[Script Info]
Title: 
[Events]
Format: Layer, Start, End, Style, Name, MarginL, MarginR, MarginV, Effect, Text
Dialogue: 0,0:00:00.17,0:00:01.07,Default,,0000,0000,0000,,- [Instructor] In a previous video,
Dialogue: 0,0:00:01.07,0:00:02.24,Default,,0000,0000,0000,,we saw that if you have a system
Dialogue: 0,0:00:02.24,0:00:04.90,Default,,0000,0000,0000,,of three equations with\Nthree unknowns like this,
Dialogue: 0,0:00:04.90,0:00:08.36,Default,,0000,0000,0000,,you can represent it as\Na matrix vector equation,
Dialogue: 0,0:00:08.36,0:00:10.52,Default,,0000,0000,0000,,where this matrix right over here
Dialogue: 0,0:00:10.52,0:00:13.74,Default,,0000,0000,0000,,is a three-by-three matrix.
Dialogue: 0,0:00:13.74,0:00:16.18,Default,,0000,0000,0000,,That is essentially a coefficient matrix.
Dialogue: 0,0:00:16.18,0:00:18.30,Default,,0000,0000,0000,,It has all of the coefficients of the Xs,
Dialogue: 0,0:00:18.30,0:00:21.35,Default,,0000,0000,0000,,the Ys, and the Zs as its various columns.
Dialogue: 0,0:00:21.35,0:00:24.66,Default,,0000,0000,0000,,and then you're going to\Nmultiply that times this vector,
Dialogue: 0,0:00:24.66,0:00:27.07,Default,,0000,0000,0000,,which is really the vector\Nof the unknown variables,
Dialogue: 0,0:00:27.07,0:00:29.57,Default,,0000,0000,0000,,and this is a three-by-one vector.
Dialogue: 0,0:00:29.57,0:00:30.83,Default,,0000,0000,0000,,And then you would result
Dialogue: 0,0:00:30.83,0:00:35.45,Default,,0000,0000,0000,,in this other three-by-one\Nvector, which is a vector
Dialogue: 0,0:00:35.45,0:00:40.05,Default,,0000,0000,0000,,that contains these constant\Nterms right over here.
Dialogue: 0,0:00:40.05,0:00:42.16,Default,,0000,0000,0000,,What we're gonna do in\Nthis video is recognize
Dialogue: 0,0:00:42.16,0:00:44.57,Default,,0000,0000,0000,,that you can generalize this phenomenon.
Dialogue: 0,0:00:44.57,0:00:46.34,Default,,0000,0000,0000,,It's not just true with a system
Dialogue: 0,0:00:46.34,0:00:48.58,Default,,0000,0000,0000,,of three equations with three unknowns.
Dialogue: 0,0:00:48.58,0:00:52.49,Default,,0000,0000,0000,,It actually generalizes to\NN equations with N unknowns.
Dialogue: 0,0:00:52.49,0:00:55.60,Default,,0000,0000,0000,,But just to appreciate that\Nthat is indeed the case,
Dialogue: 0,0:00:55.60,0:00:59.61,Default,,0000,0000,0000,,let us look at a system of two\Nequations with two unknowns.
Dialogue: 0,0:00:59.61,0:01:04.02,Default,,0000,0000,0000,,So let's say you had 2x\Nplus y is equal to nine,
Dialogue: 0,0:01:04.02,0:01:09.02,Default,,0000,0000,0000,,and we had 3x minus y is equal to five.
Dialogue: 0,0:01:09.23,0:01:10.63,Default,,0000,0000,0000,,I encourage you, pause this video
Dialogue: 0,0:01:10.63,0:01:12.61,Default,,0000,0000,0000,,and think about how that\Nwould be represented
Dialogue: 0,0:01:12.61,0:01:15.25,Default,,0000,0000,0000,,as a matrix vector equation.
Dialogue: 0,0:01:16.52,0:01:19.52,Default,,0000,0000,0000,,All right, now let's\Nwork on this together.
Dialogue: 0,0:01:19.52,0:01:23.71,Default,,0000,0000,0000,,So this is a system of two\Nequations with two unknowns.
Dialogue: 0,0:01:23.71,0:01:27.01,Default,,0000,0000,0000,,So the matrix that represents\Nthe coefficients is going
Dialogue: 0,0:01:27.01,0:01:29.39,Default,,0000,0000,0000,,to be a two-by-two matrix
Dialogue: 0,0:01:29.39,0:01:31.68,Default,,0000,0000,0000,,and then that's going to be multiplied
Dialogue: 0,0:01:31.68,0:01:35.20,Default,,0000,0000,0000,,by a vector that represents\Nthe unknown variables.
Dialogue: 0,0:01:35.20,0:01:37.09,Default,,0000,0000,0000,,We have two unknown variables over here.
Dialogue: 0,0:01:37.09,0:01:40.12,Default,,0000,0000,0000,,So this is going to be\Na two-by-one vector,
Dialogue: 0,0:01:40.12,0:01:42.91,Default,,0000,0000,0000,,and then that's going\Nto be equal to a vector
Dialogue: 0,0:01:42.91,0:01:45.52,Default,,0000,0000,0000,,that represents the constants\Non the right-hand side,
Dialogue: 0,0:01:45.52,0:01:46.81,Default,,0000,0000,0000,,and obviously we have two of those.
Dialogue: 0,0:01:46.81,0:01:49.43,Default,,0000,0000,0000,,So that's going to be a\Ntwo-by-one vector as well.
Dialogue: 0,0:01:49.43,0:01:51.87,Default,,0000,0000,0000,,And then we can do exactly what we did
Dialogue: 0,0:01:51.87,0:01:54.41,Default,,0000,0000,0000,,in that previous example\Nin a previous video.
Dialogue: 0,0:01:54.41,0:01:58.85,Default,,0000,0000,0000,,The coefficients on the\NX terms, two and three,
Dialogue: 0,0:01:58.85,0:02:02.57,Default,,0000,0000,0000,,and then we have the\Ncoefficients on the Y terms.
Dialogue: 0,0:02:02.57,0:02:04.53,Default,,0000,0000,0000,,This would be a positive one
Dialogue: 0,0:02:04.53,0:02:06.93,Default,,0000,0000,0000,,and then this would be a negative one.
Dialogue: 0,0:02:06.93,0:02:09.57,Default,,0000,0000,0000,,And then we multiply it times the vector
Dialogue: 0,0:02:09.57,0:02:12.49,Default,,0000,0000,0000,,of the variables, X, Y,
Dialogue: 0,0:02:12.49,0:02:14.70,Default,,0000,0000,0000,,and then last but not least,
Dialogue: 0,0:02:14.70,0:02:19.20,Default,,0000,0000,0000,,you have this nine and this\Nfive over here, nine and five.
Dialogue: 0,0:02:19.20,0:02:22.04,Default,,0000,0000,0000,,And I encourage you and multiply this out.
Dialogue: 0,0:02:22.04,0:02:25.39,Default,,0000,0000,0000,,Multiply this matrix times this vector.
Dialogue: 0,0:02:25.39,0:02:28.56,Default,,0000,0000,0000,,And when you do that and you\Nstill set up this equality,
Dialogue: 0,0:02:28.56,0:02:30.62,Default,,0000,0000,0000,,you're going to see that\Nit essentially turns
Dialogue: 0,0:02:30.62,0:02:32.77,Default,,0000,0000,0000,,into this exact same system
Dialogue: 0,0:02:32.77,0:02:36.01,Default,,0000,0000,0000,,of two equations and two unknowns.
Dialogue: 0,0:02:36.01,0:02:37.65,Default,,0000,0000,0000,,Now, what's interesting about this is
Dialogue: 0,0:02:37.65,0:02:40.73,Default,,0000,0000,0000,,that we see a generalizable form.
Dialogue: 0,0:02:40.73,0:02:43.86,Default,,0000,0000,0000,,In general, you can represent a system
Dialogue: 0,0:02:43.86,0:02:47.89,Default,,0000,0000,0000,,of N equations and N unknowns in the form.
Dialogue: 0,0:02:47.89,0:02:51.74,Default,,0000,0000,0000,,Sum N-by-N matrix A,
Dialogue: 0,0:02:51.74,0:02:53.40,Default,,0000,0000,0000,,N by N,
Dialogue: 0,0:02:53.40,0:02:58.34,Default,,0000,0000,0000,,times sum N-by-one vector X.
Dialogue: 0,0:02:58.34,0:02:59.52,Default,,0000,0000,0000,,This isn't just the variable X.
Dialogue: 0,0:02:59.52,0:03:03.95,Default,,0000,0000,0000,,This is a vector X that\Nhas N dimensions to it.
Dialogue: 0,0:03:03.95,0:03:08.61,Default,,0000,0000,0000,,So times sum N-by-one vector X is going
Dialogue: 0,0:03:08.61,0:03:13.53,Default,,0000,0000,0000,,to be equal to sum N-by-one vector B.
Dialogue: 0,0:03:14.48,0:03:17.36,Default,,0000,0000,0000,,These are the letters that\Npeople use by convention.
Dialogue: 0,0:03:17.36,0:03:19.03,Default,,0000,0000,0000,,This is going to be N by one.
Dialogue: 0,0:03:19.03,0:03:21.89,Default,,0000,0000,0000,,And so you can see in\Nthese different scenarios.
Dialogue: 0,0:03:21.89,0:03:24.85,Default,,0000,0000,0000,,In that first one, this is\Na three-by-three matrix.
Dialogue: 0,0:03:24.85,0:03:26.87,Default,,0000,0000,0000,,We could call that A,
Dialogue: 0,0:03:26.87,0:03:29.58,Default,,0000,0000,0000,,and then we could call this the vector X,
Dialogue: 0,0:03:29.58,0:03:31.76,Default,,0000,0000,0000,,and then we could call this the vector B.
Dialogue: 0,0:03:31.76,0:03:32.85,Default,,0000,0000,0000,,Now in that second scenario,
Dialogue: 0,0:03:32.85,0:03:34.80,Default,,0000,0000,0000,,we could call this the matrix A,
Dialogue: 0,0:03:34.80,0:03:36.85,Default,,0000,0000,0000,,we could call this the vector X,
Dialogue: 0,0:03:36.85,0:03:39.32,Default,,0000,0000,0000,,and then we could call this the vector B,
Dialogue: 0,0:03:39.32,0:03:42.82,Default,,0000,0000,0000,,but we can generalize\Nthat to N dimensions.
Dialogue: 0,0:03:42.82,0:03:44.76,Default,,0000,0000,0000,,And as I talked about\Nin the previous video,
Dialogue: 0,0:03:44.76,0:03:47.88,Default,,0000,0000,0000,,what's interesting about this\Nis you could think about,
Dialogue: 0,0:03:47.88,0:03:49.89,Default,,0000,0000,0000,,for example, in this\Nsystem of two equations
Dialogue: 0,0:03:49.89,0:03:52.30,Default,,0000,0000,0000,,with two unknowns, as all\Nright, I have a line here,
Dialogue: 0,0:03:52.30,0:03:53.49,Default,,0000,0000,0000,,I have a line here,
Dialogue: 0,0:03:53.49,0:03:57.64,Default,,0000,0000,0000,,and X and Y represent the\Nintersection of those lines.
Dialogue: 0,0:03:57.64,0:03:58.85,Default,,0000,0000,0000,,But when you represent it this way,
Dialogue: 0,0:03:58.85,0:04:00.58,Default,,0000,0000,0000,,you could also imagine it as saying,
Dialogue: 0,0:04:00.58,0:04:04.47,Default,,0000,0000,0000,,okay, I have some unknown\Nvector in the coordinate plane
Dialogue: 0,0:04:04.47,0:04:07.46,Default,,0000,0000,0000,,and I'm transforming it using this matrix
Dialogue: 0,0:04:07.46,0:04:10.03,Default,,0000,0000,0000,,to get this vector nine five.
Dialogue: 0,0:04:10.03,0:04:12.41,Default,,0000,0000,0000,,And so I have to figure out what vector,
Dialogue: 0,0:04:12.41,0:04:15.51,Default,,0000,0000,0000,,when transformed in this\Nway, gets us to nine five,
Dialogue: 0,0:04:15.51,0:04:18.16,Default,,0000,0000,0000,,and we also thought about it\Nin the three-by-three case.
Dialogue: 0,0:04:18.16,0:04:20.95,Default,,0000,0000,0000,,What three-dimensional vector,\Nwhen transformed in this way,
Dialogue: 0,0:04:20.95,0:04:23.36,Default,,0000,0000,0000,,gets us to this vector right over here?
Dialogue: 0,0:04:23.36,0:04:24.91,Default,,0000,0000,0000,,And so that hints,
Dialogue: 0,0:04:24.91,0:04:27.41,Default,,0000,0000,0000,,that foreshadows where\Nwe might be able to go.
Dialogue: 0,0:04:27.41,0:04:30.53,Default,,0000,0000,0000,,If we can unwind this\Ntransformation somehow,
Dialogue: 0,0:04:30.53,0:04:33.85,Default,,0000,0000,0000,,then we can figure out what\Nthese unknown vectors are.
Dialogue: 0,0:04:33.85,0:04:37.27,Default,,0000,0000,0000,,And if we can do it in two\Ndimensions or three dimensions,
Dialogue: 0,0:04:37.27,0:04:40.01,Default,,0000,0000,0000,,why not be able to do it in N dimensions?
Dialogue: 0,0:04:40.01,0:04:41.98,Default,,0000,0000,0000,,Which you'll see is actually very useful
Dialogue: 0,0:04:41.98,0:04:43.65,Default,,0000,0000,0000,,if you ever become a data scientist,
Dialogue: 0,0:04:43.65,0:04:45.22,Default,,0000,0000,0000,,or you go into computer science,
Dialogue: 0,0:04:45.22,0:04:47.67,Default,,0000,0000,0000,,or if you go into computer\Ngraphics of some kind.