[Script Info]
Title: 
[Events]
Format: Layer, Start, End, Style, Name, MarginL, MarginR, MarginV, Effect, Text
Dialogue: 0,0:00:00.00,0:00:00.52,Default,,0000,0000,0000,,
Dialogue: 0,0:00:00.52,0:00:04.36,Default,,0000,0000,0000,,A couple of videos ago, I made\Nthe statement that the rank of
Dialogue: 0,0:00:04.36,0:00:08.28,Default,,0000,0000,0000,,a matrix A is equal to the\Nrank of its transpose.
Dialogue: 0,0:00:08.28,0:00:09.91,Default,,0000,0000,0000,,And I made a bit of a\Nhand wavy argument.
Dialogue: 0,0:00:09.91,0:00:12.31,Default,,0000,0000,0000,,It was at the end of the\Nvideo, and I was tired.
Dialogue: 0,0:00:12.31,0:00:13.71,Default,,0000,0000,0000,,It was actually the\Nend of the day.
Dialogue: 0,0:00:13.71,0:00:16.81,Default,,0000,0000,0000,,And I thought it'd be worthwhile\Nto maybe flush this
Dialogue: 0,0:00:16.81,0:00:17.30,Default,,0000,0000,0000,,out a little bit.
Dialogue: 0,0:00:17.30,0:00:18.62,Default,,0000,0000,0000,,Because it's an important\Ntake away.
Dialogue: 0,0:00:18.62,0:00:21.55,Default,,0000,0000,0000,,It'll help us understand\Neverything we've learned a
Dialogue: 0,0:00:21.55,0:00:23.20,Default,,0000,0000,0000,,little bit better.
Dialogue: 0,0:00:23.20,0:00:25.56,Default,,0000,0000,0000,,So, let's understand-- I'm\Nactually going to start with
Dialogue: 0,0:00:25.56,0:00:26.83,Default,,0000,0000,0000,,the rank of A transpose.
Dialogue: 0,0:00:26.83,0:00:31.63,Default,,0000,0000,0000,,
Dialogue: 0,0:00:31.63,0:00:37.08,Default,,0000,0000,0000,,The rank of A transpose is equal\Nto the dimension of the
Dialogue: 0,0:00:37.08,0:00:40.02,Default,,0000,0000,0000,,column space of A transpose.
Dialogue: 0,0:00:40.02,0:00:42.69,Default,,0000,0000,0000,,That's the definition\Nof the rank.
Dialogue: 0,0:00:42.69,0:00:46.81,Default,,0000,0000,0000,,The dimension of the column\Nspace of A transpose is the
Dialogue: 0,0:00:46.81,0:00:53.95,Default,,0000,0000,0000,,number of basis vectors for the
Dialogue: 0,0:00:53.95,0:00:55.33,Default,,0000,0000,0000,,column space of A transpose.
Dialogue: 0,0:00:55.33,0:00:56.81,Default,,0000,0000,0000,,That's what dimension is.
Dialogue: 0,0:00:56.81,0:00:59.90,Default,,0000,0000,0000,,For any subspace, you figure out\Nhow many basis vectors you
Dialogue: 0,0:00:59.90,0:01:01.91,Default,,0000,0000,0000,,need in that subspace,\Nand you count them,
Dialogue: 0,0:01:01.91,0:01:02.83,Default,,0000,0000,0000,,and that's your dimension.
Dialogue: 0,0:01:02.83,0:01:07.18,Default,,0000,0000,0000,,So, it's the number of basis\Nvectors for the column space
Dialogue: 0,0:01:07.18,0:01:10.15,Default,,0000,0000,0000,,of A transpose, which is, of\Ncourse, the same thing.
Dialogue: 0,0:01:10.15,0:01:12.53,Default,,0000,0000,0000,,This thing we've seen multiple\Ntimes, is the same thing as
Dialogue: 0,0:01:12.53,0:01:13.78,Default,,0000,0000,0000,,the row space of A.
Dialogue: 0,0:01:13.78,0:01:17.69,Default,,0000,0000,0000,,
Dialogue: 0,0:01:17.69,0:01:17.95,Default,,0000,0000,0000,,Right?
Dialogue: 0,0:01:17.95,0:01:20.22,Default,,0000,0000,0000,,The columns of A transpose\Nare the same thing
Dialogue: 0,0:01:20.22,0:01:21.78,Default,,0000,0000,0000,,as the rows of A.
Dialogue: 0,0:01:21.78,0:01:24.31,Default,,0000,0000,0000,,It's because You switch the\Nrows and the columns.
Dialogue: 0,0:01:24.31,0:01:27.48,Default,,0000,0000,0000,,Now, how can we figure out the\Nnumber of basis vectors we
Dialogue: 0,0:01:27.48,0:01:30.39,Default,,0000,0000,0000,,need for the column space\Nof A transpose, or the
Dialogue: 0,0:01:30.39,0:01:32.04,Default,,0000,0000,0000,,row space of A?
Dialogue: 0,0:01:32.04,0:01:34.16,Default,,0000,0000,0000,,Let's just think about what\Nthe column space of A
Dialogue: 0,0:01:34.16,0:01:36.33,Default,,0000,0000,0000,,transpose is telling us.
Dialogue: 0,0:01:36.33,0:01:38.29,Default,,0000,0000,0000,,So, it's equivalent to--\Nso let's say, let
Dialogue: 0,0:01:38.29,0:01:39.69,Default,,0000,0000,0000,,me draw A like this.
Dialogue: 0,0:01:39.69,0:01:43.40,Default,,0000,0000,0000,,
Dialogue: 0,0:01:43.40,0:01:44.42,Default,,0000,0000,0000,,That's a matrix A.
Dialogue: 0,0:01:44.42,0:01:47.16,Default,,0000,0000,0000,,Let's say it's an\Nm by n matrix.
Dialogue: 0,0:01:47.16,0:01:49.21,Default,,0000,0000,0000,,Let me just write it as a\Nbunch of row vectors.
Dialogue: 0,0:01:49.21,0:01:51.04,Default,,0000,0000,0000,,I could also write it as a bunch\Nof column vectors, but
Dialogue: 0,0:01:51.04,0:01:53.15,Default,,0000,0000,0000,,right now let's stick\Nto the row vectors.
Dialogue: 0,0:01:53.15,0:01:55.42,Default,,0000,0000,0000,,So we have row one.
Dialogue: 0,0:01:55.42,0:01:57.42,Default,,0000,0000,0000,,The transpose of\Ncolumn vectors.
Dialogue: 0,0:01:57.42,0:02:02.46,Default,,0000,0000,0000,,That's row one, and we're going\Nto have row two, and
Dialogue: 0,0:02:02.46,0:02:05.71,Default,,0000,0000,0000,,we're going to go all the\Nway down to row m.
Dialogue: 0,0:02:05.71,0:02:06.01,Default,,0000,0000,0000,,Right?
Dialogue: 0,0:02:06.01,0:02:06.97,Default,,0000,0000,0000,,It's an m by n matrix.
Dialogue: 0,0:02:06.97,0:02:10.29,Default,,0000,0000,0000,,Each of these vectors are\Nmembers of rn, because they're
Dialogue: 0,0:02:10.29,0:02:11.69,Default,,0000,0000,0000,,going to have n entries in them
Dialogue: 0,0:02:11.69,0:02:13.80,Default,,0000,0000,0000,,because we have n columns.
Dialogue: 0,0:02:13.80,0:02:15.73,Default,,0000,0000,0000,,So, that's what A is\Ngoing to look like.
Dialogue: 0,0:02:15.73,0:02:17.05,Default,,0000,0000,0000,,A is going look like that.
Dialogue: 0,0:02:17.05,0:02:20.66,Default,,0000,0000,0000,,And then A transpose, all\Nof these rows are
Dialogue: 0,0:02:20.66,0:02:22.48,Default,,0000,0000,0000,,going to become columns.
Dialogue: 0,0:02:22.48,0:02:27.67,Default,,0000,0000,0000,,A transpose is going to look\Nlike this. r1, r2,
Dialogue: 0,0:02:27.67,0:02:30.89,Default,,0000,0000,0000,,all the way to rm.
Dialogue: 0,0:02:30.89,0:02:33.88,Default,,0000,0000,0000,,And this is of course going\Nto be an n by m matrix.
Dialogue: 0,0:02:33.88,0:02:35.48,Default,,0000,0000,0000,,You swap these out.
Dialogue: 0,0:02:35.48,0:02:38.67,Default,,0000,0000,0000,,So, all these rows are\Ngoing to be columns.
Dialogue: 0,0:02:38.67,0:02:39.65,Default,,0000,0000,0000,,Right?
Dialogue: 0,0:02:39.65,0:02:41.59,Default,,0000,0000,0000,,And, obviously the column\Nspace-- or maybe not so
Dialogue: 0,0:02:41.59,0:02:47.35,Default,,0000,0000,0000,,obviously-- the column space of\NA transpose is equal to the
Dialogue: 0,0:02:47.35,0:02:56.27,Default,,0000,0000,0000,,span of r1, r2, all\Nthe way to rm.
Dialogue: 0,0:02:56.27,0:02:56.90,Default,,0000,0000,0000,,Right?
Dialogue: 0,0:02:56.90,0:02:58.39,Default,,0000,0000,0000,,It's equal to the span\Nof these things.
Dialogue: 0,0:02:58.39,0:03:00.78,Default,,0000,0000,0000,,Or you could equivocally call\Nit, it's equal to the span of
Dialogue: 0,0:03:00.78,0:03:01.47,Default,,0000,0000,0000,,the rows of A.
Dialogue: 0,0:03:01.47,0:03:03.74,Default,,0000,0000,0000,,That's why it's also called\Nthe row space.
Dialogue: 0,0:03:03.74,0:03:12.56,Default,,0000,0000,0000,,This is equal to the span\Nof the rows of A.
Dialogue: 0,0:03:12.56,0:03:14.53,Default,,0000,0000,0000,,These two things\Nare equivalent.
Dialogue: 0,0:03:14.53,0:03:16.10,Default,,0000,0000,0000,,Now, these are the span.
Dialogue: 0,0:03:16.10,0:03:18.26,Default,,0000,0000,0000,,That means this is some subspace\Nthat's all of the
Dialogue: 0,0:03:18.26,0:03:20.70,Default,,0000,0000,0000,,linear combinations of these\Ncolumns, or all of the linear
Dialogue: 0,0:03:20.70,0:03:22.09,Default,,0000,0000,0000,,combinations of these rows.
Dialogue: 0,0:03:22.09,0:03:26.03,Default,,0000,0000,0000,,If we want the basis for it, we\Nwant to find a minimum set
Dialogue: 0,0:03:26.03,0:03:29.28,Default,,0000,0000,0000,,of linearly independent vectors\Nthat we could use to
Dialogue: 0,0:03:29.28,0:03:30.88,Default,,0000,0000,0000,,construct any of\Nthese columns.
Dialogue: 0,0:03:30.88,0:03:34.83,Default,,0000,0000,0000,,Or that we could use construct\Nany of these rows, right here.
Dialogue: 0,0:03:34.83,0:03:37.27,Default,,0000,0000,0000,,Now, what happens when\Nwe put A into
Dialogue: 0,0:03:37.27,0:03:40.07,Default,,0000,0000,0000,,reduced row echelon form?
Dialogue: 0,0:03:40.07,0:03:46.29,Default,,0000,0000,0000,,We do a bunch of row operations\Nto put it into
Dialogue: 0,0:03:46.29,0:03:49.02,Default,,0000,0000,0000,,reduced row echelon form.
Dialogue: 0,0:03:49.02,0:03:49.14,Default,,0000,0000,0000,,Right?
Dialogue: 0,0:03:49.14,0:03:52.15,Default,,0000,0000,0000,,Do a bunch of row operations\Nand you eventually get
Dialogue: 0,0:03:52.15,0:03:53.05,Default,,0000,0000,0000,,something like this.
Dialogue: 0,0:03:53.05,0:03:57.41,Default,,0000,0000,0000,,You'll get the reduced row\Nechelon form of A.
Dialogue: 0,0:03:57.41,0:03:59.61,Default,,0000,0000,0000,,The reduced row echelon form\Nof A is going to look
Dialogue: 0,0:03:59.61,0:04:00.84,Default,,0000,0000,0000,,something like this.
Dialogue: 0,0:04:00.84,0:04:04.18,Default,,0000,0000,0000,,You're going to have some pivot\Nrows, some rows that
Dialogue: 0,0:04:04.18,0:04:05.65,Default,,0000,0000,0000,,have pivot entries.
Dialogue: 0,0:04:05.65,0:04:08.01,Default,,0000,0000,0000,,Let's say that's one of them.
Dialogue: 0,0:04:08.01,0:04:08.98,Default,,0000,0000,0000,,Let's say that's one of them.
Dialogue: 0,0:04:08.98,0:04:11.39,Default,,0000,0000,0000,,This will all have 0's\Nall the way down.
Dialogue: 0,0:04:11.39,0:04:12.77,Default,,0000,0000,0000,,This one will have 0's.
Dialogue: 0,0:04:12.77,0:04:14.76,Default,,0000,0000,0000,,Your pivot entry has to\Nbe the only non-zero
Dialogue: 0,0:04:14.76,0:04:16.18,Default,,0000,0000,0000,,entry in it's column.
Dialogue: 0,0:04:16.18,0:04:18.22,Default,,0000,0000,0000,,And everything to the left\Nof it all has to be 0.
Dialogue: 0,0:04:18.22,0:04:19.79,Default,,0000,0000,0000,,Let's say that this one isn't.
Dialogue: 0,0:04:19.79,0:04:21.36,Default,,0000,0000,0000,,These are some non-zero\Nvalues.
Dialogue: 0,0:04:21.36,0:04:22.60,Default,,0000,0000,0000,,These are 0.
Dialogue: 0,0:04:22.60,0:04:24.69,Default,,0000,0000,0000,,We have another pivot\Nentry over here.
Dialogue: 0,0:04:24.69,0:04:25.34,Default,,0000,0000,0000,,Everything else is 0.
Dialogue: 0,0:04:25.34,0:04:29.35,Default,,0000,0000,0000,,Let's say everything else\Nare non-pivot entries.
Dialogue: 0,0:04:29.35,0:04:32.51,Default,,0000,0000,0000,,So you come here and you have\Na certain number of pivot
Dialogue: 0,0:04:32.51,0:04:35.35,Default,,0000,0000,0000,,rows, or a certain number\Nof pivot entries, right?
Dialogue: 0,0:04:35.35,0:04:37.63,Default,,0000,0000,0000,,And you got there by performing\Nlinear row
Dialogue: 0,0:04:37.63,0:04:38.88,Default,,0000,0000,0000,,operations on these guys.
Dialogue: 0,0:04:38.88,0:04:41.67,Default,,0000,0000,0000,,So those linear row operations--\Nyou know, I take
Dialogue: 0,0:04:41.67,0:04:44.66,Default,,0000,0000,0000,,3 times row two, and I add it\Nto row one, and that's going
Dialogue: 0,0:04:44.66,0:04:45.79,Default,,0000,0000,0000,,to become my new row two.
Dialogue: 0,0:04:45.79,0:04:47.84,Default,,0000,0000,0000,,And you keep doing that and\Nyou get these things here.
Dialogue: 0,0:04:47.84,0:04:49.17,Default,,0000,0000,0000,,So, these things\Nhere are linear
Dialogue: 0,0:04:49.17,0:04:50.89,Default,,0000,0000,0000,,combinations of those guys.
Dialogue: 0,0:04:50.89,0:04:52.83,Default,,0000,0000,0000,,Or another way to do it, you\Ncan reverse those row
Dialogue: 0,0:04:52.83,0:04:53.38,Default,,0000,0000,0000,,operations.
Dialogue: 0,0:04:53.38,0:04:56.17,Default,,0000,0000,0000,,I could start with these\Nguys right here.
Dialogue: 0,0:04:56.17,0:04:58.99,Default,,0000,0000,0000,,And I could just as easily\Nperform the reverse row
Dialogue: 0,0:04:58.99,0:05:00.42,Default,,0000,0000,0000,,operations.
Dialogue: 0,0:05:00.42,0:05:02.47,Default,,0000,0000,0000,,Any linear operation, you can\Nperform the reverse of it.
Dialogue: 0,0:05:02.47,0:05:04.04,Default,,0000,0000,0000,,We've seen that multiple\Ntimes.
Dialogue: 0,0:05:04.04,0:05:09.69,Default,,0000,0000,0000,,You could perform row operations\Nwith these guys to
Dialogue: 0,0:05:09.69,0:05:11.42,Default,,0000,0000,0000,,get all of these guys.
Dialogue: 0,0:05:11.42,0:05:15.07,Default,,0000,0000,0000,,Or another way to view it is,\Nthese vectors here, these row
Dialogue: 0,0:05:15.07,0:05:20.40,Default,,0000,0000,0000,,vectors right here, they span\Nall of these-- or all of these
Dialogue: 0,0:05:20.40,0:05:23.17,Default,,0000,0000,0000,,row vectors can be represented\Nof linear combinations of your
Dialogue: 0,0:05:23.17,0:05:24.19,Default,,0000,0000,0000,,pivot rows right here.
Dialogue: 0,0:05:24.19,0:05:29.28,Default,,0000,0000,0000,,Obviously, your non-pivot rows\Nare going to be all 0's.
Dialogue: 0,0:05:29.28,0:05:31.38,Default,,0000,0000,0000,,And those are useless.
Dialogue: 0,0:05:31.38,0:05:33.67,Default,,0000,0000,0000,,But, your pivot rows, if you\Ntake linear combinations of
Dialogue: 0,0:05:33.67,0:05:37.87,Default,,0000,0000,0000,,them, you can clearly do reverse\Nrow echelon form and
Dialogue: 0,0:05:37.87,0:05:39.19,Default,,0000,0000,0000,,get back to your matrix.
Dialogue: 0,0:05:39.19,0:05:41.28,Default,,0000,0000,0000,,So, all of these guys can\Nbe represented as linear
Dialogue: 0,0:05:41.28,0:05:42.73,Default,,0000,0000,0000,,combinations of them.
Dialogue: 0,0:05:42.73,0:05:47.14,Default,,0000,0000,0000,,And all of these pivot entries\Nare by definition-- well,
Dialogue: 0,0:05:47.14,0:05:48.59,Default,,0000,0000,0000,,almost by definition--\Nthey are linearly
Dialogue: 0,0:05:48.59,0:05:49.90,Default,,0000,0000,0000,,independent, right?
Dialogue: 0,0:05:49.90,0:05:50.97,Default,,0000,0000,0000,,Because I've got a 1 here.
Dialogue: 0,0:05:50.97,0:05:53.32,Default,,0000,0000,0000,,No one else has a 1 there.
Dialogue: 0,0:05:53.32,0:05:55.88,Default,,0000,0000,0000,,So this guy can definitely not\Nbe represented as a linear
Dialogue: 0,0:05:55.88,0:05:57.99,Default,,0000,0000,0000,,combination of the other guy.
Dialogue: 0,0:05:57.99,0:06:00.71,Default,,0000,0000,0000,,So why am I going through\Nthis whole exercise?
Dialogue: 0,0:06:00.71,0:06:02.30,Default,,0000,0000,0000,,Well, we started off\Nsaying we wanted a
Dialogue: 0,0:06:02.30,0:06:05.47,Default,,0000,0000,0000,,basis for the row space.
Dialogue: 0,0:06:05.47,0:06:09.60,Default,,0000,0000,0000,,We wanted some minimum set of\Nlinearly independent vectors
Dialogue: 0,0:06:09.60,0:06:12.61,Default,,0000,0000,0000,,that spans everything that\Nthese guys can span.
Dialogue: 0,0:06:12.61,0:06:14.92,Default,,0000,0000,0000,,Well, if all of these guys can\Nbe represented as linear
Dialogue: 0,0:06:14.92,0:06:17.66,Default,,0000,0000,0000,,combinations of these row\Nvectors in reduced row echelon
Dialogue: 0,0:06:17.66,0:06:23.09,Default,,0000,0000,0000,,form-- or these pivot rows in\Nreduced row echelon form-- and
Dialogue: 0,0:06:23.09,0:06:25.91,Default,,0000,0000,0000,,these guys are all linearly\Nindependent, then they are a
Dialogue: 0,0:06:25.91,0:06:27.98,Default,,0000,0000,0000,,reasonable basis.
Dialogue: 0,0:06:27.98,0:06:30.81,Default,,0000,0000,0000,,So these pivot rows right here,\Nthat's one of them, this
Dialogue: 0,0:06:30.81,0:06:33.75,Default,,0000,0000,0000,,is the second one, this is the\Nthird one, maybe they're the
Dialogue: 0,0:06:33.75,0:06:34.38,Default,,0000,0000,0000,,only three.
Dialogue: 0,0:06:34.38,0:06:36.05,Default,,0000,0000,0000,,This is just my particular\Nexample.
Dialogue: 0,0:06:36.05,0:06:38.72,Default,,0000,0000,0000,,That would be a suitable basis\Nfor the row space.
Dialogue: 0,0:06:38.72,0:06:40.52,Default,,0000,0000,0000,,So let me write this down.
Dialogue: 0,0:06:40.52,0:06:57.48,Default,,0000,0000,0000,,The pivot rows in reduced row\Nechelon form of A are a basis
Dialogue: 0,0:06:57.48,0:07:03.47,Default,,0000,0000,0000,,for the row space of A.
Dialogue: 0,0:07:03.47,0:07:07.18,Default,,0000,0000,0000,,And the row space of A is the\Nsame thing, or the column
Dialogue: 0,0:07:07.18,0:07:08.23,Default,,0000,0000,0000,,space of A transpose.
Dialogue: 0,0:07:08.23,0:07:10.37,Default,,0000,0000,0000,,The row space of A is the\Nsame thing as the
Dialogue: 0,0:07:10.37,0:07:11.49,Default,,0000,0000,0000,,column space of A transpose.
Dialogue: 0,0:07:11.49,0:07:13.15,Default,,0000,0000,0000,,We've see that multiple times.
Dialogue: 0,0:07:13.15,0:07:16.87,Default,,0000,0000,0000,,Now, if we want to know the\Ndimension of your column
Dialogue: 0,0:07:16.87,0:07:20.77,Default,,0000,0000,0000,,space, we just count the number\Nof pivot rows you have.
Dialogue: 0,0:07:20.77,0:07:22.53,Default,,0000,0000,0000,,So you just count the number\Nof pivot rows.
Dialogue: 0,0:07:22.53,0:07:25.74,Default,,0000,0000,0000,,So the dimension of your row\Nspace, which is the same thing
Dialogue: 0,0:07:25.74,0:07:28.36,Default,,0000,0000,0000,,as the column space of A\Ntranspose, is going to be the
Dialogue: 0,0:07:28.36,0:07:32.42,Default,,0000,0000,0000,,number of pivot rows you have\Nin reduced row echelon form.
Dialogue: 0,0:07:32.42,0:07:35.01,Default,,0000,0000,0000,,Or, even simpler, the number\Nof pivot entries you have
Dialogue: 0,0:07:35.01,0:07:37.43,Default,,0000,0000,0000,,because every pivot entry\Nhas a pivot row.
Dialogue: 0,0:07:37.43,0:07:46.76,Default,,0000,0000,0000,,So we can write that the rank of\NA transpose is equal to the
Dialogue: 0,0:07:46.76,0:07:57.18,Default,,0000,0000,0000,,number of pivot entries in\Nreduced row echelon form of A.
Dialogue: 0,0:07:57.18,0:07:57.49,Default,,0000,0000,0000,,Right?
Dialogue: 0,0:07:57.49,0:07:59.95,Default,,0000,0000,0000,,Because every pivot entry\Ncorresponds to a pivot row.
Dialogue: 0,0:07:59.95,0:08:03.84,Default,,0000,0000,0000,,Those pivot rows are a suitable\Nbasis for the entire
Dialogue: 0,0:08:03.84,0:08:06.26,Default,,0000,0000,0000,,row space, because every row\Ncould be made with a linear
Dialogue: 0,0:08:06.26,0:08:07.91,Default,,0000,0000,0000,,combination of these guys.
Dialogue: 0,0:08:07.91,0:08:10.27,Default,,0000,0000,0000,,And since all these can be, then\Nanything that these guys
Dialogue: 0,0:08:10.27,0:08:12.97,Default,,0000,0000,0000,,can construct, these\Nguys can construct.
Dialogue: 0,0:08:12.97,0:08:13.93,Default,,0000,0000,0000,,Fair enough.
Dialogue: 0,0:08:13.93,0:08:16.35,Default,,0000,0000,0000,,Now, what is the rank of A?
Dialogue: 0,0:08:16.35,0:08:18.16,Default,,0000,0000,0000,,This is the rank of A transpose\Nthat we've been
Dialogue: 0,0:08:18.16,0:08:20.44,Default,,0000,0000,0000,,dealing with so far.
Dialogue: 0,0:08:20.44,0:08:30.35,Default,,0000,0000,0000,,The rank of A is equal to\Nthe dimension of the
Dialogue: 0,0:08:30.35,0:08:32.62,Default,,0000,0000,0000,,column space of A.
Dialogue: 0,0:08:32.62,0:08:41.67,Default,,0000,0000,0000,,Or, you could say it's the\Nnumber of vectors in the basis
Dialogue: 0,0:08:41.67,0:08:44.45,Default,,0000,0000,0000,,for the column space of A.
Dialogue: 0,0:08:44.45,0:08:50.91,Default,,0000,0000,0000,,So if we take that same matrix\NA that we used above, and we
Dialogue: 0,0:08:50.91,0:08:55.86,Default,,0000,0000,0000,,instead we write it as a bunch\Nof column vectors, so c1, c2,
Dialogue: 0,0:08:55.86,0:08:57.72,Default,,0000,0000,0000,,all the way to cn.
Dialogue: 0,0:08:57.72,0:09:00.44,Default,,0000,0000,0000,,We have n columns right there.
Dialogue: 0,0:09:00.44,0:09:02.49,Default,,0000,0000,0000,,The column space is essentially\Nthe subspace
Dialogue: 0,0:09:02.49,0:09:05.15,Default,,0000,0000,0000,,that's spanned by all of these\Ncharacters right here, right?
Dialogue: 0,0:09:05.15,0:09:06.79,Default,,0000,0000,0000,,Spanned by each of these\Ncolumn vectors.
Dialogue: 0,0:09:06.79,0:09:13.81,Default,,0000,0000,0000,,So the column space of A is\Nequal to the span of c1, c2,
Dialogue: 0,0:09:13.81,0:09:15.81,Default,,0000,0000,0000,,all the way to cn.
Dialogue: 0,0:09:15.81,0:09:17.41,Default,,0000,0000,0000,,That's the definition of it.
Dialogue: 0,0:09:17.41,0:09:19.28,Default,,0000,0000,0000,,But we want to know the number\Nof basis vectors.
Dialogue: 0,0:09:19.28,0:09:23.02,Default,,0000,0000,0000,,And we've seen before-- we've\Ndone this multiple times--
Dialogue: 0,0:09:23.02,0:09:25.17,Default,,0000,0000,0000,,what suitable basis\Nvectors could be.
Dialogue: 0,0:09:25.17,0:09:28.80,Default,,0000,0000,0000,,If you put this into reduced row\Nechelon form, and you have
Dialogue: 0,0:09:28.80,0:09:33.48,Default,,0000,0000,0000,,some pivot entries and their\Ncorresponding pivot columns,
Dialogue: 0,0:09:33.48,0:09:35.82,Default,,0000,0000,0000,,so some pivot entries with\Ntheir corresponding pivot
Dialogue: 0,0:09:35.82,0:09:37.38,Default,,0000,0000,0000,,columns just like that.
Dialogue: 0,0:09:37.38,0:09:41.54,Default,,0000,0000,0000,,Maybe that's like that, and then\Nmaybe this one isn't one,
Dialogue: 0,0:09:41.54,0:09:42.62,Default,,0000,0000,0000,,and then this one is.
Dialogue: 0,0:09:42.62,0:09:44.21,Default,,0000,0000,0000,,So you have a certain number\Nof pivot columns.
Dialogue: 0,0:09:44.21,0:09:47.04,Default,,0000,0000,0000,,
Dialogue: 0,0:09:47.04,0:09:49.45,Default,,0000,0000,0000,,Let me do it with another\Ncolor right here.
Dialogue: 0,0:09:49.45,0:09:53.19,Default,,0000,0000,0000,,When you put A into reduced row\Nechelon form, we learned
Dialogue: 0,0:09:53.19,0:09:56.66,Default,,0000,0000,0000,,that the basis vectors, or the\Nbasis columns that form a
Dialogue: 0,0:09:56.66,0:09:59.09,Default,,0000,0000,0000,,basis for your column space,\Nare the columns that
Dialogue: 0,0:09:59.09,0:10:02.00,Default,,0000,0000,0000,,correspond to the\Npivot columns.
Dialogue: 0,0:10:02.00,0:10:04.75,Default,,0000,0000,0000,,So the first column here is a\Npivot column, so this guy
Dialogue: 0,0:10:04.75,0:10:05.78,Default,,0000,0000,0000,,could be a basis vector.
Dialogue: 0,0:10:05.78,0:10:08.01,Default,,0000,0000,0000,,The second column is, so this\Nguy could be a pivot vector.
Dialogue: 0,0:10:08.01,0:10:10.72,Default,,0000,0000,0000,,Or maybe the fourth one right\Nhere, so this guy could be a
Dialogue: 0,0:10:10.72,0:10:11.88,Default,,0000,0000,0000,,pivot vector.
Dialogue: 0,0:10:11.88,0:10:15.69,Default,,0000,0000,0000,,So, in general, you just say\Nhey, if you want to count the
Dialogue: 0,0:10:15.69,0:10:17.29,Default,,0000,0000,0000,,number basis vectors-- because\Nwe don't even have to know
Dialogue: 0,0:10:17.29,0:10:18.40,Default,,0000,0000,0000,,what they are to figure\Nout the rank.
Dialogue: 0,0:10:18.40,0:10:20.23,Default,,0000,0000,0000,,We just have to know the\Nnumber they are.
Dialogue: 0,0:10:20.23,0:10:22.96,Default,,0000,0000,0000,,Well you say, well for every\Npivot column here, we have a
Dialogue: 0,0:10:22.96,0:10:24.53,Default,,0000,0000,0000,,basis vector over there.
Dialogue: 0,0:10:24.53,0:10:26.99,Default,,0000,0000,0000,,So we could just count the\Nnumber pivot columns.
Dialogue: 0,0:10:26.99,0:10:29.51,Default,,0000,0000,0000,,But the number of pivot columns\Nis equivalent to just
Dialogue: 0,0:10:29.51,0:10:31.51,Default,,0000,0000,0000,,the number of pivot entries we\Nhave. Because every pivot
Dialogue: 0,0:10:31.51,0:10:33.20,Default,,0000,0000,0000,,entry gets its own column.
Dialogue: 0,0:10:33.20,0:10:42.22,Default,,0000,0000,0000,,So we could say that the rank of\NA is equal to the number of
Dialogue: 0,0:10:42.22,0:10:49.87,Default,,0000,0000,0000,,pivot entries in the reduced\Nrow echelon form of A.
Dialogue: 0,0:10:49.87,0:10:53.00,Default,,0000,0000,0000,,And, as you can see very\Nclearly, that's the exact same
Dialogue: 0,0:10:53.00,0:10:55.94,Default,,0000,0000,0000,,thing that we deduced was\Nequivalent to the rank of A
Dialogue: 0,0:10:55.94,0:10:57.48,Default,,0000,0000,0000,,transpose-- the dimension\Nof the
Dialogue: 0,0:10:57.48,0:10:59.72,Default,,0000,0000,0000,,columns space of A transpose.
Dialogue: 0,0:10:59.72,0:11:02.24,Default,,0000,0000,0000,,Or the dimension of the\Nrow space of A.
Dialogue: 0,0:11:02.24,0:11:04.45,Default,,0000,0000,0000,,So we can now write\Nour conclusion.
Dialogue: 0,0:11:04.45,0:11:11.10,Default,,0000,0000,0000,,The rank of A is definitely the\Nsame thing as the rank of
Dialogue: 0,0:11:11.10,0:11:12.35,Default,,0000,0000,0000,,A transpose.
Dialogue: 0,0:11:12.35,0:11:13.30,Default,,0000,0000,0000,,