[Script Info]
Title: 
[Events]
Format: Layer, Start, End, Style, Name, MarginL, MarginR, MarginV, Effect, Text
Dialogue: 0,0:00:00.49,0:00:03.32,Default,,0000,0000,0000,,I'm going to do one more video\Nwhere we compare old and new
Dialogue: 0,0:00:03.32,0:00:05.32,Default,,0000,0000,0000,,definitions of a projection.
Dialogue: 0,0:00:05.32,0:00:10.56,Default,,0000,0000,0000,,Our old definition of a\Nprojection onto some line, l,
Dialogue: 0,0:00:10.56,0:00:18.38,Default,,0000,0000,0000,,of the vector, x, is the vector\Nin l, or that's a
Dialogue: 0,0:00:18.38,0:00:26.50,Default,,0000,0000,0000,,member of l, such that x minus\Nthat vector, minus the
Dialogue: 0,0:00:26.50,0:00:38.54,Default,,0000,0000,0000,,projection onto l of x,\Nis orthogonal to l.
Dialogue: 0,0:00:38.54,0:00:41.99,Default,,0000,0000,0000,,So the visualization is, if\Nyou have your line l like
Dialogue: 0,0:00:41.99,0:00:46.36,Default,,0000,0000,0000,,this, that is your line\Nl right there.
Dialogue: 0,0:00:46.36,0:00:49.95,Default,,0000,0000,0000,,And then you have some other\Nvector x that we're take the
Dialogue: 0,0:00:49.95,0:00:52.01,Default,,0000,0000,0000,,projection of it on to l.
Dialogue: 0,0:00:52.01,0:00:53.45,Default,,0000,0000,0000,,So that's x.
Dialogue: 0,0:00:53.45,0:00:56.77,Default,,0000,0000,0000,,The projection of x onto l,\Nthis thing right here, is
Dialogue: 0,0:00:56.77,0:01:00.49,Default,,0000,0000,0000,,going to be some vector in l.
Dialogue: 0,0:01:00.49,0:01:03.06,Default,,0000,0000,0000,,Such that when I take the\Ndifference between x and that
Dialogue: 0,0:01:03.06,0:01:05.88,Default,,0000,0000,0000,,vector, it's going to\Nbe orthogonal to l.
Dialogue: 0,0:01:05.88,0:01:07.94,Default,,0000,0000,0000,,So it's going to be\Nsome vector in l.
Dialogue: 0,0:01:07.94,0:01:10.19,Default,,0000,0000,0000,,This was our old definition when\Nwe took the projection
Dialogue: 0,0:01:10.19,0:01:10.99,Default,,0000,0000,0000,,onto a line.
Dialogue: 0,0:01:10.99,0:01:12.46,Default,,0000,0000,0000,,Some vector in l.
Dialogue: 0,0:01:12.46,0:01:13.46,Default,,0000,0000,0000,,Maybe it's there.
Dialogue: 0,0:01:13.46,0:01:16.09,Default,,0000,0000,0000,,And if I take the difference\Nbetween that and that, this
Dialogue: 0,0:01:16.09,0:01:18.09,Default,,0000,0000,0000,,difference vector's going\Nto be orthogonal to
Dialogue: 0,0:01:18.09,0:01:19.34,Default,,0000,0000,0000,,everything in l.
Dialogue: 0,0:01:23.08,0:01:26.11,Default,,0000,0000,0000,,Just like that.
Dialogue: 0,0:01:26.11,0:01:28.70,Default,,0000,0000,0000,,So, this right here would be\Nit's difference vector.
Dialogue: 0,0:01:28.70,0:01:33.65,Default,,0000,0000,0000,,That would be x minus the\Nprojection of x onto l.
Dialogue: 0,0:01:33.65,0:01:36.20,Default,,0000,0000,0000,,And then, of course, this\Nvector right here.
Dialogue: 0,0:01:36.20,0:01:39.95,Default,,0000,0000,0000,,This is the one we\Nwere defining it.
Dialogue: 0,0:01:39.95,0:01:44.35,Default,,0000,0000,0000,,That was the projection\Nonto l of x.
Dialogue: 0,0:01:44.35,0:01:46.27,Default,,0000,0000,0000,,Now, what's a different\Nway that we could
Dialogue: 0,0:01:46.27,0:01:48.64,Default,,0000,0000,0000,,have written this?
Dialogue: 0,0:01:48.64,0:01:51.23,Default,,0000,0000,0000,,We could have written this\Nexact same definition.
Dialogue: 0,0:01:51.23,0:01:56.20,Default,,0000,0000,0000,,We could have said it is the\Nvector in l such that-- so we
Dialogue: 0,0:01:56.20,0:01:58.46,Default,,0000,0000,0000,,could say, let me write\Nit here in purple.
Dialogue: 0,0:01:58.46,0:02:14.01,Default,,0000,0000,0000,,Is the vector v in l such that\Nv-- let me write it this way--
Dialogue: 0,0:02:14.01,0:02:18.22,Default,,0000,0000,0000,,such that x minus v, right?
Dialogue: 0,0:02:18.22,0:02:24.34,Default,,0000,0000,0000,,x minus the projection of l is\Northogonal is equal to w,
Dialogue: 0,0:02:24.34,0:02:36.66,Default,,0000,0000,0000,,which is orthogonal to\Neverything in l.
Dialogue: 0,0:02:36.66,0:02:38.73,Default,,0000,0000,0000,,Being orthogonal to l literally\Nmeans being
Dialogue: 0,0:02:38.73,0:02:41.05,Default,,0000,0000,0000,,orthogonal to every\Nvector in l.
Dialogue: 0,0:02:41.05,0:02:43.18,Default,,0000,0000,0000,,So I just rewrote it a little\Nbit different, instead of just
Dialogue: 0,0:02:43.18,0:02:45.80,Default,,0000,0000,0000,,leaving it as a projection\Nof x onto l.
Dialogue: 0,0:02:45.80,0:02:51.78,Default,,0000,0000,0000,,I said hey, that's some vector,\Nv, in l such that x
Dialogue: 0,0:02:51.78,0:02:58.31,Default,,0000,0000,0000,,minus v is equal to some other\Nvector, w, which is orthogonal
Dialogue: 0,0:02:58.31,0:02:59.66,Default,,0000,0000,0000,,to everything in l.
Dialogue: 0,0:02:59.66,0:03:05.09,Default,,0000,0000,0000,,Or we can rewrite that statement\Nright there as x is
Dialogue: 0,0:03:05.09,0:03:08.25,Default,,0000,0000,0000,,equal to v plus w.
Dialogue: 0,0:03:08.25,0:03:12.67,Default,,0000,0000,0000,,So we can just say that the\Nprojection of x onto l is the
Dialogue: 0,0:03:12.67,0:03:19.47,Default,,0000,0000,0000,,unique vector v in l, such that\Nx is equal to v plus w,
Dialogue: 0,0:03:19.47,0:03:23.59,Default,,0000,0000,0000,,where w is a unique vector-- I\Nmean it is going to be unique
Dialogue: 0,0:03:23.59,0:03:27.46,Default,,0000,0000,0000,,vector-- in the orthogonal\Ncomplement of l.
Dialogue: 0,0:03:27.46,0:03:28.08,Default,,0000,0000,0000,,Right?
Dialogue: 0,0:03:28.08,0:03:31.07,Default,,0000,0000,0000,,This is got to be orthogonal\Nto everything in l.
Dialogue: 0,0:03:31.07,0:03:33.41,Default,,0000,0000,0000,,So that's going to be a member\Nof the orthogonal
Dialogue: 0,0:03:33.41,0:03:38.07,Default,,0000,0000,0000,,complement of l.
Dialogue: 0,0:03:38.07,0:03:40.91,Default,,0000,0000,0000,,So this definition is actually\Ncompletely consistent with our
Dialogue: 0,0:03:40.91,0:03:42.95,Default,,0000,0000,0000,,new subspace definition.
Dialogue: 0,0:03:42.95,0:03:45.82,Default,,0000,0000,0000,,And we could just extend\Nit to arbitrary
Dialogue: 0,0:03:45.82,0:03:47.41,Default,,0000,0000,0000,,subspaces, not just lines.
Dialogue: 0,0:03:47.41,0:03:51.39,Default,,0000,0000,0000,,Let me help you visualize\Nthat.
Dialogue: 0,0:03:51.39,0:03:55.63,Default,,0000,0000,0000,,So let's say we're dealing\Nwith R3 right here.
Dialogue: 0,0:03:59.59,0:04:02.89,Default,,0000,0000,0000,,And I've got some\Nsubspace in R3.
Dialogue: 0,0:04:02.89,0:04:04.95,Default,,0000,0000,0000,,And let's say that subspace\Nhappens to be a plane.
Dialogue: 0,0:04:04.95,0:04:08.14,Default,,0000,0000,0000,,I'm going to make it a plane\Njust so that it becomes clear
Dialogue: 0,0:04:08.14,0:04:10.29,Default,,0000,0000,0000,,that we don't have to take\Nprojections just onto lines.
Dialogue: 0,0:04:10.29,0:04:13.92,Default,,0000,0000,0000,,So this is my subspace\Nv right there.
Dialogue: 0,0:04:13.92,0:04:16.14,Default,,0000,0000,0000,,Let me draw its orthogonal\Ncompliment.
Dialogue: 0,0:04:16.14,0:04:18.50,Default,,0000,0000,0000,,Let's say its orthogonal\Ncomplement looks
Dialogue: 0,0:04:18.50,0:04:20.61,Default,,0000,0000,0000,,something like that.
Dialogue: 0,0:04:20.61,0:04:22.10,Default,,0000,0000,0000,,Let's say it's a line.
Dialogue: 0,0:04:22.10,0:04:24.30,Default,,0000,0000,0000,,And then it goes-- it intersects\Nright there.
Dialogue: 0,0:04:24.30,0:04:25.52,Default,,0000,0000,0000,,Then it goes back.
Dialogue: 0,0:04:25.52,0:04:28.36,Default,,0000,0000,0000,,And, of course, it would have to\Nintersect at the 0 vector.
Dialogue: 0,0:04:28.36,0:04:33.17,Default,,0000,0000,0000,,That's the only place where a\Nsubspace and its orthogonal
Dialogue: 0,0:04:33.17,0:04:34.16,Default,,0000,0000,0000,,complement overlap.
Dialogue: 0,0:04:34.16,0:04:36.71,Default,,0000,0000,0000,,And then it goes behind\Nand you see it again.
Dialogue: 0,0:04:36.71,0:04:38.86,Default,,0000,0000,0000,,Obviously you wouldn't be able\Nto you again because this
Dialogue: 0,0:04:38.86,0:04:40.54,Default,,0000,0000,0000,,plane would extend in\Nevery direction.
Dialogue: 0,0:04:40.54,0:04:41.77,Default,,0000,0000,0000,,But you get the idea.
Dialogue: 0,0:04:41.77,0:04:45.33,Default,,0000,0000,0000,,So this right here\Nis the orthogonal
Dialogue: 0,0:04:45.33,0:04:48.55,Default,,0000,0000,0000,,complement of v, that line.
Dialogue: 0,0:04:48.55,0:04:53.51,Default,,0000,0000,0000,,Now, let's have some other\Narbitrary vector in R3 here.
Dialogue: 0,0:04:53.51,0:04:56.85,Default,,0000,0000,0000,,So let's say I have some vector\Nthat looks like that.
Dialogue: 0,0:04:56.85,0:04:59.18,Default,,0000,0000,0000,,Let's say that that is x.
Dialogue: 0,0:04:59.18,0:05:08.67,Default,,0000,0000,0000,,Now our new definition for the\Nprojection of x onto v is
Dialogue: 0,0:05:08.67,0:05:16.54,Default,,0000,0000,0000,,equal to the unique vector v.
Dialogue: 0,0:05:16.54,0:05:17.39,Default,,0000,0000,0000,,This is a vector v.
Dialogue: 0,0:05:17.39,0:05:18.40,Default,,0000,0000,0000,,That's a subspace v.
Dialogue: 0,0:05:18.40,0:05:27.24,Default,,0000,0000,0000,,The unique vector v, that is a\Nmember of v, such that x is
Dialogue: 0,0:05:27.24,0:05:43.12,Default,,0000,0000,0000,,equal to v plus w, where w\Nis a unique member of the
Dialogue: 0,0:05:43.12,0:05:45.61,Default,,0000,0000,0000,,orthogonal complement of v.
Dialogue: 0,0:05:45.61,0:05:47.12,Default,,0000,0000,0000,,This is our new definition.
Dialogue: 0,0:05:47.12,0:05:51.25,Default,,0000,0000,0000,,So, if we say x is equal to\Nsome member of v and some
Dialogue: 0,0:05:51.25,0:05:53.49,Default,,0000,0000,0000,,member of its orthogonal\Ncomplement-- we can visually
Dialogue: 0,0:05:53.49,0:05:54.71,Default,,0000,0000,0000,,understand that here.
Dialogue: 0,0:05:54.71,0:05:59.28,Default,,0000,0000,0000,,We could say, OK it's going to\Nbe equal to, on v, it'll be
Dialogue: 0,0:05:59.28,0:06:01.47,Default,,0000,0000,0000,,equal to that vector\Nto right there.
Dialogue: 0,0:06:01.47,0:06:04.41,Default,,0000,0000,0000,,And then on v's orthogonal\Ncomplement,
Dialogue: 0,0:06:04.41,0:06:05.67,Default,,0000,0000,0000,,you add that to it.
Dialogue: 0,0:06:05.67,0:06:07.47,Default,,0000,0000,0000,,So, if you were to shift it\Nover, you would get that
Dialogue: 0,0:06:07.47,0:06:09.01,Default,,0000,0000,0000,,vector, just like that.
Dialogue: 0,0:06:09.01,0:06:11.31,Default,,0000,0000,0000,,This right here is v.
Dialogue: 0,0:06:11.31,0:06:12.75,Default,,0000,0000,0000,,That right there is v.
Dialogue: 0,0:06:12.75,0:06:15.95,Default,,0000,0000,0000,,And then this is vector that\Ngoes up like this, out of the
Dialogue: 0,0:06:15.95,0:06:18.34,Default,,0000,0000,0000,,plane, orthogonal to\Nthe plane, is w.
Dialogue: 0,0:06:18.34,0:06:21.94,Default,,0000,0000,0000,,You could see if you take v plus\Nw, you're going to get x.
Dialogue: 0,0:06:21.94,0:06:29.40,Default,,0000,0000,0000,,And you could see that v is\Nthe projection onto the
Dialogue: 0,0:06:29.40,0:06:33.61,Default,,0000,0000,0000,,subspace capital v-- so this\Nis a vector, v-- is the
Dialogue: 0,0:06:33.61,0:06:38.60,Default,,0000,0000,0000,,projection onto the subspace\Ncapital V of the vector x.
Dialogue: 0,0:06:38.60,0:06:40.71,Default,,0000,0000,0000,,So the analogy to a shadow\Nstill holds.
Dialogue: 0,0:06:40.71,0:06:43.64,Default,,0000,0000,0000,,If you imagine kind of a light\Nsource coming straight down
Dialogue: 0,0:06:43.64,0:06:47.06,Default,,0000,0000,0000,,onto our subspace, kind of\Northogonal to our subspace,
Dialogue: 0,0:06:47.06,0:06:50.57,Default,,0000,0000,0000,,the projection onto our subspace\Nis kind of the shadow
Dialogue: 0,0:06:50.57,0:06:51.32,Default,,0000,0000,0000,,of our vector x.
Dialogue: 0,0:06:51.32,0:06:53.88,Default,,0000,0000,0000,,Hopefully that help you\Nvisualize it a little better.
Dialogue: 0,0:06:53.88,0:06:57.43,Default,,0000,0000,0000,,But what we're doing here is\Nwe're going to generalize it.
Dialogue: 0,0:06:57.43,0:06:58.96,Default,,0000,0000,0000,,Earlier in this video\NI showed you a line.
Dialogue: 0,0:06:58.96,0:07:00.31,Default,,0000,0000,0000,,This is a plane.
Dialogue: 0,0:07:00.31,0:07:02.00,Default,,0000,0000,0000,,But we can generalize\Nit to any subspace.
Dialogue: 0,0:07:02.00,0:07:02.92,Default,,0000,0000,0000,,This is in R3.
Dialogue: 0,0:07:02.92,0:07:06.23,Default,,0000,0000,0000,,We can generalize it\Nto Rn, to R100.
Dialogue: 0,0:07:06.23,0:07:08.63,Default,,0000,0000,0000,,And that's really the power\Nof what we're doing here.
Dialogue: 0,0:07:08.63,0:07:11.36,Default,,0000,0000,0000,,It's easy to visualize it here,\Nbut it's not so easy to
Dialogue: 0,0:07:11.36,0:07:13.53,Default,,0000,0000,0000,,visualize it once you get\Nto higher dimensions.
Dialogue: 0,0:07:13.53,0:07:14.90,Default,,0000,0000,0000,,And actually, one other thing.
Dialogue: 0,0:07:14.90,0:07:17.37,Default,,0000,0000,0000,,Let me show that this new\Ndefinition is pretty much
Dialogue: 0,0:07:17.37,0:07:20.13,Default,,0000,0000,0000,,almost identical to exactly\Nwhat we did with lines.
Dialogue: 0,0:07:20.13,0:07:24.95,Default,,0000,0000,0000,,This is identical to saying that\Nthe projection onto the
Dialogue: 0,0:07:24.95,0:07:45.82,Default,,0000,0000,0000,,subspace x is equal to some\Nunique vector in V such that x
Dialogue: 0,0:07:45.82,0:07:58.29,Default,,0000,0000,0000,,minus the projection onto v of\Nx is orthogonal to every
Dialogue: 0,0:07:58.29,0:08:01.90,Default,,0000,0000,0000,,member of V.
Dialogue: 0,0:08:01.90,0:08:05.23,Default,,0000,0000,0000,,Because this statement, right\Nhere, is saying any vector
Dialogue: 0,0:08:05.23,0:08:08.25,Default,,0000,0000,0000,,that's orthogonal to any member\Nof v says that it's a
Dialogue: 0,0:08:08.25,0:08:10.61,Default,,0000,0000,0000,,member of the orthogonal\Ncomplement of v.
Dialogue: 0,0:08:10.61,0:08:13.83,Default,,0000,0000,0000,,So that statement could be\Nwritten as x minus the
Dialogue: 0,0:08:13.83,0:08:20.45,Default,,0000,0000,0000,,projection onto v of x is a\Nmember of v's orthogonal
Dialogue: 0,0:08:20.45,0:08:21.03,Default,,0000,0000,0000,,complement.
Dialogue: 0,0:08:21.03,0:08:23.31,Default,,0000,0000,0000,,Or we could call some w.
Dialogue: 0,0:08:23.31,0:08:27.21,Default,,0000,0000,0000,,So if you call this your v,\Nand if you call this whole
Dialogue: 0,0:08:27.21,0:08:31.64,Default,,0000,0000,0000,,thing your w, you get this exact\Ndefinition right there.
Dialogue: 0,0:08:31.64,0:08:35.82,Default,,0000,0000,0000,,You would have w is equal\Nto x minus v.
Dialogue: 0,0:08:35.82,0:08:39.84,Default,,0000,0000,0000,,And then if you add v to both\Nsides, you get w plus v is
Dialogue: 0,0:08:39.84,0:08:40.95,Default,,0000,0000,0000,,equal to x.
Dialogue: 0,0:08:40.95,0:08:44.25,Default,,0000,0000,0000,,We defined v to be, the\Northogonal-- the
Dialogue: 0,0:08:44.25,0:08:46.25,Default,,0000,0000,0000,,projection of x onto v.
Dialogue: 0,0:08:46.25,0:08:51.05,Default,,0000,0000,0000,,w is a member of our orthogonal\Ncomplement.
Dialogue: 0,0:08:51.05,0:08:53.99,Default,,0000,0000,0000,,And I don't want you\Nto get confused.
Dialogue: 0,0:08:53.99,0:08:57.56,Default,,0000,0000,0000,,The vector v is the orthogonal\Nprojection of our vector x
Dialogue: 0,0:08:57.56,0:09:00.08,Default,,0000,0000,0000,,onto the subspace capital V.
Dialogue: 0,0:09:00.08,0:09:02.17,Default,,0000,0000,0000,,I probably should use different\Nletters instead of
Dialogue: 0,0:09:02.17,0:09:03.74,Default,,0000,0000,0000,,using a lowercase and\Na uppercase v.
Dialogue: 0,0:09:03.74,0:09:05.54,Default,,0000,0000,0000,,It makes the language\Na little difficult.
Dialogue: 0,0:09:05.54,0:09:07.89,Default,,0000,0000,0000,,But I just wanted to give you\Nanother video to give you a
Dialogue: 0,0:09:07.89,0:09:10.04,Default,,0000,0000,0000,,visualization of projections\Nonto
Dialogue: 0,0:09:10.04,0:09:11.65,Default,,0000,0000,0000,,subspaces other than lines.
Dialogue: 0,0:09:11.65,0:09:14.57,Default,,0000,0000,0000,,And to show you that our old\Ndefinition, with just a
Dialogue: 0,0:09:14.57,0:09:17.07,Default,,0000,0000,0000,,projection onto a line which was\Na linear transformation,
Dialogue: 0,0:09:17.07,0:09:19.40,Default,,0000,0000,0000,,is essentially equivalent\Nto this new definition.
Dialogue: 0,0:09:19.40,0:09:23.40,Default,,0000,0000,0000,,On the next video, I'll show\Nyou that this, for any
Dialogue: 0,0:09:23.40,0:09:26.74,Default,,0000,0000,0000,,subspace is, indeed, a linear\Ntransformation.