are linearly 0:05:48.590,0:05:49.900 independent, right? 0:05:49.900,0:05:50.970 Because I've got a 1 here. 0:05:50.970,0:05:53.320 No one else has a 1 there. 0:05:53.320,0:05:55.880 So this guy can definitely not[br]be represented as a linear 0:05:55.880,0:05:57.990 combination of the other guy. 0:05:57.990,0:06:00.710 So why am I going through[br]this whole exercise? 0:06:00.710,0:06:02.300 Well, we started off[br]saying we wanted a 0:06:02.300,0:06:05.470 basis for the row space. 0:06:05.470,0:06:09.600 We wanted some minimum set of[br]linearly independent vectors 0:06:09.600,0:06:12.610 that spans everything that[br]these guys can span. 0:06:12.610,0:06:14.920 Well, if all of these guys can[br]be represented as linear 0:06:14.920,0:06:17.660 combinations of these row[br]vectors in reduced row echelon 0:06:17.660,0:06:23.090 form-- or these pivot rows in[br]reduced row echelon form-- and 0:06:23.090,0:06:25.910 these guys are all linearly[br]independent, then they are a 0:06:25.910,0:06:27.980 reasonable basis. 0:06:27.980,0:06:30.810 So these pivot rows right here,[br]that's one of them, this 0:06:30.810,0:06:33.750 is the second one, this is the[br]third one, maybe they're the 0:06:33.750,0:06:34.380 only three. 0:06:34.380,0:06:36.050 This is just my particular[br]example. 0:06:36.050,0:06:38.715 That would be a suitable basis[br]for the row space. 0:06:38.715,0:06:40.520 So let me write this down. 0:06:40.520,0:06:57.480 The pivot rows in reduced row[br]echelon form of A are a basis 0:06:57.480,0:07:03.470 for the row space of A. 0:07:03.470,0:07:07.180 And the row space of A is the[br]same thing, or the column 0:07:07.180,0:07:08.230 space of A transpose. 0:07:08.230,0:07:10.370 The row space of A is the[br]same thing as the 0:07:10.370,0:07:11.490 column space of A transpose. 0:07:11.490,0:07:13.150 We've see that multiple times. 0:07:13.150,0:07:16.870 Now, if we want to know the[br]dimension of your column 0:07:16.870,0:07:20.770 space, we just count the number[br]of pivot rows you have. 0:07:20.770,0:07:22.530 So you just count the number[br]of pivot rows. 0:07:22.530,0:07:25.740 So the dimension of your row[br]space, which is the same thing 0:07:25.740,0:07:28.360 as the column space of A[br]transpose, is going to be the 0:07:28.360,0:07:32.420 number of pivot rows you have[br]in reduced row echelon form. 0:07:32.420,0:07:35.010 Or, even simpler, the number[br]of pivot entries you have 0:07:35.010,0:07:37.430 because every pivot entry[br]has a pivot row. 0:07:37.430,0:07:46.760 So we can write that the rank of[br]A transpose is equal to the 0:07:46.760,0:07:57.180 number of pivot entries in[br]reduced row echelon form of A. 0:07:57.180,0:07:57.490 Right? 0:07:57.490,0:07:59.950 Because every pivot entry[br]corresponds to a pivot row. 0:07:59.950,0:08:03.840 Those pivot rows are a suitable[br]basis for the entire 0:08:03.840,0:08:06.260 row space, because every row[br]could be made with a linear 0:08:06.260,0:08:07.910 combination of these guys. 0:08:07.910,0:08:10.270 And since all these can be, then[br]anything that these guys 0:08:10.270,0:08:12.970 can construct, these[br]guys can construct. 0:08:12.970,0:08:13.930 Fair enough. 0:08:13.930,0:08:16.350 Now, what is the rank of A? 0:08:16.350,0:08:18.160 This is the rank of A transpose[br]that we've been 0:08:18.160,0:08:20.440 dealing with so far. 0:08:20.440,0:08:30.350 The rank of A is equal to[br]the dimension of the 0:08:30.350,0:08:32.620 column space of A. 0:08:32.620,0:08:41.669 Or, you could say it's the[br]number of vectors in the basis 0:08:41.669,0:08:44.450 for the column space of A. 0:08:44.450,0:08:50.910 So if we take that same matrix[br]A that we used above, and we 0:08:50.910,0:08:55.860 instead we write it as a bunch[br]of column vectors, so c1, c2, 0:08:55.860,0:08:57.720 all the way to cn. 0:08:57.720,0:09:00.440 We have n columns right there. 0:09:00.440,0:09:02.490 The column space is essentially[br]the subspace 0:09:02.490,0:09:05.150 that's spanned by all of these[br]characters right here, right? 0:09:05.150,0:09:06.790 Spanned by each of these[br]column vectors. 0:09:06.790,0:09:13.810 So the column space of A is[br]equal to the span of c1, c2, 0:09:13.810,0:09:15.810 all the way to cn. 0:09:15.810,0:09:17.410 That's the definition of it. 0:09:17.410,0:09:19.280 But we want to know the number[br]of basis vectors. 0:09:19.280,0:09:23.020 And we've seen before-- we've[br]done this multiple times-- 0:09:23.020,0:09:25.170 what suitable basis[br]vectors could be. 0:09:25.170,0:09:28.800 If you put this into reduced row[br]echelon form, and you have 0:09:28.800,0:09:33.480 some pivot entries and their[br]corresponding pivot columns, 0:09:33.480,0:09:35.820 so some pivot entries with[br]their corresponding pivot 0:09:35.820,0:09:37.380 columns just like that. 0:09:37.380,0:09:41.540 Maybe that's like that, and then[br]maybe this one isn't one, 0:09:41.540,0:09:42.620 and then this one is. 0:09:42.620,0:09:44.210 So you have a certain number[br]of pivot columns. 0:09:47.040,0:09:49.450 Let me do it with another[br]color right here. 0:09:49.450,0:09:53.190 When you put A into reduced row[br]echelon form, we learned 0:09:53.190,0:09:56.660 that the basis vectors, or the[br]basis columns that form a 0:09:56.660,0:09:59.090 basis for your column space,[br]are the columns that 0:09:59.090,0:10:02.000 correspond to the[br]pivot columns. 0:10:02.000,0:10:04.750 So the first column here is a[br]pivot column, so this guy 0:10:04.750,0:10:05.780 could be a basis vector. 0:10:05.780,0:10:08.010 The second column is, so this[br]guy could be a pivot vector. 0:10:08.010,0:10:10.720 Or maybe the fourth one right[br]here, so this guy could be a 0:10:10.720,0:10:11.880 pivot vector. 0:10:11.880,0:10:15.690 So, in general, you just say[br]hey, if you want to count the 0:10:15.690,0:10:17.290 number basis vectors-- because[br]we don't even have to know 0:10:17.290,0:10:18.400 what they are to figure[br]out the rank. 0:10:18.400,0:10:20.230 We just have to know the[br]number they are. 0:10:20.230,0:10:22.960 Well you say, well for every[br]pivot column here, we have a 0:10:22.960,0:10:24.530 basis vector over there. 0:10:24.530,0:10:26.990 So we could just count the[br]number pivot columns. 0:10:26.990,0:10:29.510 But the number of pivot columns[br]is equivalent to just 0:10:29.510,0:10:31.510 the number of pivot entries we[br]have. Because every pivot 0:10:31.510,0:10:33.200 entry gets its own column. 0:10:33.200,0:10:42.220 So we could say that the rank of[br]A is equal to the number of 0:10:42.220,0:10:49.870 pivot entries in the reduced[br]row echelon form of A. 0:10:49.870,0:10:53.000 And, as you can see very[br]clearly, that's the exact same 0:10:53.000,0:10:55.940 thing that we deduced was[br]equivalent to the rank of A 0:10:55.940,0:10:57.480 transpose-- the dimension[br]of the 0:10:57.480,0:10:59.720 columns space of A transpose. 0:10:59.720,0:11:02.240 Or the dimension of the[br]row space of A. 0:11:02.240,0:11:04.450 So we can now write[br]our conclusion. 0:11:04.450,0:11:11.100 The rank of A is definitely the[br]same thing as the rank of 0:11:11.100,0:11:12.350 A transpose.