WEBVTT

00:00:00.170 --> 00:00:01.070
- [Instructor] In a previous video,

00:00:01.070 --> 00:00:02.240
we saw that if you have a system

00:00:02.240 --> 00:00:04.900
of three equations with
three unknowns like this,

00:00:04.900 --> 00:00:08.360
you can represent it as
a matrix vector equation,

00:00:08.360 --> 00:00:10.520
where this matrix right over here

00:00:10.520 --> 00:00:13.740
is a three-by-three matrix.

00:00:13.740 --> 00:00:16.180
That is essentially a coefficient matrix.

00:00:16.180 --> 00:00:18.300
It has all of the coefficients of the Xs,

00:00:18.300 --> 00:00:21.350
the Ys, and the Zs as its various columns.

00:00:21.350 --> 00:00:24.660
and then you're going to
multiply that times this vector,

00:00:24.660 --> 00:00:27.070
which is really the vector
of the unknown variables,

00:00:27.070 --> 00:00:29.570
and this is a three-by-one vector.

00:00:29.570 --> 00:00:30.830
And then you would result

00:00:30.830 --> 00:00:35.450
in this other three-by-one
vector, which is a vector

00:00:35.450 --> 00:00:40.050
that contains these constant
terms right over here.

00:00:40.050 --> 00:00:42.160
What we're gonna do in
this video is recognize

00:00:42.160 --> 00:00:44.570
that you can generalize this phenomenon.

00:00:44.570 --> 00:00:46.340
It's not just true with a system

00:00:46.340 --> 00:00:48.580
of three equations with three unknowns.

00:00:48.580 --> 00:00:52.490
It actually generalizes to
N equations with N unknowns.

00:00:52.490 --> 00:00:55.600
But just to appreciate that
that is indeed the case,

00:00:55.600 --> 00:00:59.610
let us look at a system of two
equations with two unknowns.

00:00:59.610 --> 00:01:04.020
So let's say you had 2x
plus y is equal to nine,

00:01:04.020 --> 00:01:09.020
and we had 3x minus y is equal to five.

00:01:09.230 --> 00:01:10.630
I encourage you, pause this video

00:01:10.630 --> 00:01:12.610
and think about how that
would be represented

00:01:12.610 --> 00:01:15.253
as a matrix vector equation.

00:01:16.520 --> 00:01:19.520
All right, now let's
work on this together.

00:01:19.520 --> 00:01:23.710
So this is a system of two
equations with two unknowns.

00:01:23.710 --> 00:01:27.010
So the matrix that represents
the coefficients is going

00:01:27.010 --> 00:01:29.390
to be a two-by-two matrix

00:01:29.390 --> 00:01:31.680
and then that's going to be multiplied

00:01:31.680 --> 00:01:35.200
by a vector that represents
the unknown variables.

00:01:35.200 --> 00:01:37.090
We have two unknown variables over here.

00:01:37.090 --> 00:01:40.120
So this is going to be
a two-by-one vector,

00:01:40.120 --> 00:01:42.910
and then that's going
to be equal to a vector

00:01:42.910 --> 00:01:45.520
that represents the constants
on the right-hand side,

00:01:45.520 --> 00:01:46.810
and obviously we have two of those.

00:01:46.810 --> 00:01:49.430
So that's going to be a
two-by-one vector as well.

00:01:49.430 --> 00:01:51.870
And then we can do exactly what we did

00:01:51.870 --> 00:01:54.410
in that previous example
in a previous video.

00:01:54.410 --> 00:01:58.850
The coefficients on the
X terms, two and three,

00:01:58.850 --> 00:02:02.570
and then we have the
coefficients on the Y terms.

00:02:02.570 --> 00:02:04.530
This would be a positive one

00:02:04.530 --> 00:02:06.930
and then this would be a negative one.

00:02:06.930 --> 00:02:09.570
And then we multiply it times the vector

00:02:09.570 --> 00:02:12.490
of the variables, X, Y,

00:02:12.490 --> 00:02:14.700
and then last but not least,

00:02:14.700 --> 00:02:19.200
you have this nine and this
five over here, nine and five.

00:02:19.200 --> 00:02:22.040
And I encourage you and multiply this out.

00:02:22.040 --> 00:02:25.390
Multiply this matrix times this vector.

00:02:25.390 --> 00:02:28.560
And when you do that and you
still set up this equality,

00:02:28.560 --> 00:02:30.620
you're going to see that
it essentially turns

00:02:30.620 --> 00:02:32.770
into this exact same system

00:02:32.770 --> 00:02:36.010
of two equations and two unknowns.

00:02:36.010 --> 00:02:37.650
Now, what's interesting about this is

00:02:37.650 --> 00:02:40.730
that we see a generalizable form.

00:02:40.730 --> 00:02:43.860
In general, you can represent a system

00:02:43.860 --> 00:02:47.890
of N equations and N unknowns in the form.

00:02:47.890 --> 00:02:51.740
Sum N-by-N matrix A,

00:02:51.740 --> 00:02:53.400
N by N,

00:02:53.400 --> 00:02:58.340
times sum N-by-one vector X.

00:02:58.340 --> 00:02:59.523
This isn't just the variable X.

00:02:59.523 --> 00:03:03.950
This is a vector X that
has N dimensions to it.

00:03:03.950 --> 00:03:08.610
So times sum N-by-one vector X is going

00:03:08.610 --> 00:03:13.530
to be equal to sum N-by-one vector B.

00:03:14.480 --> 00:03:17.360
These are the letters that
people use by convention.

00:03:17.360 --> 00:03:19.030
This is going to be N by one.

00:03:19.030 --> 00:03:21.890
And so you can see in
these different scenarios.

00:03:21.890 --> 00:03:24.850
In that first one, this is
a three-by-three matrix.

00:03:24.850 --> 00:03:26.870
We could call that A,

00:03:26.870 --> 00:03:29.580
and then we could call this the vector X,

00:03:29.580 --> 00:03:31.760
and then we could call this the vector B.

00:03:31.760 --> 00:03:32.850
Now in that second scenario,

00:03:32.850 --> 00:03:34.800
we could call this the matrix A,

00:03:34.800 --> 00:03:36.850
we could call this the vector X,

00:03:36.850 --> 00:03:39.320
and then we could call this the vector B,

00:03:39.320 --> 00:03:42.820
but we can generalize
that to N dimensions.

00:03:42.820 --> 00:03:44.760
And as I talked about
in the previous video,

00:03:44.760 --> 00:03:47.880
what's interesting about this
is you could think about,

00:03:47.880 --> 00:03:49.890
for example, in this
system of two equations

00:03:49.890 --> 00:03:52.300
with two unknowns, as all
right, I have a line here,

00:03:52.300 --> 00:03:53.490
I have a line here,

00:03:53.490 --> 00:03:57.640
and X and Y represent the
intersection of those lines.

00:03:57.640 --> 00:03:58.850
But when you represent it this way,

00:03:58.850 --> 00:04:00.580
you could also imagine it as saying,

00:04:00.580 --> 00:04:04.470
okay, I have some unknown
vector in the coordinate plane

00:04:04.470 --> 00:04:07.460
and I'm transforming it using this matrix

00:04:07.460 --> 00:04:10.030
to get this vector nine five.

00:04:10.030 --> 00:04:12.410
And so I have to figure out what vector,

00:04:12.410 --> 00:04:15.510
when transformed in this
way, gets us to nine five,

00:04:15.510 --> 00:04:18.160
and we also thought about it
in the three-by-three case.

00:04:18.160 --> 00:04:20.950
What three-dimensional vector,
when transformed in this way,

00:04:20.950 --> 00:04:23.360
gets us to this vector right over here?

00:04:23.360 --> 00:04:24.910
And so that hints,

00:04:24.910 --> 00:04:27.410
that foreshadows where
we might be able to go.

00:04:27.410 --> 00:04:30.530
If we can unwind this
transformation somehow,

00:04:30.530 --> 00:04:33.850
then we can figure out what
these unknown vectors are.

00:04:33.850 --> 00:04:37.270
And if we can do it in two
dimensions or three dimensions,

00:04:37.270 --> 00:04:40.010
why not be able to do it in N dimensions?

00:04:40.010 --> 00:04:41.980
Which you'll see is actually very useful

00:04:41.980 --> 00:04:43.650
if you ever become a data scientist,

00:04:43.650 --> 00:04:45.220
or you go into computer science,

00:04:45.220 --> 00:04:47.670
or if you go into computer
graphics of some kind.