Derivative as slope of a tangent line

WEBVTT
Kind: captions
Language: en

00:00:00.610 --> 00:00:05.130
We're first exposed to the idea
of a slope of a line early on

00:00:05.130 --> 00:00:07.280
in our algebra careers, but I
figure it never hurts

00:00:07.280 --> 00:00:08.600
to review it a bit.

00:00:08.600 --> 00:00:10.450
So let me draw some axes.

00:00:10.450 --> 00:00:12.800
That is my y-axis.

00:00:12.800 --> 00:00:15.170
Maybe I should call
it my f of x-axis.

00:00:15.170 --> 00:00:17.320
y is equal to f of x.

00:00:17.320 --> 00:00:23.380
Let me draw my x-axis, just
like that, that is my x-axis.

00:00:23.380 --> 00:00:28.970
And let me draw a line, let
me draw a line like this.

00:00:28.970 --> 00:00:31.770
And what we want to do is
remind ourselves, how do we

00:00:31.770 --> 00:00:36.935
find the slope of that line?

00:00:36.935 --> 00:00:42.260
And what we do is, we take
two points on the line,

00:00:42.260 --> 00:00:45.030
so let's say we take
this point, right here.

00:00:45.030 --> 00:00:48.070
Let's say that that is the
point x is equal to a.

00:00:48.070 --> 00:00:49.260
And then what would this be?

00:00:49.260 --> 00:00:53.480
This would be the point f of
a, where the function is

00:00:53.480 --> 00:00:54.440
going to be some line.

00:00:54.440 --> 00:00:59.170
We could write f of x is going
to be equal to mx plus b.

00:00:59.170 --> 00:01:01.200
We don't know what m and
b are, but this is all

00:01:01.200 --> 00:01:02.410
a little bit of review.

00:01:02.410 --> 00:01:04.050
So this is a.

00:01:04.050 --> 00:01:06.880
And then the y-value is what
happens to the function when

00:01:06.880 --> 00:01:09.890
you evaluated it at a, so
that's that point right there.

00:01:09.890 --> 00:01:11.950
And then we could take
another point on this line.

00:01:11.950 --> 00:01:14.760
Let's say we take
point b, right there.

00:01:14.760 --> 00:01:17.110
And then this coordinate
up here is going to be

00:01:17.110 --> 00:01:22.010
the point b, f of b.

00:01:22.010 --> 00:01:22.290
Right?

00:01:22.290 --> 00:01:23.980
Because this is just the
point when you evaluate

00:01:23.980 --> 00:01:24.780
the function at b.

00:01:24.780 --> 00:01:27.360
You put b in here, you're going
to get that point right there.

00:01:27.360 --> 00:01:29.270
So let me just draw a
little line right there.

00:01:29.270 --> 00:01:32.630
So that is f of b, right there.

00:01:32.630 --> 00:01:35.420
Actually, let me make it clear
that this coordinate right

00:01:35.420 --> 00:01:38.780
is the point a, f of a.

00:01:38.780 --> 00:01:42.280
So how do we find the slope
between these 2 points, or more

00:01:42.280 --> 00:01:43.760
generally, of this entire line?

00:01:43.760 --> 00:01:46.890
Because whole the slope
is consistent the

00:01:46.890 --> 00:01:48.350
whole way through it.

00:01:48.350 --> 00:01:50.060
And we know that once we
find the slope, that's

00:01:50.060 --> 00:01:51.610
actually going to be
the value of this m.

00:01:51.610 --> 00:01:54.420
That's all a review of your
algebra, but how do we do it?

00:01:54.420 --> 00:01:56.890
Well, a couple of ways
to think about it.

00:01:56.890 --> 00:02:05.130
Slope is equal to
rise over run.

00:02:05.130 --> 00:02:07.050
You might have seen that when
you first learned algebra.

00:02:07.050 --> 00:02:09.670
Or another way of writing
it, it's change in

00:02:09.670 --> 00:02:13.570
y over change in x.

00:02:13.570 --> 00:02:16.240
So let's figure out what the
change in why over the change

00:02:16.240 --> 00:02:18.810
in x is for this
particular case.

00:02:18.810 --> 00:02:22.230
So the change in y
is equal to what?

00:02:22.230 --> 00:02:25.180
Well, let's just take, you can
take this guy as being the

00:02:25.180 --> 00:02:26.990
first point, or that guy
as being the first point.

00:02:26.990 --> 00:02:29.980
But since this guy has a
larger x and a larger y,

00:02:29.980 --> 00:02:30.870
let's start with him.

00:02:30.870 --> 00:02:33.960
The change in y between that
guy and that guy is this

00:02:33.960 --> 00:02:35.070
distance, right here.

00:02:35.070 --> 00:02:36.800
So let me draw a
little triangle.

00:02:36.800 --> 00:02:39.210
That distance right
there is a change in y.

00:02:39.210 --> 00:02:42.190
Or I could just transfer
it to the y-axis.

00:02:42.190 --> 00:02:44.190
This is the change in y.

00:02:44.190 --> 00:02:46.340
That is your change
in y, that distance.

00:02:46.340 --> 00:02:47.980
So what is that distance?

00:02:47.980 --> 00:02:51.030
It's f of b minus f of a.

00:02:51.030 --> 00:02:57.880
So it equals f of
b minus f of a.

00:02:57.880 --> 00:03:01.990
That is your change in y.

00:03:01.990 --> 00:03:05.480
Now what is your change
in x The slope is change

00:03:05.480 --> 00:03:09.010
in y over change in x.

00:03:09.010 --> 00:03:10.760
So what our change in x?

00:03:10.760 --> 00:03:11.770
What's this distance?

00:03:11.770 --> 00:03:13.490
Remember, we're taking this
to be the first point,

00:03:13.490 --> 00:03:16.370
so we took its y minus
the other point's y.

00:03:16.370 --> 00:03:17.880
So to be consistent, we're
going to have to take this

00:03:17.880 --> 00:03:20.660
point x minus this point x.

00:03:20.660 --> 00:03:23.460
So this point's
x-coordinate is b.

00:03:23.460 --> 00:03:29.130
So it's going to be b minus a.

00:03:29.130 --> 00:03:31.580
And just like that, if you knew
the equation of this line, or

00:03:31.580 --> 00:03:34.610
if you had the coordinates of
these 2 points, you would just

00:03:34.610 --> 00:03:39.460
plug them in right here and
you would get your slope.

00:03:39.460 --> 00:03:40.990
That straightforward.

00:03:40.990 --> 00:03:43.150
And that comes straight out
of your Algebra 1 class.

00:03:43.150 --> 00:03:46.060
And let me just, just to make
sure it's concrete for you, if

00:03:46.060 --> 00:03:55.250
this was the point 2, 3, and
let's say that this, up here,

00:03:55.250 --> 00:04:00.640
was the point 5, 7, then if we
wanted to find the slope of

00:04:00.640 --> 00:04:06.460
this line, we would do 7 minus
3, that would be our change in

00:04:06.460 --> 00:04:10.370
y, this would be 7 and this
would be 3, and then we

00:04:10.370 --> 00:04:14.950
do that over 5 minus 2.

00:04:14.950 --> 00:04:18.460
Because this would be a 5, and
this would be a 2, and so this

00:04:18.460 --> 00:04:19.620
would be your change in x.

00:04:19.620 --> 00:04:20.690
5 minus 2.

00:04:20.690 --> 00:04:24.830
So 7 minus 3 is 4, and 5
minus 2 is 3. so your

00:04:24.830 --> 00:04:30.760
slope would be 4/3.

00:04:30.760 --> 00:04:33.010
Now let's see if we
can generalize this.

00:04:33.010 --> 00:04:35.370
And this is what the new
concept that we're going

00:04:35.370 --> 00:04:38.320
to be learning as we
delve into calculus.

00:04:38.320 --> 00:04:41.970
Let's see if we can generalize
this somehow to a curve.

00:04:41.970 --> 00:04:44.310
So let's say I have a curve.

00:04:44.310 --> 00:04:46.340
We have to have a curve
before we can generalize

00:04:46.340 --> 00:04:47.180
it to a curve.

00:04:47.180 --> 00:04:47.900
Let me scroll down a little.

00:04:47.900 --> 00:04:49.850
Well, actually, I want to
leave this up here, show

00:04:49.850 --> 00:04:51.820
you the similarity.

00:04:51.820 --> 00:04:56.390
Let's say I have, I'll keep
it pretty general right now.

00:04:56.390 --> 00:04:58.290
Let's say I have a curve.

00:04:58.290 --> 00:05:01.080
I'll make it a
familiar-looking curve.

00:05:01.080 --> 00:05:03.390
Let's say it's the curve y is
equal to x squared, which

00:05:03.390 --> 00:05:06.860
looks something like that.

00:05:06.860 --> 00:05:10.010
And I want to find the slope.

00:05:10.010 --> 00:05:13.280
Let's say I want to find
the slope at some point.

00:05:13.280 --> 00:05:15.680
And actually, before even
talking about it, let's even

00:05:15.680 --> 00:05:18.930
think about what it means to
find the slope of a curve.

00:05:18.930 --> 00:05:22.450
Here, the slope was the same
the whole time, right?

00:05:22.450 --> 00:05:24.920
But on a curve your
slope is changing.

00:05:24.920 --> 00:05:27.350
And just to get an intuition
for that means, is, what's

00:05:27.350 --> 00:05:28.240
the slope over here?

00:05:28.240 --> 00:05:31.610
Your slope over here is the
slope of the tangent line.

00:05:31.610 --> 00:05:33.150
The line just
barely touches it.

00:05:33.150 --> 00:05:34.180
That's the slope over there.

00:05:34.180 --> 00:05:35.760
It's a negative slope.

00:05:35.760 --> 00:05:37.790
Then over here, your slope is
still negative, but it's a

00:05:37.790 --> 00:05:39.360
little bit less negative.

00:05:39.360 --> 00:05:40.250
It goes like that.

00:05:40.250 --> 00:05:41.570
I don't know if I did
that, drew that.

00:05:41.570 --> 00:05:43.660
Let me do it in a
different color.

00:05:43.660 --> 00:05:45.790
Let me do it in purple.

00:05:45.790 --> 00:05:48.320
So over here, your slope is
slightly less negative.

00:05:48.320 --> 00:05:51.590
It's a slightly less
downward-sloping line.

00:05:51.590 --> 00:05:56.130
And then when you go over here,
at the 0 point, right here,

00:05:56.130 --> 00:06:00.030
your slope is pretty much flat,
because the horizontal line, y

00:06:00.030 --> 00:06:02.610
equals 0, is tangent
to this curve.

00:06:02.610 --> 00:06:05.330
And then as you go to more
positive x's, then your

00:06:05.330 --> 00:06:08.660
slope starts increasing.

00:06:08.660 --> 00:06:10.340
I'm trying to draw
a tangent line.

00:06:10.340 --> 00:06:14.170
And here it's increasing even
more, it's increased even more.

00:06:14.170 --> 00:06:17.240
So your slope is changing the
entire time, and this is kind

00:06:17.240 --> 00:06:19.930
of the big change that happens
when you go from a

00:06:19.930 --> 00:06:21.530
line to a curve.

00:06:21.530 --> 00:06:24.450
A line, your slope is the
same the entire time.

00:06:24.450 --> 00:06:27.600
You could take any two points
of a line, take the change in y

00:06:27.600 --> 00:06:29.170
over the change in x, and you
get the slope for

00:06:29.170 --> 00:06:30.620
the entire line.

00:06:30.620 --> 00:06:32.980
But as you can see already,
it's going to be a little

00:06:32.980 --> 00:06:35.290
bit more nuanced when
we do it for a curve.

00:06:35.290 --> 00:06:37.570
Because it depends what
point we're talking about.

00:06:37.570 --> 00:06:40.200
We can't just say, what is
the slope for this curve?

00:06:40.200 --> 00:06:43.760
The slope is different at
every point along the curve.

00:06:43.760 --> 00:06:44.120
It changes.

00:06:44.120 --> 00:06:46.500
If we go up here, it's
going to be even steeper.

00:06:46.500 --> 00:06:50.450
It's going to look
something like that.

00:06:50.450 --> 00:06:54.780
So let's try a bit
of an experiment.

00:06:54.780 --> 00:06:57.870
And I know how this experiment
turns out, so it won't

00:06:57.870 --> 00:06:59.450
be too much of a risk.

00:06:59.450 --> 00:07:02.300
Let me draw better than that.

00:07:02.300 --> 00:07:11.800
So that is my y-axis,
and that's my x-axis.

00:07:11.800 --> 00:07:13.690
Let's call this, we can call
this y, or we can call

00:07:13.690 --> 00:07:15.660
this the f of x axis.

00:07:15.660 --> 00:07:16.890
Either way.

00:07:16.890 --> 00:07:20.780
And let me draw my curve again.

00:07:20.780 --> 00:07:24.740
And I'll just draw it in the
positive coordinate, like that.

00:07:24.740 --> 00:07:26.890
That's my curve.

00:07:26.890 --> 00:07:33.400
And what if I want to find
the slope right there?

00:07:33.400 --> 00:07:34.680
What can I do?

00:07:34.680 --> 00:07:38.010
Well, based on our definition
of a slope, we need 2 points

00:07:38.010 --> 00:07:39.380
to find a slope, right?

00:07:39.380 --> 00:07:41.905
Here, I don't know how to
find the slope with 1 point.

00:07:41.905 --> 00:07:49.210
So let's just call this
point right here,

00:07:49.210 --> 00:07:50.240
that's going to be x.

00:07:50.240 --> 00:07:51.500
We're going to be general.

00:07:51.500 --> 00:07:53.120
This is going to
be our point x.

00:07:53.120 --> 00:07:55.300
But to find our slope,
according to our traditional

00:07:55.300 --> 00:07:58.140
algebra 1 definition of a
slope, we need 2 points.

00:07:58.140 --> 00:08:02.100
So let's get another
point in here.

00:08:02.100 --> 00:08:05.950
Let's just take a slightly
larger version of this x.

00:08:05.950 --> 00:08:10.430
So let's say, we want to take,
actually, let's do it even

00:08:10.430 --> 00:08:13.590
further out, just because it's
going to get messy otherwise.

00:08:13.590 --> 00:08:15.680
So let's say we have
this point right here.

00:08:15.680 --> 00:08:18.750
And the difference, it's
just h bigger than x.

00:08:18.750 --> 00:08:20.565
Or actually, instead of saying
h bigger, let's just, well

00:08:20.565 --> 00:08:22.120
let me just say h bigger.

00:08:22.120 --> 00:08:25.630
So this is x plus h.

00:08:25.630 --> 00:08:27.910
That's what that point
is right there.

00:08:27.910 --> 00:08:30.430
So what going to be their
corresponding y-coordinates

00:08:30.430 --> 00:08:31.820
on the curve?

00:08:31.820 --> 00:08:35.030
Well, this is the curve
of y is equal to f of x.

00:08:35.030 --> 00:08:39.910
So this point right here
is going to be f of our

00:08:39.910 --> 00:08:41.590
particular x right here.

00:08:41.590 --> 00:08:43.930
And maybe to show you that I'm
taking a particular x, maybe

00:08:43.930 --> 00:08:45.305
I'll do a little 0 here.

00:08:45.305 --> 00:08:48.400
This is x naught, this
is x naught plus h.

00:08:48.400 --> 00:08:51.430
This is f of x naught.

00:08:51.430 --> 00:08:56.870
And then what is this going
to be up here, this point up

00:08:56.870 --> 00:08:59.140
here, that point up here?

00:08:59.140 --> 00:09:04.430
Its y-coordinate is going to be
f of f of this x-coordinate,

00:09:04.430 --> 00:09:05.630
which I shifted
over a little bit.

00:09:05.630 --> 00:09:08.170
It's right there. f of this
x-coordinate, which is

00:09:08.170 --> 00:09:10.950
f of x naught plus h.

00:09:10.950 --> 00:09:12.310
That's its y-coordinate.

00:09:12.310 --> 00:09:15.840
So what is a slope going to be
between these two points that

00:09:15.840 --> 00:09:17.410
are relatively close
to each other?

00:09:17.410 --> 00:09:18.810
Remember, this isn't
going to be the slope

00:09:18.810 --> 00:09:20.070
just at this point.

00:09:20.070 --> 00:09:23.950
This is the slope of the line
between these two points.

00:09:23.950 --> 00:09:27.080
And if I were to actually draw
it out, it would actually be a

00:09:27.080 --> 00:09:29.680
secant line between,
to the curve.

00:09:29.680 --> 00:09:31.860
So it would intersect the curve
twice, once at this point,

00:09:31.860 --> 00:09:32.450
once at this point.

00:09:32.450 --> 00:09:34.180
You can't see it.

00:09:34.180 --> 00:09:35.510
If I blew it up a little
bit, it would look

00:09:35.510 --> 00:09:36.330
something like this.

00:09:41.360 --> 00:09:47.510
This is our coordinate x naught
f of x naught, and up here

00:09:47.510 --> 00:09:51.805
is our coordinate for this
point, which would be, the

00:09:51.805 --> 00:09:56.080
x-coordinate would be x naught
plus h, and the y-coordinate

00:09:56.080 --> 00:09:59.180
would be f of x naught plus h.

00:09:59.180 --> 00:10:02.135
Just whatever this function is,
we're evaluating it at this

00:10:02.135 --> 00:10:03.770
x-coordinate That's all it is.

00:10:03.770 --> 00:10:05.380
So these are the 2 points.

00:10:05.380 --> 00:10:07.740
So maybe a good start is to
just say, hey, what is the

00:10:07.740 --> 00:10:10.090
slope of this secant line?

00:10:10.090 --> 00:10:12.760
And just like we did in the
previous example, you find the

00:10:12.760 --> 00:10:22.470
change in y, and you divide
that by your change in x.

00:10:22.470 --> 00:10:23.220
Let me draw it here.

00:10:23.220 --> 00:10:26.940
Your change in y would be that
right here, change in y, and

00:10:26.940 --> 00:10:29.840
then your change in x would
be that right there.

00:10:29.840 --> 00:10:33.040
So what is the slope going
to be of the secant line?

00:10:33.040 --> 00:10:37.250
The slope is going to be equal
to, let's start with this

00:10:37.250 --> 00:10:39.440
point up here, just because
it seems to be larger.

00:10:39.440 --> 00:10:43.320
So we want a change in y. so
this value right here, this

00:10:43.320 --> 00:10:46.230
y-value, is f of
x naught plus h.

00:10:46.230 --> 00:10:47.740
I just evaluated
this guy up here.

00:10:47.740 --> 00:10:49.630
Looks like a fancy term,
but all it means is, look.

00:10:49.630 --> 00:10:52.970
The slightly larger x
evaluate its y-coordinate.

00:10:52.970 --> 00:10:57.250
Where the curve is
at that value of x.

00:10:57.250 --> 00:11:00.290
So that is going to be, so the
change in y is going to be

00:11:00.290 --> 00:11:03.400
a f of x naught plus h.

00:11:03.400 --> 00:11:05.880
That's just the
y-coordinate up here.

00:11:05.880 --> 00:11:08.830
Minus this y-coordinate
over here.

00:11:08.830 --> 00:11:13.600
So minus f of x naught.

00:11:13.600 --> 00:11:16.500
So that equals our change in y.

00:11:16.500 --> 00:11:18.543
And you want to divide
that by your change in x.

00:11:21.090 --> 00:11:21.800
So what is this?

00:11:21.800 --> 00:11:23.340
This is the larger x-value.

00:11:23.340 --> 00:11:25.030
We started with this
coordinate, so we start

00:11:25.030 --> 00:11:26.460
with its x-coordinate.

00:11:26.460 --> 00:11:30.450
So it's x naught plus
h, x naught plus h.

00:11:30.450 --> 00:11:32.610
Minus this x-coordinate.

00:11:32.610 --> 00:11:33.970
Well, we just picked
a general number.

00:11:33.970 --> 00:11:36.330
It's x naught.

00:11:36.330 --> 00:11:39.300
So that is over
your change in x.

00:11:39.300 --> 00:11:39.990
Just like that.

00:11:39.990 --> 00:11:41.510
So this is the slope
of the secant line.

00:11:41.510 --> 00:11:43.670
We still haven't answered what
the slope is right at that

00:11:43.670 --> 00:11:47.100
point, but maybe this
will help us get there.

00:11:47.100 --> 00:11:51.250
If we simplify this, so let
me write it down like this.

00:11:51.250 --> 00:11:57.630
The slope of the secant, let
me write that properly.

00:11:57.630 --> 00:12:03.930
The slope of the secant line is
equal to the value of the

00:12:03.930 --> 00:12:08.840
function at this point, f of x
naught plus h, minus the value

00:12:08.840 --> 00:12:12.750
of the function here,
mine f of x naught.

00:12:12.750 --> 00:12:14.370
So that just tells
us the change in y.

00:12:14.370 --> 00:12:18.250
It's the exact same definition
of slope we've always used.

00:12:18.250 --> 00:12:19.890
Over the change in x.

00:12:19.890 --> 00:12:20.920
And we can simplify this.

00:12:20.920 --> 00:12:23.980
We have x naught plus
h minus x naught.

00:12:23.980 --> 00:12:26.450
So x naught minus x naught
cancel out, so you

00:12:26.450 --> 00:12:27.700
have that over h.

00:12:27.700 --> 00:12:32.530
So this is equal to our change
in y over change in x.

00:12:32.530 --> 00:12:33.380
Fair enough.

00:12:33.380 --> 00:12:36.500
But I started off saying, I
want to find the slope of

00:12:36.500 --> 00:12:38.880
the line at that point, at
this point, right here.

00:12:38.880 --> 00:12:40.860
This is the zoomed-out
version of it.

00:12:40.860 --> 00:12:42.260
So what can I do?

00:12:42.260 --> 00:12:45.830
Well, I defined second point
here as just the first

00:12:45.830 --> 00:12:49.370
point plus some h.

00:12:49.370 --> 00:12:52.300
And we have something in our
toolkit called a limit.

00:12:52.300 --> 00:12:55.840
This h is just a
general number.

00:12:55.840 --> 00:12:59.400
It could be 10, it could be 2,
it could be 0.02, it could be 1

00:12:59.400 --> 00:13:01.050
times 10 to the negative 100.

00:13:01.050 --> 00:13:03.690
It could be an arbitrarily
small number.

00:13:03.690 --> 00:13:07.210
So what happens, what would
happen, at least theoretically,

00:13:07.210 --> 00:13:09.620
if I were take the limit
as h approaches 0?

00:13:09.620 --> 00:13:13.360
So, you know, first, maybe h is
this fairly large number over

00:13:13.360 --> 00:13:15.710
here, and then if I take h a
little bit smaller, then I'd be

00:13:15.710 --> 00:13:17.740
finding the slope of
this secant line.

00:13:17.740 --> 00:13:20.300
If I took h to be even a little
bit smaller, I'd be finding the

00:13:20.300 --> 00:13:21.790
slope of that secant line.

00:13:21.790 --> 00:13:23.200
If h is a little bit
smaller, I'd be finding

00:13:23.200 --> 00:13:24.390
the slope of that line.

00:13:24.390 --> 00:13:28.430
So as h approaches 0, I'll be
getting closer and closer to

00:13:28.430 --> 00:13:34.230
finding the slope of the line
right at my point in question.

00:13:34.230 --> 00:13:36.720
Obviously, if h is a large
number, my secant line is going

00:13:36.720 --> 00:13:41.110
to be way off from the slope at
exactly that point right there.

00:13:41.110 --> 00:13:45.430
But if h is 0.0000001, if it's
an infinitesimally small

00:13:45.430 --> 00:13:47.810
number, then I'm going
to get pretty close.

00:13:47.810 --> 00:13:51.250
So what happens if I
take the limit as h

00:13:51.250 --> 00:13:53.530
approaches zero of this?

00:13:53.530 --> 00:14:00.220
So the limit as h approaches
0 of my secant slope.

00:14:00.220 --> 00:14:03.180
Of, let me switch to green.

00:14:03.180 --> 00:14:08.560
f of x naught plus h minus f of
x naught, that was my change in

00:14:08.560 --> 00:14:15.490
y, over h, which is
my change in x.

00:14:15.490 --> 00:14:17.720
And now just to clarify
something, and sometimes you'll

00:14:17.720 --> 00:14:20.810
see it in different calculus
books, sometimes instead of an

00:14:20.810 --> 00:14:22.930
h, they'll write
a delta x here.

00:14:22.930 --> 00:14:25.390
Where this second point would
have been defined as x naught

00:14:25.390 --> 00:14:28.660
plus delta x, and then, this
would have simplified to just

00:14:28.660 --> 00:14:30.685
delta x over there, and we'd
be taking the limit as

00:14:30.685 --> 00:14:31.870
delta x approaches 0.

00:14:31.870 --> 00:14:33.200
The exact same thing.

00:14:33.200 --> 00:14:34.780
h, delta x, doesn't matter.

00:14:34.780 --> 00:14:37.630
We're taking h as the
difference between one x point

00:14:37.630 --> 00:14:39.710
and then the higher x point,
and then we're just going to

00:14:39.710 --> 00:14:40.980
take the limit as that
approaches zero.

00:14:40.980 --> 00:14:44.010
We could have called that
delta x just as easily.

00:14:44.010 --> 00:14:52.050
But I'm going to call this
thing, which equals the slope

00:14:52.050 --> 00:14:54.260
of the tangent line, and it
does equal the slope of the

00:14:54.260 --> 00:14:58.720
tangent line, I'm going to call
this the derivative of f.

00:14:58.720 --> 00:15:00.105
Let me write that down.

00:15:05.580 --> 00:15:10.650
And I'm going to say that this
is equal to f prime of x.

00:15:10.650 --> 00:15:12.680
And this is going to
be another function.

00:15:12.680 --> 00:15:16.320
Because remember, the slope
changes at every x-value.

00:15:16.320 --> 00:15:18.640
No matter what x-value
you pick, the slope is

00:15:18.640 --> 00:15:19.260
going to be different.

00:15:19.260 --> 00:15:20.880
Doesn't have to be, but the
way I drew this curve,

00:15:20.880 --> 00:15:21.500
it is different.

00:15:21.500 --> 00:15:22.950
It can be different.

00:15:22.950 --> 00:15:27.070
So now, you give me an x-value
in here, I'll apply this

00:15:27.070 --> 00:15:30.090
formula over here, and
then I can tell you the

00:15:30.090 --> 00:15:31.810
slope at that point.

00:15:31.810 --> 00:15:34.180
And it all seems very
confusing and maybe

00:15:34.180 --> 00:15:35.810
abstract at this point.

00:15:35.810 --> 00:15:38.030
In the next video, I'll
actually do an example of

00:15:38.030 --> 00:15:40.300
calculating a slope, and
it'll make it everything a

00:15:40.300 --> 00:15:42.400
little bit more concrete.


The derivative of f(x)=x^2 for any x

WEBVTT
Kind: captions
Language: en

00:00:00.580 --> 00:00:04.900
In the last video, we found the
slope at a particular point of

00:00:04.900 --> 00:00:07.035
the curve y is equal
to x squared.

00:00:07.035 --> 00:00:09.700
But let's see if we can
generalize this and come up

00:00:09.700 --> 00:00:12.780
with a formula that finds us
the slope at any point of the

00:00:12.780 --> 00:00:15.350
curve y is equal to x squared.

00:00:15.350 --> 00:00:18.000
So let me redraw
my function here.

00:00:18.000 --> 00:00:21.720
It never hurts to
have a nice drawing.

00:00:21.720 --> 00:00:24.530
So that is my y-axis.

00:00:24.530 --> 00:00:28.420
That is my x-axis right there.

00:00:28.420 --> 00:00:29.580
My x-axis.

00:00:29.580 --> 00:00:30.670
Let me draw my curve.

00:00:30.670 --> 00:00:32.900
It looks something like that.

00:00:32.900 --> 00:00:34.210
You've seen that
multiple times.

00:00:34.210 --> 00:00:37.970
This is y is equal
to x squared.

00:00:37.970 --> 00:00:39.740
So let's be very
general right now.

00:00:39.740 --> 00:00:42.600
Remember, if we want to find--
let me just write the

00:00:42.600 --> 00:00:44.600
definition of our derivative.

00:00:44.600 --> 00:00:48.450
So if we have some point right
here-- let's call that x.

00:00:48.450 --> 00:00:49.630
So we want to be very general.

00:00:49.630 --> 00:00:52.470
We want to find the
slope at the point x.

00:00:52.470 --> 00:00:55.190
We want to find a function
where you give me an x

00:00:55.190 --> 00:00:57.010
and I'll tell you the
slope at that point.

00:00:57.010 --> 00:00:59.800
We're going to call
that f prime of x.

00:00:59.800 --> 00:01:06.210
That's going to be the
derivative of f of x.

00:01:06.210 --> 00:01:09.830
But all it does is, look, f of
x, you give-- it's a function

00:01:09.830 --> 00:01:12.670
that you give it an x, and it
tells you the value of that.

00:01:12.670 --> 00:01:14.810
And we draw the curve here.

00:01:14.810 --> 00:01:19.470
With f of x, you give that same
x but it's not going to tell

00:01:19.470 --> 00:01:21.050
you the value of the curve.

00:01:21.050 --> 00:01:22.760
It's not going to say,
oh, this is your f of x.

00:01:22.760 --> 00:01:24.855
It's going to give you the
value of the slope of

00:01:24.855 --> 00:01:26.280
the curve at that point.

00:01:26.280 --> 00:01:28.790
So f of x, if you put it into
that function, it should tell

00:01:28.790 --> 00:01:31.540
you, oh, the slope at that
point, is equal to-- you know,

00:01:31.540 --> 00:01:33.360
if you put 3 there, you'll say,
oh, the slope there

00:01:33.360 --> 00:01:34.220
is equal to 6.

00:01:34.220 --> 00:01:35.590
We saw that in the
last example.

00:01:35.590 --> 00:01:37.410
So that's what we want to do.

00:01:37.410 --> 00:01:40.600
And we saw on the last-- I
think it was 2-- videos ago,

00:01:40.600 --> 00:01:47.130
that we defined f prime of x
to be equal to-- just the--

00:01:47.130 --> 00:01:48.470
well, I'll write it this way.

00:01:48.470 --> 00:01:51.210
It's the slope of the secant
line between x and some

00:01:51.210 --> 00:01:53.540
point that's a little
bit further away from x.

00:01:53.540 --> 00:01:56.640
So the slope of the secant
line is change in y.

00:01:56.640 --> 00:02:00.940
So it's the y value of the
point that's a little

00:02:00.940 --> 00:02:02.290
bit further away from x.

00:02:02.290 --> 00:02:07.630
So f of x plus h minus
the y value at x, right?

00:02:07.630 --> 00:02:08.600
Because this is right here.

00:02:08.600 --> 00:02:11.200
This is f of x.

00:02:11.200 --> 00:02:13.820
So minus f of x.

00:02:13.820 --> 00:02:15.830
All of that over
the change in x.

00:02:15.830 --> 00:02:19.620
So if this is x plus h
here, the change in x

00:02:19.620 --> 00:02:21.740
is x plus h minus x.

00:02:21.740 --> 00:02:23.470
Or this distance right
here is just h.

00:02:23.470 --> 00:02:26.830
The change in x is going
to be equal to h.

00:02:26.830 --> 00:02:29.150
So that's just slope of
the secant line, between

00:02:29.150 --> 00:02:31.240
any 2 points like that.

00:02:31.240 --> 00:02:33.880
And we said, hey, we could find
the slope of the tangent line

00:02:33.880 --> 00:02:36.390
if we just take the limit of
this as it approaches--

00:02:36.390 --> 00:02:40.060
as h approaches 0.

00:02:40.060 --> 00:02:42.930
And then we'll be finding the
slope of the tangent line.

00:02:42.930 --> 00:02:48.060
Now let's apply this idea to
a particular function, f of

00:02:48.060 --> 00:02:49.970
x is equal to x squared.

00:02:49.970 --> 00:02:52.430
Or y is equal to x squared.

00:02:52.430 --> 00:02:55.800
So here, we could have the
point-- we could consider this

00:02:55.800 --> 00:02:58.910
to be the point x-- x squared.

00:02:58.910 --> 00:03:01.380
So f of x is just
equal to x squared.

00:03:01.380 --> 00:03:05.040
And then this would be the
point-- let me do it in

00:03:05.040 --> 00:03:07.140
a more vibrant color.

00:03:07.140 --> 00:03:11.510
This is the point x plus h--
that's this point right here.

00:03:11.510 --> 00:03:13.040
It's a little bit further down.

00:03:13.040 --> 00:03:14.770
And then x plus h squared.

00:03:19.450 --> 00:03:20.880
And you know, in the
last video, we did this

00:03:20.880 --> 00:03:21.500
for a particular x.

00:03:21.500 --> 00:03:22.590
We did it for 3.

00:03:22.590 --> 00:03:24.330
But now I want a
general formula.

00:03:24.330 --> 00:03:26.300
You give me any x and I won't
have to do what I did in the

00:03:26.300 --> 00:03:28.330
last video for any
particular number.

00:03:28.330 --> 00:03:29.490
I'll have a general function.

00:03:29.490 --> 00:03:31.570
You give me 7, I'll tell you
what the slope is at 7.

00:03:31.570 --> 00:03:33.260
You give me negative 3, I'll
tell you what the slope

00:03:33.260 --> 00:03:34.300
is at negative 3.

00:03:34.300 --> 00:03:36.450
You give me 100,000,
I'll tell you what the

00:03:36.450 --> 00:03:37.690
slope is at 100,000.

00:03:37.690 --> 00:03:39.310
So let's apply it here.

00:03:39.310 --> 00:03:42.740
So we want to find the change
in y over the change in x.

00:03:45.410 --> 00:03:51.190
So first of all, the change
in y is this guy's y value,

00:03:51.190 --> 00:03:52.975
which is x plus h squared.

00:03:57.410 --> 00:03:59.290
That's this guy's y
value right here.

00:03:59.290 --> 00:04:00.270
That's this right here.

00:04:00.270 --> 00:04:02.730
That's x plus h squared.

00:04:02.730 --> 00:04:06.380
I just took x plus h,
evaluated, I squared it, and

00:04:06.380 --> 00:04:07.820
that's its point on the curve.

00:04:07.820 --> 00:04:08.930
So it's x plus h squared.

00:04:08.930 --> 00:04:10.630
So that's there right there.

00:04:10.630 --> 00:04:12.080
And then what's this value?

00:04:12.080 --> 00:04:15.230
f of x right here is equal to--
I know it's getting messy--

00:04:15.230 --> 00:04:16.310
is equal to x squared.

00:04:16.310 --> 00:04:18.530
If you take your x, you
evaluate the function

00:04:18.530 --> 00:04:20.600
at that point, you're
going to get x squared.

00:04:20.600 --> 00:04:23.300
So it's equal to
minus x squared.

00:04:23.300 --> 00:04:24.900
This is your change in y.

00:04:24.900 --> 00:04:29.090
That's this distance
right there.

00:04:29.090 --> 00:04:33.110
And just to relate it to our
definition of a derivative,

00:04:33.110 --> 00:04:35.820
this blue thing right here is
equivalent to this

00:04:35.820 --> 00:04:36.570
thing right here.

00:04:36.570 --> 00:04:39.140
We just evaluated our function.

00:04:39.140 --> 00:04:42.850
Our function is f of x
is equal to x squared.

00:04:42.850 --> 00:04:46.310
We just evaluated when x
is equal to x plus h.

00:04:46.310 --> 00:04:48.920
So if you have to square
it, if I put an a there,

00:04:48.920 --> 00:04:49.540
it'd be a squared.

00:04:49.540 --> 00:04:51.200
If I put an apple there,
it'd be apple squared.

00:04:51.200 --> 00:04:52.970
If I put an x plus h in
there, it's going to

00:04:52.970 --> 00:04:54.580
be x plus h squared.

00:04:54.580 --> 00:04:56.210
So this is that thing.

00:04:56.210 --> 00:05:00.420
And then, this thing right here
is just the function evaluated

00:05:00.420 --> 00:05:01.810
at the point in question.

00:05:01.810 --> 00:05:02.440
Right there.

00:05:02.440 --> 00:05:05.120
So this is our change in y.

00:05:05.120 --> 00:05:07.920
And let's divide that
by our change in x.

00:05:07.920 --> 00:05:10.730
Our change in x-- if this is x
plus h and this is just x, our

00:05:10.730 --> 00:05:13.360
change in x is just
going to be h.

00:05:13.360 --> 00:05:14.880
So that's where we
get that term from.

00:05:14.880 --> 00:05:17.670
So this is just a slope
between these 2 points.

00:05:17.670 --> 00:05:19.350
This is just a slope
between those two points.

00:05:19.350 --> 00:05:22.690
But, of course, we want to
find-- the limit at this point

00:05:22.690 --> 00:05:25.500
gets closer and closer to this
point, and this point gets

00:05:25.500 --> 00:05:26.790
closer and closer
to that point.

00:05:26.790 --> 00:05:29.530
So this becomes a tangent line.

00:05:29.530 --> 00:05:34.480
So we're going to take the
limit as h approaches 0, and

00:05:34.480 --> 00:05:37.250
this is our f prime of x.

00:05:37.250 --> 00:05:39.580
And this is the exact same
definition of this, instead

00:05:39.580 --> 00:05:42.400
of being general and saying,
for any function, we know

00:05:42.400 --> 00:05:43.380
what the function was.

00:05:43.380 --> 00:05:45.680
It was f of x is
equal to x squared.

00:05:45.680 --> 00:05:46.660
So we actually applied it.

00:05:46.660 --> 00:05:48.350
Instead of f of x,
we wrote x squared.

00:05:48.350 --> 00:05:52.950
Instead of f of x plus h,
we wrote x plus h squared.

00:05:52.950 --> 00:05:56.180
So let's see if we can
evaluate this limit.

00:05:56.180 --> 00:06:04.680
So this is going to be equal
to the limit as h approaches

00:06:04.680 --> 00:06:06.680
0 to square this out.

00:06:06.680 --> 00:06:09.160
I'll do it in the same color.

00:06:09.160 --> 00:06:19.230
That's x squared plus 2xh plus
h squared, and then we have

00:06:19.230 --> 00:06:22.320
this minus x squared over here.

00:06:22.320 --> 00:06:24.790
I just multiplied this
guy out over here.

00:06:24.790 --> 00:06:27.635
And then all of that
is divided by h.

00:06:27.635 --> 00:06:30.850
Now let's see if we can
simplify this a little bit.

00:06:30.850 --> 00:06:32.610
Well, you immediately see you
have an x squared and you

00:06:32.610 --> 00:06:35.390
have a minus x squared,
so those cancel out.

00:06:35.390 --> 00:06:36.570
And then we can divide
the numerator and the

00:06:36.570 --> 00:06:38.185
denominator by h.

00:06:38.185 --> 00:06:44.780
So this simplifies to-- so we
get f prime of x is equal to--

00:06:44.780 --> 00:06:46.590
if we divide the numerator and
the denominator by h--

00:06:46.590 --> 00:06:50.260
we get 2x plus h.

00:06:50.260 --> 00:06:52.070
I'm sorry, I forgot my limit.

00:06:52.070 --> 00:06:53.420
It equals the limit.

00:06:53.420 --> 00:06:54.330
Very important.

00:06:54.330 --> 00:07:01.200
Limit as h approaches 0 of
divide everything by h, and

00:07:01.200 --> 00:07:08.640
you get 2x plus h squared
divided by h is h.

00:07:08.640 --> 00:07:10.440
And if you remember the last
video, when we did it with a

00:07:10.440 --> 00:07:13.780
particular x, we said x is
equal to 3, we got 6

00:07:13.780 --> 00:07:14.920
plus delta x here.

00:07:14.920 --> 00:07:17.420
Or 6 plus h here, so
it's very similar.

00:07:17.420 --> 00:07:20.130
So if you take the limited h
approaches 0 here, that's

00:07:20.130 --> 00:07:21.010
just going to disappear.

00:07:21.010 --> 00:07:24.240
So this is just going
to be equal to 2x.

00:07:24.240 --> 00:07:28.540
So we just figured out that if
f of x-- this is a big result.

00:07:28.540 --> 00:07:30.180
This is exciting!

00:07:30.180 --> 00:07:35.130
That if f of x is equal
to x squared, f prime

00:07:35.130 --> 00:07:37.910
of x is equal to 2x.

00:07:37.910 --> 00:07:39.470
That's what we
just figured out.

00:07:39.470 --> 00:07:41.570
And I wanted to make
sure you understand

00:07:41.570 --> 00:07:42.880
how to interpret this.

00:07:42.880 --> 00:07:45.780
f of x, if you give me a value,
is going to tell you the value

00:07:45.780 --> 00:07:47.560
of the function at that point.

00:07:47.560 --> 00:07:48.920
At prime of x it's
going to tell you the

00:07:48.920 --> 00:07:49.960
slope at that point.

00:07:49.960 --> 00:07:51.180
Let me draw that.

00:07:51.180 --> 00:07:53.490
Because this is a
key realization.

00:07:53.490 --> 00:07:56.220
And you might, you know, it's
kind of maybe initially

00:07:56.220 --> 00:07:58.590
unintuitive to think of a
function that gives us the

00:07:58.590 --> 00:08:02.040
slope, at any point,
of another function.

00:08:02.040 --> 00:08:05.070
So it looks like this.

00:08:05.070 --> 00:08:07.650
Let me draw a little
neater than that.

00:08:07.650 --> 00:08:10.290
Ah, it's still not that neat.

00:08:10.290 --> 00:08:11.280
That's satisfactory.

00:08:11.280 --> 00:08:13.650
Let me just draw it in
the positive coordinate.

00:08:13.650 --> 00:08:15.640
Well, I'll just draw the
whole-- the curve looks

00:08:15.640 --> 00:08:17.450
something like that.

00:08:17.450 --> 00:08:20.090
Now this is the
curve of f of x.

00:08:20.090 --> 00:08:23.050
This is the curve of f of
x is equal to x squared.

00:08:23.050 --> 00:08:24.230
Just like that.

00:08:24.230 --> 00:08:26.490
So if you give me a point.

00:08:26.490 --> 00:08:27.840
You give me the point 7.

00:08:27.840 --> 00:08:30.610
You apply, you put it in
here, you square it.

00:08:30.610 --> 00:08:34.460
And it is mapped
to the number 49.

00:08:34.460 --> 00:08:36.170
So you get the number
49 right there.

00:08:36.170 --> 00:08:38.620
This is the number 7, 49.

00:08:38.620 --> 00:08:40.940
You're used to dealing with
functions right there.

00:08:40.940 --> 00:08:43.630
But what is f prime of 7?

00:08:43.630 --> 00:08:45.000
f prime of 7.

00:08:45.000 --> 00:08:47.740
You say, 2 times 7
is equal to 14.

00:08:47.740 --> 00:08:49.990
What is this 14 number here?

00:08:49.990 --> 00:08:51.380
What is this thing?

00:08:51.380 --> 00:08:53.440
Well, this is the slope
of the tangent line

00:08:53.440 --> 00:08:55.030
at x is equal to 7.

00:08:55.030 --> 00:08:59.330
So if I were to take that point
and draw a tangent line-- a

00:08:59.330 --> 00:09:02.930
point that just grazes our
curve-- if I were to just

00:09:02.930 --> 00:09:03.890
draw a tangent line.

00:09:03.890 --> 00:09:06.410
That wasn't tangent
enough for me.

00:09:06.410 --> 00:09:08.790
So that's my tangent
line right there.

00:09:08.790 --> 00:09:10.200
You get the idea.

00:09:10.200 --> 00:09:13.660
The slope of this guy-- you do
your change in y over your

00:09:13.660 --> 00:09:16.260
change in x-- is going
to be equal to 14.

00:09:16.260 --> 00:09:18.880
The slope of the curve
at y is equal to 7-- is

00:09:18.880 --> 00:09:20.270
a pretty steep curve.

00:09:20.270 --> 00:09:21.990
If you wanted to find the
slope, let's say that this

00:09:21.990 --> 00:09:25.910
is y-- let's say it's
x is equal to 2.

00:09:25.910 --> 00:09:30.290
I said at x is equal to
7, the slope is 14.

00:09:30.290 --> 00:09:32.900
At x is equal to 2,
what is the slope?

00:09:32.900 --> 00:09:37.230
Well, you figure out f prime of
2, which is equal to 2 times

00:09:37.230 --> 00:09:38.880
2, which is equal to 4.

00:09:38.880 --> 00:09:43.990
So the slope here is 4.

00:09:43.990 --> 00:09:46.520
You could say m is equal
to 4. m for slope.

00:09:46.520 --> 00:09:50.560
What is f prime of 0?

00:09:50.560 --> 00:09:51.170
f prime.

00:09:51.170 --> 00:09:53.120
We know that f of
0 is 0, right?

00:09:53.120 --> 00:09:55.120
0 squared is 0.

00:09:55.120 --> 00:09:57.710
But what is f prime of 0?

00:09:57.710 --> 00:09:59.180
Well, 2 times 0 is 0.

00:09:59.180 --> 00:10:00.480
That's also equal to 0.

00:10:00.480 --> 00:10:01.350
But what does that mean?

00:10:01.350 --> 00:10:03.100
What's the interpretation?

00:10:03.100 --> 00:10:05.850
It means the slope of
the tangent line is 0.

00:10:05.850 --> 00:10:08.390
So a 0 sloped line
looks like this.

00:10:08.390 --> 00:10:09.920
Looks just like a
horizontal line.

00:10:09.920 --> 00:10:11.370
And that looks about right.

00:10:11.370 --> 00:10:14.220
A horizontal line would
be tangent to the

00:10:14.220 --> 00:10:15.910
curve at y equals 0.

00:10:15.910 --> 00:10:17.620
Let's try another one.

00:10:17.620 --> 00:10:23.190
Let's try the point minus 1.

00:10:23.190 --> 00:10:26.290
So let's say we're right
there. x is equal to minus 1.

00:10:26.290 --> 00:10:29.220
So f of minus 1, you
just square it.

00:10:29.220 --> 00:10:30.380
Because we're dealing
with x squared.

00:10:30.380 --> 00:10:31.570
So it's equal to 1.

00:10:31.570 --> 00:10:33.180
That's that point right there.

00:10:33.180 --> 00:10:36.680
What is f prime of minus 1?

00:10:36.680 --> 00:10:40.050
f prime of minus 1
is 2 times minus 1.

00:10:40.050 --> 00:10:42.340
2 times minus is minus 2.

00:10:42.340 --> 00:10:43.650
What does that mean?

00:10:43.650 --> 00:10:47.680
It means that the slope of the
tangent line at x is equal to

00:10:47.680 --> 00:10:51.050
1, to this curve, to the
function, is minus 2.

00:10:51.050 --> 00:10:54.600
So if I were to draw the
tangent line here-- the tangent

00:10:54.600 --> 00:10:57.140
line looks like that-- and
look, it is a downward

00:10:57.140 --> 00:10:57.700
sloping line.

00:10:57.700 --> 00:10:58.450
And it makes sense.

00:10:58.450 --> 00:11:04.700
The slope here is
equal to minus 2.


Calculating slope of tangent line using derivative definition

WEBVTT
Kind: captions
Language: en

00:00:00.660 --> 00:00:04.110
In the last video we tried to
figure out the slope of a

00:00:04.110 --> 00:00:07.570
point or the slope of a
curve at a certain point.

00:00:07.570 --> 00:00:10.310
And the way we did, we said OK,
well let's find the slope

00:00:10.310 --> 00:00:12.970
between that point and then
another point that's not too

00:00:12.970 --> 00:00:14.220
far away from that point.

00:00:14.220 --> 00:00:15.630
And we got the slope
of the secant

00:00:15.630 --> 00:00:15.920
line.

00:00:15.920 --> 00:00:19.530
And it looks all fancy, but
this is just the y value of the

00:00:19.530 --> 00:00:23.570
point that's not too far away,
and this is just the y value

00:00:23.570 --> 00:00:25.700
point of the point in
question, so this is

00:00:25.700 --> 00:00:27.100
just your change in y.

00:00:27.100 --> 00:00:29.020
And then you divide that
by your change in x.

00:00:29.020 --> 00:00:31.580
So in the example we did,
h was the difference

00:00:31.580 --> 00:00:33.610
between our 2 x values.

00:00:33.610 --> 00:00:35.420
This distance was h.

00:00:35.420 --> 00:00:39.210
And that gave us the
slope of that line.

00:00:39.210 --> 00:00:42.940
We said hey, what if we take
the limit as this point

00:00:42.940 --> 00:00:45.510
right here gets closer
and closer to this point.

00:00:45.510 --> 00:00:48.420
If this point essentially
almost becomes this point, then

00:00:48.420 --> 00:00:51.230
our slope is going to be the
slope of our tangent line.

00:00:51.230 --> 00:00:54.620
And we define that as the
derivative of our function.

00:00:54.620 --> 00:00:57.280
We said that's equal
to f prime of x.

00:00:57.280 --> 00:01:00.520
So let's if we can apply this
in this video to maybe make

00:01:00.520 --> 00:01:03.750
things a little bit more
concrete in your head.

00:01:03.750 --> 00:01:04.910
So let me do one.

00:01:04.910 --> 00:01:07.140
First I'll do a particular
case where I want to find the

00:01:07.140 --> 00:01:09.980
slope at exactly some point.

00:01:09.980 --> 00:01:14.670
So let me draw my axes again.

00:01:14.670 --> 00:01:17.520
Let's draw some
axes right there.

00:01:17.520 --> 00:01:25.400
Let's say I have the curve--
this is the curve-- y

00:01:25.400 --> 00:01:27.110
is equal to x squared.

00:01:27.110 --> 00:01:32.790
So this is my y-axis, this is
my x-axis, and I want to know

00:01:32.790 --> 00:01:37.270
the slope at the point
x is equal to 3.

00:01:37.270 --> 00:01:41.660
When I say the slope you can
imagine a tangent line here.

00:01:41.660 --> 00:01:44.760
You can imagine a tangent line
that goes just like that, and

00:01:44.760 --> 00:01:47.710
it would just barely graze
the curve at that point.

00:01:47.710 --> 00:01:51.260
But what is the slope
of that tangent line?

00:01:51.260 --> 00:01:53.570
What is the slope of that
tangent line which is the same

00:01:53.570 --> 00:01:56.360
as the slope of the curve
right at that point.

00:01:56.360 --> 00:01:58.770
So to do it, I'm actually going
to do this exact technique that

00:01:58.770 --> 00:02:01.270
we did before, then we'll
generalize it so you don't have

00:02:01.270 --> 00:02:03.910
to do it every time for
a particular number.

00:02:03.910 --> 00:02:06.805
So let's take some
other point here.

00:02:09.610 --> 00:02:14.110
Let's call this 3 plus delta x.

00:02:14.110 --> 00:02:16.380
I'm changing the notation
because in some books you'll

00:02:16.380 --> 00:02:19.530
see an h, some books you'll see
a delta x, doesn't hurt to

00:02:19.530 --> 00:02:20.660
be exposed to both of them.

00:02:20.660 --> 00:02:22.500
So this is 3 plus delta x.

00:02:22.500 --> 00:02:25.770
So first of all what is
this point right here?

00:02:25.770 --> 00:02:31.890
This is a curve y is equal to
x squared, so f of x is 3

00:02:31.890 --> 00:02:33.720
squared-- this is the point 9.

00:02:33.720 --> 00:02:36.805
This is the point
3,9 right here.

00:02:39.350 --> 00:02:40.870
And what is this
point right here?

00:02:40.870 --> 00:02:45.180
So if we were go all the way
up here, what is that point?

00:02:45.180 --> 00:02:49.120
Well here our x is
3 plus delta x.

00:02:49.120 --> 00:02:50.310
It's the same thing as
this one right here,

00:02:50.310 --> 00:02:50.990
as x naught

00:02:50.990 --> 00:02:51.300
plus h.

00:02:51.300 --> 00:02:54.235
I could have called this
3 plus h just as easily.

00:02:54.235 --> 00:02:58.110
So it's 3 plus
delta x up there.

00:02:58.110 --> 00:02:59.930
So what's the y
value going to be?

00:02:59.930 --> 00:03:02.080
Well whatever x value is, it's
on the curve, it's going

00:03:02.080 --> 00:03:03.550
to be that squared.

00:03:03.550 --> 00:03:12.310
So it's going to be the point
3 plus delta x squared.

00:03:12.310 --> 00:03:14.790
So let's figure out the
slope of this secant

00:03:14.790 --> 00:03:15.240
line.

00:03:18.080 --> 00:03:20.270
And let me zoom in a little
bit, because that might help.

00:03:20.270 --> 00:03:22.950
So if I zoom in on just
this part of the curve,

00:03:22.950 --> 00:03:25.250
it might look like that.

00:03:25.250 --> 00:03:28.330
And then I have one point
here, and then I have the

00:03:28.330 --> 00:03:29.776
other point is up here.

00:03:32.330 --> 00:03:33.070
That's the secant

00:03:33.070 --> 00:03:33.510
line.

00:03:39.100 --> 00:03:40.210
Just like that.

00:03:40.210 --> 00:03:45.510
This was the point over
here, the point 3,9.

00:03:45.510 --> 00:03:50.230
And then this point up here is
the point 3 plus delta x, so

00:03:50.230 --> 00:03:54.340
just some larger number than
3, and then it's going to

00:03:54.340 --> 00:03:55.790
be that number squared.

00:03:55.790 --> 00:03:58.110
So it's going to be 3
plus delta x squared.

00:03:58.110 --> 00:03:59.070
What is that?

00:03:59.070 --> 00:04:01.310
That's going to be 9.

00:04:01.310 --> 00:04:03.720
I'm just foiling this out,
or you do the distribute

00:04:03.720 --> 00:04:04.940
property twice.

00:04:04.940 --> 00:04:09.280
a plus b squared is a squared
plus 2 a b plus b squared, so

00:04:09.280 --> 00:04:11.430
it's going to be 9 plus two
times the product

00:04:11.430 --> 00:04:12.020
of these things.

00:04:12.020 --> 00:04:17.345
So plus 6 delta x, and then
plus delta x squared.

00:04:21.070 --> 00:04:22.530
That's the coordinate
of the second line.

00:04:22.530 --> 00:04:25.470
This looks complicated, but I
just took this x value and I

00:04:25.470 --> 00:04:29.230
squared it, because it's on the
line y is equal to x squared.

00:04:29.230 --> 00:04:32.430
So the slope of the secant

00:04:32.430 --> 00:04:35.500
line is going to be the
change in y divided

00:04:35.500 --> 00:04:37.050
by the change in x.

00:04:37.050 --> 00:04:42.500
So the change in y is just
going to be this guy's y value,

00:04:42.500 --> 00:04:49.430
which is 9 plus 6 delta
x plus delta x squared.

00:04:49.430 --> 00:04:52.610
That's this guy's y value,
minus this guy's y value.

00:04:55.310 --> 00:04:58.000
So minus 9.

00:04:58.000 --> 00:04:59.510
That's your change in y.

00:04:59.510 --> 00:05:03.630
And you want to divide
that by your change in x.

00:05:03.630 --> 00:05:05.020
Well what is your change in x?

00:05:05.020 --> 00:05:07.210
This is actually going to
be pretty convenient.

00:05:07.210 --> 00:05:09.490
This larger x value-- we
started with this point on the

00:05:09.490 --> 00:05:11.460
top, so we have to start with
this point on the bottom.

00:05:11.460 --> 00:05:15.680
So it's going to be
3 plus delta x.

00:05:15.680 --> 00:05:17.340
And then what's this x value?

00:05:17.340 --> 00:05:20.700
What is minus 3?

00:05:20.700 --> 00:05:22.070
That's his x value.

00:05:22.070 --> 00:05:23.760
So what does this simplify to?

00:05:23.760 --> 00:05:27.020
The numerator-- this 9
and that 9 cancel out,

00:05:27.020 --> 00:05:28.320
we get a 9 minus 9.

00:05:28.320 --> 00:05:30.780
And in the denominator
what happens?

00:05:30.780 --> 00:05:32.560
This 3 and minus 3 cancel out.

00:05:32.560 --> 00:05:35.550
So the change in x actually end
up becoming this delta x, which

00:05:35.550 --> 00:05:38.490
makes sense, because this delta
x is essentially how much more

00:05:38.490 --> 00:05:39.990
this guy is then that guy.

00:05:39.990 --> 00:05:43.360
So that should be the
change in x, delta x.

00:05:43.360 --> 00:05:44.810
So the slope of my secant

00:05:44.810 --> 00:05:53.860
line has simplified to 6 times
my change in x, plus my change

00:05:53.860 --> 00:06:00.130
in x squared, all of that
over my change in x.

00:06:00.130 --> 00:06:02.640
And now we can simplify
this even more.

00:06:02.640 --> 00:06:05.140
Let's divide the numerator
and the denominator

00:06:05.140 --> 00:06:06.790
by our change in x.

00:06:06.790 --> 00:06:09.900
And I'll switch colors just
to ease the monotony.

00:06:09.900 --> 00:06:13.110
So my slope of my
tangent of my secant

00:06:13.110 --> 00:06:15.580
line-- the one that goes
through both of these-- is

00:06:15.580 --> 00:06:17.483
going to be equal if you
divide the numerator and

00:06:17.483 --> 00:06:19.650
denominator this becomes 6.

00:06:19.650 --> 00:06:21.420
I'm just dividing numerator
and denominator by delta

00:06:21.420 --> 00:06:26.130
x plus six plus delta x.

00:06:26.130 --> 00:06:28.080
So that is the slope
of this secant

00:06:28.080 --> 00:06:34.630
line So slope is equal
to 6 plus delta x.

00:06:34.630 --> 00:06:36.280
That's this one right here.

00:06:36.280 --> 00:06:38.990
That's this reddish line that
I've drawn right there.

00:06:38.990 --> 00:06:42.340
So this number right here, if
the delta x was one, if

00:06:42.340 --> 00:06:46.390
these were the points 3 and 4,
then my slope would be 6 plus

00:06:46.390 --> 00:06:49.640
1, because I'm picking a point
4 where the delta x here

00:06:49.640 --> 00:06:50.480
would have to be 1.

00:06:50.480 --> 00:06:53.780
So the slope would be 7.

00:06:53.780 --> 00:06:56.730
So we have a general formula
for no matter what my delta

00:06:56.730 --> 00:07:00.790
x is, I can find the slope
between 3 and 3 plus delta x.

00:07:00.790 --> 00:07:02.640
Between those two points.

00:07:02.640 --> 00:07:08.010
Now we wanted to find the
slope at exactly that

00:07:08.010 --> 00:07:09.120
point right there.

00:07:09.120 --> 00:07:12.100
So let's see what happens
when delta x get

00:07:12.100 --> 00:07:12.810
smaller and smaller.

00:07:12.810 --> 00:07:14.470
This is what delta
x is right now.

00:07:14.470 --> 00:07:15.580
It's this distance.

00:07:15.580 --> 00:07:18.510
But if delta x got a little
bit smaller, then the secant

00:07:18.510 --> 00:07:20.160
line would look like that.

00:07:20.160 --> 00:07:21.360
Got even smaller, the secant

00:07:21.360 --> 00:07:23.680
line would look like that,
it gets even smaller.

00:07:23.680 --> 00:07:25.240
Then we're getting pretty
close to the slope

00:07:25.240 --> 00:07:26.860
of the tangent line.

00:07:26.860 --> 00:07:28.970
The tangent line is this
thing right here that I

00:07:28.970 --> 00:07:30.910
want to find the slope of.

00:07:30.910 --> 00:07:34.970
Let's find a limit as our
delta x approaches 0.

00:07:34.970 --> 00:07:39.400
So the limit as delta
x approaches 0 of our

00:07:39.400 --> 00:07:40.240
slope of the secant

00:07:40.240 --> 00:07:43.700
line of 6 plus delta
x is equal to what?

00:07:43.700 --> 00:07:44.830
This is pretty straightforward.

00:07:44.830 --> 00:07:48.350
You can just set this equal
to 0 and it's equal to 6.

00:07:48.350 --> 00:07:54.280
So the slope of our tangent
line at the point x is equal to

00:07:54.280 --> 00:07:57.615
3 right there is equal to 6.

00:07:57.615 --> 00:08:01.280
And another way we could write
this if we wrote that f of

00:08:01.280 --> 00:08:04.450
x is equal to x squared.

00:08:04.450 --> 00:08:08.770
We now know that the derivative
or the slope of the tangent

00:08:08.770 --> 00:08:13.350
line of this function at the
point 3-- I just only evaluated

00:08:13.350 --> 00:08:18.410
it at the point 3 right there--
that that is equal to 6.

00:08:18.410 --> 00:08:22.670
I haven't yet come up with a
general formula for the slope

00:08:22.670 --> 00:08:25.790
of this line at any point, and
I'm going to do that

00:08:25.790 --> 00:08:26.330
in the next video.


Example of finding matrix inverse

WEBVTT
Kind: captions
Language: en

00:00:00.540 --> 00:00:04.250
In the last video, we stumbled
upon a way to figure out the

00:00:04.250 --> 00:00:06.940
inverse for an invertible
matrix.

00:00:06.940 --> 00:00:09.090
So, let's actually use
that method in

00:00:09.090 --> 00:00:10.310
this video right here.

00:00:10.310 --> 00:00:12.390
I'm going to use the same matrix
that we started off

00:00:12.390 --> 00:00:13.820
with in the last video.

00:00:13.820 --> 00:00:15.860
It seems like a fairly
good matrix.

00:00:15.860 --> 00:00:18.360
We know that it's reduced row
echelon form is the identity

00:00:18.360 --> 00:00:19.890
matrix, so we know
it's invertable.

00:00:19.890 --> 00:00:21.800
So, let's find its inverse.

00:00:21.800 --> 00:00:24.040
The technique is pretty
straightforward.

00:00:24.040 --> 00:00:26.700
You literally just apply the
same transformations you would

00:00:26.700 --> 00:00:30.220
apply to this guy to get you
to the identity matrix, and

00:00:30.220 --> 00:00:32.170
you would apply those same
transformations to the

00:00:32.170 --> 00:00:33.370
identity matrix.

00:00:33.370 --> 00:00:35.440
That's because the collection
of those transformations, if

00:00:35.440 --> 00:00:38.240
you represent them as matrixes,
are really just the

00:00:38.240 --> 00:00:39.820
inverse of this guy.

00:00:39.820 --> 00:00:42.480
Let's just do it.

00:00:42.480 --> 00:00:44.280
So I'll create an augmented
matrix here.

00:00:44.280 --> 00:00:46.420
Maybe I'll do it right here.

00:00:46.420 --> 00:00:48.800
Let me make it a little
bit neater.

00:00:48.800 --> 00:00:49.940
First, I'll write a.

00:00:49.940 --> 00:00:52.490
It's 1, minus 1, 1.

00:00:52.490 --> 00:00:55.310
And then minus 1, 2, 1.

00:00:55.310 --> 00:00:58.340
Minus 1, 3, 4.

00:00:58.340 --> 00:01:02.180
And then I'll augment it with
the identity matrix, with 1,

00:01:02.180 --> 00:01:08.040
0, 0, 0, 1, 0, 0, 0, 1.

00:01:08.040 --> 00:01:11.600
Now, if I want to get a into
reduced row echelon form,

00:01:11.600 --> 00:01:13.480
maybe I'll replace
the second row.

00:01:13.480 --> 00:01:15.280
I'll keep the first row
the same for now.

00:01:15.280 --> 00:01:19.100
Let me just draw it like this.

00:01:19.100 --> 00:01:24.000
The entire first row:
1, minus 1, minus 1.

00:01:24.000 --> 00:01:28.680
It's going to be augmented
with 1, 0, 0.

00:01:28.680 --> 00:01:30.630
Keep the whole first
row the same.

00:01:30.630 --> 00:01:32.960
Let's replace the second
row with the second row

00:01:32.960 --> 00:01:34.230
plus the first row.

00:01:34.230 --> 00:01:38.220
Minus 1 plus 1 is 0.

00:01:38.220 --> 00:01:40.830
2 plus minus 1 is 1.

00:01:40.830 --> 00:01:43.950
3 plus minus 1 is 2.

00:01:43.950 --> 00:01:46.570
0 plus 1 is 0.

00:01:46.570 --> 00:01:48.660
1 plus-- oh, sorry.

00:01:48.660 --> 00:01:49.780
That was a tricky one.

00:01:49.780 --> 00:01:52.520
0 plus 1 is 1.

00:01:52.520 --> 00:01:55.050
1 plus 0 is 1.

00:01:55.050 --> 00:02:00.170
0 plus 0 is 0.

00:02:00.170 --> 00:02:03.290
All I did is I added
these two rows up.

00:02:03.290 --> 00:02:04.450
Now, this third row.

00:02:04.450 --> 00:02:06.190
Let me replace-- I want
to get a zero here.

00:02:06.190 --> 00:02:09.220
Let me replace the third
row with the third row

00:02:09.220 --> 00:02:11.730
minus the first row.

00:02:11.730 --> 00:02:14.080
1 minus 1 is 0.

00:02:14.080 --> 00:02:17.930
1 minus minus 1 is 2.

00:02:17.930 --> 00:02:21.850
4 minus minus 1 is 5.

00:02:21.850 --> 00:02:25.260
0 minus 1 is minus 1.

00:02:25.260 --> 00:02:28.270
0 minus 0 is 0.

00:02:28.270 --> 00:02:32.110
And then 1 minus 0 is 1.

00:02:32.110 --> 00:02:33.880
Just like that.

00:02:33.880 --> 00:02:35.300
Now, what do we want to do?

00:02:35.300 --> 00:02:36.750
Well, we've gotten this far.

00:02:36.750 --> 00:02:40.170
We want to zero out that
guy and that guy.

00:02:40.170 --> 00:02:44.760
Let's keep our second
row the same.

00:02:44.760 --> 00:02:46.600
Let me write it down here.

00:02:49.840 --> 00:02:56.410
It's 0, 1, 2, and then you
augmented it with 1, 1, 0.

00:02:56.410 --> 00:02:58.680
Just like that.

00:02:58.680 --> 00:03:01.900
And let's replace my first row
with the first row plus the

00:03:01.900 --> 00:03:02.590
second row.

00:03:02.590 --> 00:03:04.880
1 plus 0 is 1.

00:03:04.880 --> 00:03:07.890
Minus 1 plus 1 is 0.

00:03:07.890 --> 00:03:09.980
That's why I did that,
to get a zero there.

00:03:09.980 --> 00:03:14.050
Minus 1 plus 2 is 1.

00:03:14.050 --> 00:03:16.970
1 plus 1 is 2.

00:03:16.970 --> 00:03:19.150
0 plus 1 is 1.

00:03:19.150 --> 00:03:22.160
0 plus 0 is 0.

00:03:22.160 --> 00:03:25.730
And now, I also want to zero
out this guy right here.

00:03:25.730 --> 00:03:28.770
Let's replace the third row
with the third row minus 2

00:03:28.770 --> 00:03:30.000
times the second row.

00:03:30.000 --> 00:03:33.360
0 minus 2 times 0 is 0.

00:03:33.360 --> 00:03:37.010
2 minus 2 times 1 is 0.

00:03:37.010 --> 00:03:43.340
5 minus 2 times 2 is 5
minus 4, that's 1.

00:03:43.340 --> 00:03:47.820
Minus 1 minus 2 times 1--
that's minus 1 minus

00:03:47.820 --> 00:03:50.570
2-- is minus 3.

00:03:50.570 --> 00:03:54.660
0 minus 2 times 1,
that's minus 2.

00:03:54.660 --> 00:03:59.030
And then, 1 minus 2 times
0 is just 1 again.

00:03:59.030 --> 00:04:00.250
All right, home stretch.

00:04:00.250 --> 00:04:05.070
Now, I just want to zero out
these guys right here.

00:04:05.070 --> 00:04:07.685
All right, so just let me keep
my third row the same.

00:04:07.685 --> 00:04:10.736
Let me switch colors, keep
things colorful.

00:04:13.330 --> 00:04:15.589
It's 0, 0, 1.

00:04:15.589 --> 00:04:21.490
We're going to augment it with
minus 3, minus 2, and 1.

00:04:21.490 --> 00:04:27.370
Now, let's replace our first row
with the first row minus

00:04:27.370 --> 00:04:28.980
the third row.

00:04:28.980 --> 00:04:31.210
1 minus 0 is 1.

00:04:31.210 --> 00:04:34.600
0 minus 0 is 0.

00:04:34.600 --> 00:04:37.730
1 minus 1 is 0.

00:04:37.730 --> 00:04:42.220
2 minus minus 3, that's 5.

00:04:42.220 --> 00:04:47.310
1 minus minus 2 is 3.

00:04:47.310 --> 00:04:51.530
0 minus 1 is minus 1.

00:04:51.530 --> 00:04:56.100
Now, let's replace the second
row with the second row minus

00:04:56.100 --> 00:04:57.610
2 times the third row.

00:04:57.610 --> 00:05:00.310
0 minus 2 times 0 is 0.

00:05:00.310 --> 00:05:02.930
1 minus 2 times 0 is 0.

00:05:02.930 --> 00:05:09.060
2 minus 2 times 1 is-- I'm
sorry, I just-- oh, whoops.

00:05:09.060 --> 00:05:13.200
Let me-- we have to be very
careful not to make any

00:05:13.200 --> 00:05:14.090
careless mistakes.

00:05:14.090 --> 00:05:16.370
0 minus 2 times 0 is 0.

00:05:16.370 --> 00:05:18.490
1 minus 2 times 0 is 1.

00:05:18.490 --> 00:05:21.960
It's not 0.

00:05:21.960 --> 00:05:26.690
2 minus 2 times 1 is 0.

00:05:26.690 --> 00:05:32.730
1 minus 2 times minus
3-- that is 1 plus 2

00:05:32.730 --> 00:05:35.780
times 3-- that is 7.

00:05:35.780 --> 00:05:42.010
1 minus 2 times minus 2, that's
1 plus 4, which is 5.

00:05:42.010 --> 00:05:47.780
And then, 0 minus 2 times
1, so that's minus 2.

00:05:47.780 --> 00:05:50.710
And just like that, we've
gotten the A part of our

00:05:50.710 --> 00:05:53.980
augmented matrix into reduced
row echelon form.

00:05:53.980 --> 00:05:58.020
This is the reduced row
echelon form of A.

00:05:58.020 --> 00:06:00.830
And when you apply those exact
same transformations-- because

00:06:00.830 --> 00:06:03.020
if you think about it, that
series of matrix products that

00:06:03.020 --> 00:06:05.420
got you from this to the
identity matrix-- that, by

00:06:05.420 --> 00:06:08.230
definition, is the
identity matrix.

00:06:08.230 --> 00:06:10.210
So you apply those same
transformations to the

00:06:10.210 --> 00:06:13.730
identity matrix, you're going
to get the inverse of A.

00:06:13.730 --> 00:06:17.030
This right here is A inverse.

00:06:17.030 --> 00:06:19.650
And we have solved for the
inverse, and it actually

00:06:19.650 --> 00:06:22.010
wasn't too painful.


Deriving a method for determining inverses

WEBVTT
Kind: captions
Language: en

00:00:00.680 --> 00:00:04.510
I have this matrix A here that I
want to put into reduced row

00:00:04.510 --> 00:00:05.450
echelon form.

00:00:05.450 --> 00:00:07.160
And we've done this
multiple times.

00:00:07.160 --> 00:00:09.670
You just perform a bunch
of row operations.

00:00:09.670 --> 00:00:12.470
But what I want to show you in
this video is that those row

00:00:12.470 --> 00:00:16.520
operations are equivalent to
linear transformations on the

00:00:16.520 --> 00:00:19.450
column vectors of A.

00:00:19.450 --> 00:00:21.490
So let me show you by example.

00:00:21.490 --> 00:00:24.450
So if we just want to put A into
reduced row echelon form,

00:00:24.450 --> 00:00:26.860
the first step that we might
want to do if we wanted to

00:00:26.860 --> 00:00:31.580
zero out these entries right
here, is-- let me do it right

00:00:31.580 --> 00:00:34.890
here-- is we'll keep our
first entry the same.

00:00:34.890 --> 00:00:36.830
So for each of these column
vectors, we're going to keep

00:00:36.830 --> 00:00:38.010
the first entry the same.

00:00:38.010 --> 00:00:41.590
So they're going to be
1, minus 1, minus 1.

00:00:41.590 --> 00:00:44.360
And actually, let me
simultaneously construct my

00:00:44.360 --> 00:00:45.800
transformation.

00:00:45.800 --> 00:00:48.340
So I'm saying that my row
operation I'm going to perform

00:00:48.340 --> 00:00:51.690
is equivalent to a linear
transformation

00:00:51.690 --> 00:00:52.630
on the column vector.

00:00:52.630 --> 00:00:55.160
So it's going to be a
transformation that's going to

00:00:55.160 --> 00:01:00.880
take some column vector,
a1, a2, and a3.

00:01:00.880 --> 00:01:03.130
It's going to take each of these
and then do something to

00:01:03.130 --> 00:01:05.239
them, do something to them
in a linear way.

00:01:05.239 --> 00:01:07.330
They'll be linear
transformations.

00:01:07.330 --> 00:01:09.470
So we're keeping the
first entry of our

00:01:09.470 --> 00:01:11.090
column vector the same.

00:01:11.090 --> 00:01:14.670
So this is just going
to be a1.

00:01:14.670 --> 00:01:16.330
This is a line right here.

00:01:16.330 --> 00:01:17.250
That's going to be a1.

00:01:17.250 --> 00:01:19.050
Now, what can we do if
we want to get to

00:01:19.050 --> 00:01:20.780
reduced row echelon form?

00:01:20.780 --> 00:01:22.610
We'd want to make
this equal to 0.

00:01:22.610 --> 00:01:26.360
So we would want to replace our
second row with the second

00:01:26.360 --> 00:01:29.230
row plus the first row, because
then these guys would

00:01:29.230 --> 00:01:30.500
turn out to be 0.

00:01:30.500 --> 00:01:32.140
So let me write that on
my transformation.

00:01:32.140 --> 00:01:35.490
I'm going to replace the second
row with the second row

00:01:35.490 --> 00:01:39.090
plus the first row.

00:01:39.090 --> 00:01:40.400
Let me write it out here.

00:01:40.400 --> 00:01:43.410
Minus 1 plus 1 is 0.

00:01:43.410 --> 00:01:45.810
2 plus minus 1 is 1.

00:01:45.810 --> 00:01:48.950
3 plus minus 1 is 2.

00:01:48.950 --> 00:01:51.070
Now, we also want
to get a 0 here.

00:01:51.070 --> 00:01:54.360
So let me replace my third
row with my third row

00:01:54.360 --> 00:01:55.900
minus my first row.

00:01:55.900 --> 00:01:59.320
So I'm going to replace my third
row with my third row

00:01:59.320 --> 00:02:01.690
minus my first row.

00:02:01.690 --> 00:02:05.240
So 1 minus 1 is 0.

00:02:05.240 --> 00:02:08.660
1 minus minus 1 is 2.

00:02:08.660 --> 00:02:14.100
4 minus minus 1 is 5,
just like that.

00:02:14.100 --> 00:02:16.790
So you see this was just a
linear transformation.

00:02:16.790 --> 00:02:19.390
And any linear transformation
you could actually represent

00:02:19.390 --> 00:02:22.280
as a matrix vector product.

00:02:22.280 --> 00:02:24.140
So for example, this
transformation, I could

00:02:24.140 --> 00:02:26.150
represent it.

00:02:26.150 --> 00:02:28.230
To figure out its transformation
matrix, so if

00:02:28.230 --> 00:02:32.600
we say that T of x is equal to,
I don't know, let's call

00:02:32.600 --> 00:02:36.220
it some matrix S times x.

00:02:36.220 --> 00:02:37.740
We already used the matrix A.

00:02:37.740 --> 00:02:40.110
So I have to pick
another letter.

00:02:40.110 --> 00:02:41.160
So how do we find S?

00:02:41.160 --> 00:02:43.570
Well, we just apply the
transformation to all of the

00:02:43.570 --> 00:02:46.370
column vectors, or the standard
basis vectors of the

00:02:46.370 --> 00:02:47.240
identity matrix.

00:02:47.240 --> 00:02:48.460
So let's do that.

00:02:48.460 --> 00:02:50.760
So the identity matrix-- I'll
draw it really small like

00:02:50.760 --> 00:02:55.080
this-- the identity matrix looks
like this, 1, 0, 0, 0,

00:02:55.080 --> 00:02:57.900
1, 0, 0, 0, 1.

00:02:57.900 --> 00:02:59.880
That's what that identity
matrix looks like.

00:02:59.880 --> 00:03:02.580
To find the transformation
matrix, we just apply this guy

00:03:02.580 --> 00:03:04.660
to each of the column
vectors of this.

00:03:04.660 --> 00:03:06.286
So what do we get?

00:03:06.286 --> 00:03:09.270
I'll do it a little
bit bigger.

00:03:09.270 --> 00:03:11.140
We apply it to each of
these column vectors.

00:03:11.140 --> 00:03:13.370
But we see the first row
always stays the same.

00:03:13.370 --> 00:03:16.250
So the first row is always going
to be the same thing.

00:03:16.250 --> 00:03:18.850
So 1, 0, 0.

00:03:18.850 --> 00:03:21.180
I'm essentially applying it
simultaneously to each of

00:03:21.180 --> 00:03:24.290
these column vectors, saying,
look, when you transform each

00:03:24.290 --> 00:03:27.710
of these column vectors, their
first entry stays the same.

00:03:27.710 --> 00:03:31.890
The second entry becomes
the second entry

00:03:31.890 --> 00:03:32.910
plus the first entry.

00:03:32.910 --> 00:03:35.730
So 0 plus 1 is 1.

00:03:35.730 --> 00:03:38.510
1 plus 0 is 1.

00:03:38.510 --> 00:03:41.350
0 plus 0 is 0.

00:03:41.350 --> 00:03:45.440
Then the third entry gets
replaced with the third entry

00:03:45.440 --> 00:03:46.690
minus the first entry.

00:03:46.690 --> 00:03:49.660
So 0 minus 1 is minus 1.

00:03:49.660 --> 00:03:52.500
0 minus 0 is 0.

00:03:52.500 --> 00:03:54.930
1 minus 0 is 1.

00:03:54.930 --> 00:03:58.510
Now notice, when I apply this
transformation to the column

00:03:58.510 --> 00:04:02.010
vectors of our identity matrix,
I essentially just

00:04:02.010 --> 00:04:03.760
performed those same
row operations

00:04:03.760 --> 00:04:04.730
that I did up there.

00:04:04.730 --> 00:04:07.160
I performed those exact same
row operations on this

00:04:07.160 --> 00:04:08.330
identity matrix.

00:04:08.330 --> 00:04:10.520
But we know that this is
actually the transformation

00:04:10.520 --> 00:04:13.110
matrix, that if we multiply
it by each of these column

00:04:13.110 --> 00:04:16.769
vectors, or by each of these
column vectors, we're going to

00:04:16.769 --> 00:04:18.430
get these column vectors.

00:04:18.430 --> 00:04:20.240
So you can view it this way.

00:04:20.240 --> 00:04:22.990
This right here, this
is equal to S.

00:04:22.990 --> 00:04:25.510
This is our transformation
matrix.

00:04:25.510 --> 00:04:32.350
So we could say that if we
create a new matrix whose

00:04:32.350 --> 00:04:37.320
columns are S times this column
vector, S times 1,

00:04:37.320 --> 00:04:39.420
minus 1, 1.

00:04:39.420 --> 00:04:47.540
And then the next column is S
times-- I wanted to do it in

00:04:47.540 --> 00:04:54.670
that other color-- S times
this guy, minus 1, 2, 1.

00:04:54.670 --> 00:05:02.930
And then the third column is
going to be S times this third

00:05:02.930 --> 00:05:09.180
column vector, minus 1, 3, 4.

00:05:09.180 --> 00:05:11.950
We now know we're applying this
transformation, this is

00:05:11.950 --> 00:05:13.920
S, times each of these
column vectors.

00:05:13.920 --> 00:05:15.890
That is the matrix
representation of this

00:05:15.890 --> 00:05:17.630
transformation.

00:05:17.630 --> 00:05:22.300
This guy right here will
be transformed

00:05:22.300 --> 00:05:25.070
to this right here.

00:05:25.070 --> 00:05:30.580
Let me do it down here.

00:05:30.580 --> 00:05:33.670
I wanted to show that stuff that
I had above here as well.

00:05:33.670 --> 00:05:35.310
Well, I'll just draw an arrow.

00:05:35.310 --> 00:05:36.440
That's probably the
simplest thing.

00:05:36.440 --> 00:05:39.900
This matrix right here
will become that

00:05:39.900 --> 00:05:41.430
matrix right there.

00:05:41.430 --> 00:05:43.610
So another way you could
write it, this is

00:05:43.610 --> 00:05:44.570
equivalent to what?

00:05:44.570 --> 00:05:45.520
What is this equivalent to?

00:05:45.520 --> 00:05:47.900
When you take a matrix and you
multiply it times each of the

00:05:47.900 --> 00:05:50.400
column vectors, when you
transform each of the column

00:05:50.400 --> 00:05:53.970
vectors by this matrix, this
is the definition of a

00:05:53.970 --> 00:05:55.270
matrix-matrix product.

00:05:55.270 --> 00:05:59.440
This is equal to our matrix S--
I'll do it in pink-- this

00:05:59.440 --> 00:06:05.920
is equal to our matrix S, which
is 1, 0, 0, 1, 1, 0,

00:06:05.920 --> 00:06:15.790
minus 1, 0, 1, times our matrix
A, times 1, minus 1, 1,

00:06:15.790 --> 00:06:22.090
minus 1, 2, 1, minus 1, 3, 4.

00:06:22.090 --> 00:06:25.630
So let me make this
very clear.

00:06:25.630 --> 00:06:28.430
This is our transformation
matrix S.

00:06:28.430 --> 00:06:30.440
This is our matrix A.

00:06:30.440 --> 00:06:34.130
And when you perform this
product you're going to get

00:06:34.130 --> 00:06:37.720
this guy right over here.

00:06:37.720 --> 00:06:40.450
I'll just copy and paste it.

00:06:40.450 --> 00:06:45.230
Edit, copy, and let
me paste it.

00:06:45.230 --> 00:06:48.210
You're going to get that
guy just like that.

00:06:48.210 --> 00:06:50.480
Now the whole reason why I'm
doing that is just to remind

00:06:50.480 --> 00:06:53.710
you that when we perform each of
these row operations, we're

00:06:53.710 --> 00:06:54.600
just multiplying.

00:06:54.600 --> 00:06:57.170
We're performing a linear
transformation on each of

00:06:57.170 --> 00:06:58.090
these columns.

00:06:58.090 --> 00:07:00.730
And it is completely equivalent
to just multiplying

00:07:00.730 --> 00:07:02.790
this guy by some matrix S.

00:07:02.790 --> 00:07:04.710
In this case, we took the
trouble of figuring out what

00:07:04.710 --> 00:07:06.150
that matrix S is.

00:07:06.150 --> 00:07:08.550
But any of these row operations
that we've been

00:07:08.550 --> 00:07:12.300
doing, you can always represent
them by a matrix

00:07:12.300 --> 00:07:13.550
multiplication.

00:07:17.060 --> 00:07:19.045
So this leads to a very
interesting idea.

00:07:22.650 --> 00:07:25.700
When you put something in
reduced row echelon form, let

00:07:25.700 --> 00:07:26.950
me do it up here.

00:07:29.730 --> 00:07:32.180
Actually, let's just finish what
we started with this guy.

00:07:32.180 --> 00:07:34.230
Let's put this guy in reduced
row echelon form.

00:07:37.440 --> 00:07:39.080
Let me call this first S.

00:07:39.080 --> 00:07:40.400
Let's call that S1.

00:07:40.400 --> 00:07:42.730
So this guy right here
is equal to that

00:07:42.730 --> 00:07:46.050
first S1 times A.

00:07:46.050 --> 00:07:47.580
We already showed that
that's true.

00:07:47.580 --> 00:07:50.040
Now let's perform another
transformation.

00:07:50.040 --> 00:07:53.170
Let's just do another set of
row operations to get us to

00:07:53.170 --> 00:07:55.000
reduced row echelon form.

00:07:55.000 --> 00:07:58.860
So let's keep our middle
row the same, 0, 1, 2.

00:07:58.860 --> 00:08:02.190
And let's replace the first row
with the first row plus

00:08:02.190 --> 00:08:04.630
the second row, because I
want to make this a 0.

00:08:04.630 --> 00:08:06.980
So 1 plus 0 is 1.

00:08:06.980 --> 00:08:10.310
Let me do it in another color.

00:08:10.310 --> 00:08:12.810
Minus 1 plus 1 is 0.

00:08:12.810 --> 00:08:15.620
Minus 1 plus 2 is 1.

00:08:15.620 --> 00:08:21.640
Now, I want to replace the third
row with, let's say the

00:08:21.640 --> 00:08:28.210
third row minus 2 times
the first row.

00:08:28.210 --> 00:08:31.300
So that's 0 minus 2,
times 0, is 0.

00:08:31.300 --> 00:08:33.820
2 minus 2, times 1, is 0.

00:08:33.820 --> 00:08:37.350
5 minus 2, times 2, is 1.

00:08:37.350 --> 00:08:40.130
5 minus 4 is 1.

00:08:40.130 --> 00:08:41.549
We're almost there.

00:08:41.549 --> 00:08:44.870
We just have to zero out
these guys right there.

00:08:44.870 --> 00:08:47.220
Let's see if we can get this
into reduced row echelon form.

00:08:47.220 --> 00:08:47.900
So what is this?

00:08:47.900 --> 00:08:49.880
I just performed another
linear transformation.

00:08:49.880 --> 00:08:50.800
Actually, let me write this.

00:08:50.800 --> 00:08:53.710
Let's say if this was our first
linear transformation,

00:08:53.710 --> 00:08:55.390
what I just did is I performed
another linear

00:08:55.390 --> 00:08:56.660
transformation, T2.

00:08:56.660 --> 00:08:59.940
I'll write it in a different
notation, where you give me

00:08:59.940 --> 00:09:04.100
some vector, some column
vector, x1, x2, x3.

00:09:04.100 --> 00:09:05.650
What did I just do?

00:09:05.650 --> 00:09:08.390
What was the transformation
that I just performed?

00:09:08.390 --> 00:09:12.380
My new vector, I made the top
row equal to the top row plus

00:09:12.380 --> 00:09:13.325
the second row.

00:09:13.325 --> 00:09:15.690
So it's x1 plus x2.

00:09:15.690 --> 00:09:17.500
I kept the second
row the same.

00:09:17.500 --> 00:09:20.580
And then the third row, I
replaced it with the third row

00:09:20.580 --> 00:09:22.920
minus 2 times the second row.

00:09:22.920 --> 00:09:25.290
That was a linear transformation
we just did.

00:09:25.290 --> 00:09:27.450
And we could represent this
linear transformation as

00:09:27.450 --> 00:09:31.300
being, we could say T2 applied
to some vector x is equal to

00:09:31.300 --> 00:09:36.120
some transformation vector
S2, times our vector x.

00:09:42.140 --> 00:09:45.340
Because if we applied this
transformation matrix to each

00:09:45.340 --> 00:09:48.590
of these columns, it's
equivalent to multiplying this

00:09:48.590 --> 00:09:50.940
guy by this transformation
matrix.

00:09:50.940 --> 00:09:53.430
So you could say that this guy
right here-- we haven't

00:09:53.430 --> 00:09:56.270
figured out what this is, but
I think you get the idea--

00:09:56.270 --> 00:09:59.200
this matrix right here is going
to be equal to this guy.

00:09:59.200 --> 00:10:03.310
It's going to be equal
to S2 times this guy.

00:10:03.310 --> 00:10:04.670
What is this guy right here?

00:10:04.670 --> 00:10:07.805
Well, this guy is equal
to S1 times A.

00:10:07.805 --> 00:10:12.510
It's going to be S2
times S1, times A.

00:10:12.510 --> 00:10:13.760
Fair enough.

00:10:16.930 --> 00:10:19.200
And you could have gotten
straight here if you just

00:10:19.200 --> 00:10:20.940
multiplied S2 times S1.

00:10:20.940 --> 00:10:22.250
This could be some
other matrix.

00:10:22.250 --> 00:10:24.610
If you just multiplied it by
A, you'd go straight from

00:10:24.610 --> 00:10:26.070
there to there.

00:10:26.070 --> 00:10:26.670
Fair enough.

00:10:26.670 --> 00:10:28.595
Now, we still haven't gotten
this guy into reduced row

00:10:28.595 --> 00:10:30.010
echelon form.

00:10:30.010 --> 00:10:31.220
So let's try to get there.

00:10:31.220 --> 00:10:33.125
I've run out of space below
him, so I'm going

00:10:33.125 --> 00:10:35.270
to have to go up.

00:10:35.270 --> 00:10:36.520
So let's go upwards.

00:10:40.620 --> 00:10:43.650
What I want to do is, I'm going
to keep the third row

00:10:43.650 --> 00:10:48.790
the same, 0, 0, 1.

00:10:48.790 --> 00:10:54.700
Let me replace the second row
with the second row minus 2

00:10:54.700 --> 00:10:56.150
times the third row.

00:10:56.150 --> 00:10:59.680
So we'll get a 0, we'll get a 1
minus 2, times 0, and we'll

00:10:59.680 --> 00:11:02.250
get a 2 minus 2, times 1.

00:11:02.250 --> 00:11:04.100
So that's a 0.

00:11:04.100 --> 00:11:06.500
Let's replaced the first
row with the first row

00:11:06.500 --> 00:11:08.280
minus the third row.

00:11:08.280 --> 00:11:10.970
So 1 minus 0 is 1.

00:11:10.970 --> 00:11:13.880
0 minus 0 is 0.

00:11:13.880 --> 00:11:19.310
1 minus 1 is 0, just
like that.

00:11:19.310 --> 00:11:21.470
Let's just actually write what
our transformation was.

00:11:21.470 --> 00:11:22.686
Let's call it T3.

00:11:22.686 --> 00:11:26.490
I'll do it in purple.

00:11:26.490 --> 00:11:30.225
T3 is the transformation of some
vector x-- let me write

00:11:30.225 --> 00:11:34.710
it like this-- of some
vector x1, x2, x3.

00:11:37.570 --> 00:11:38.290
What did we do?

00:11:38.290 --> 00:11:41.050
We replaced the first row with
the first row minus the third

00:11:41.050 --> 00:11:44.300
row, x1 minus x3.

00:11:44.300 --> 00:11:47.580
We replaced the second row with
the second row minus 2

00:11:47.580 --> 00:11:48.970
times the third row.

00:11:48.970 --> 00:11:51.870
So it's x2 minus 2 times x3.

00:11:51.870 --> 00:11:53.960
Then the third row just
stayed the same.

00:11:53.960 --> 00:11:57.510
So obviously, this could
also be represented.

00:11:57.510 --> 00:12:01.840
T3 of x could be equal to some
other transformation matrix,

00:12:01.840 --> 00:12:04.230
S3 times x.

00:12:04.230 --> 00:12:07.040
So this transformation, when
you multiply it to each of

00:12:07.040 --> 00:12:12.090
these columns, is equivalent to
multiplying this guy times

00:12:12.090 --> 00:12:14.910
this transformation matrix,
which we haven't found yet.

00:12:14.910 --> 00:12:15.560
We can write it.

00:12:15.560 --> 00:12:20.430
So this is going to be equal to
S3 times this matrix right

00:12:20.430 --> 00:12:27.150
here, which is S2, S1, A.

00:12:27.150 --> 00:12:28.330
And what do we have here?

00:12:28.330 --> 00:12:30.000
We got the identity matrix.

00:12:30.000 --> 00:12:32.070
We put it in reduced
row echelon form.

00:12:32.070 --> 00:12:33.580
We got the identity matrix.

00:12:33.580 --> 00:12:36.530
We already know from previous
videos the reduced row echelon

00:12:36.530 --> 00:12:38.750
form of something is the
identity matrix.

00:12:38.750 --> 00:12:41.830
Then we are dealing with an
invertible transformation, or

00:12:41.830 --> 00:12:44.140
an invertible matrix.

00:12:44.140 --> 00:12:46.350
Because this obviously could be
the transformation for some

00:12:46.350 --> 00:12:47.580
transformation.

00:12:47.580 --> 00:12:51.670
Let's just call this
transformation, I don't know,

00:12:51.670 --> 00:12:52.970
did I already use T?

00:12:52.970 --> 00:12:57.420
Let's just call it Tnaught for
our transformation applied to

00:12:57.420 --> 00:13:00.130
some vector x, that might
be equal to Ax.

00:13:00.130 --> 00:13:04.390
So we know that this
is invertible.

00:13:04.390 --> 00:13:06.170
We put it in reduced
row echelon form.

00:13:06.170 --> 00:13:07.850
We put its transformation
matrix in

00:13:07.850 --> 00:13:09.560
reduced row echelon form.

00:13:09.560 --> 00:13:11.130
And we got the identity
matrix.

00:13:11.130 --> 00:13:12.880
So that tells us that
this is invertible.

00:13:12.880 --> 00:13:14.990
But something even more
interesting happened.

00:13:14.990 --> 00:13:18.130
We got here by performing
some row operations.

00:13:18.130 --> 00:13:21.620
And we said those row operations
were completely

00:13:21.620 --> 00:13:26.080
equivalent to multiplying this
guy right here by multiplying

00:13:26.080 --> 00:13:29.890
our original transformation
matrix by a series of

00:13:29.890 --> 00:13:33.080
transformation matrices that
represent our row operations.

00:13:33.080 --> 00:13:37.150
And when we multiplied all this,
this was equal to the

00:13:37.150 --> 00:13:38.990
identity matrix.

00:13:38.990 --> 00:13:43.930
Now, in the last video we said
that the inverse matrix, so if

00:13:43.930 --> 00:13:48.450
this is Tnaught, Tnaught inverse
could be represented--

00:13:48.450 --> 00:13:50.850
it's also a linear
transformation-- It can be

00:13:50.850 --> 00:13:54.450
represented by some inverse
matrix that we just called A

00:13:54.450 --> 00:13:56.070
inverse times x.

00:13:56.070 --> 00:14:02.610
And we saw that the inverse
transformation matrix times

00:14:02.610 --> 00:14:06.580
our transformation matrix is
equal to the identity matrix.

00:14:06.580 --> 00:14:09.540
We saw this last time.

00:14:09.540 --> 00:14:11.060
We proved this to you.

00:14:11.060 --> 00:14:12.710
Now, something very
interesting here.

00:14:12.710 --> 00:14:16.750
We have a series of matrix
products times this guy, times

00:14:16.750 --> 00:14:20.010
this guy, that also got me
the identity matrix.

00:14:20.010 --> 00:14:23.640
So this guy right here, this
series of matrix products,

00:14:23.640 --> 00:14:29.750
this must be the same thing as
my inverse matrix, as my

00:14:29.750 --> 00:14:32.170
inverse transformation matrix.

00:14:32.170 --> 00:14:35.720
And so we could actually
calculate it if we wanted to.

00:14:35.720 --> 00:14:38.100
Just like we did, we actually
figured out what S1 was.

00:14:38.100 --> 00:14:39.560
We did it down here.

00:14:39.560 --> 00:14:41.520
We could do a similar operation
to figure out what

00:14:41.520 --> 00:14:46.370
S2 was, S3 was, and then
multiply them all out.

00:14:46.370 --> 00:14:50.810
We would have actually
constructed A inverse.

00:14:50.810 --> 00:14:53.240
I guess, something more
interesting we could do

00:14:53.240 --> 00:15:00.820
instead of doing that, what if
we applied these same matrix

00:15:00.820 --> 00:15:05.020
products to the identity
matrix.

00:15:05.020 --> 00:15:06.370
So the whole time we did
here, when we did

00:15:06.370 --> 00:15:07.950
our first row operation.

00:15:07.950 --> 00:15:10.500
So we have here, we
have the matrix A.

00:15:10.500 --> 00:15:13.120
Let's say we have an identity
matrix on the right.

00:15:13.120 --> 00:15:15.050
Let's call that I,
right there.

00:15:15.050 --> 00:15:17.930
Now, our first linear
transformation we did-- we saw

00:15:17.930 --> 00:15:20.240
that right here-- that
was equivalent to

00:15:20.240 --> 00:15:23.910
multiplying S1 times A.

00:15:23.910 --> 00:15:26.330
The first set of row operations
was this.

00:15:26.330 --> 00:15:27.510
It got us here.

00:15:27.510 --> 00:15:30.520
Now, if we perform that same set
of row operations on the

00:15:30.520 --> 00:15:32.630
identity matrix, what
are we going to get?

00:15:32.630 --> 00:15:35.050
We're going to get
the matrix S1.

00:15:35.050 --> 00:15:37.580
S1 times the identity
matrix is just S1.

00:15:37.580 --> 00:15:41.490
All of the columns of anything
times the identity times the

00:15:41.490 --> 00:15:43.760
standard basis columns, it'll
just be equal to itself.

00:15:43.760 --> 00:15:45.930
You'll just be left
with that S1.

00:15:45.930 --> 00:15:47.820
This is S1 times I.

00:15:47.820 --> 00:15:49.290
That's just S1.

00:15:49.290 --> 00:15:50.090
Fair enough.

00:15:50.090 --> 00:15:52.310
Now, you performed your next row
operation and you ended up

00:15:52.310 --> 00:15:56.320
with S2 times S1, times A.

00:15:56.320 --> 00:15:58.710
Now if you performed that same
row operation on this guy

00:15:58.710 --> 00:16:00.820
right there, what
would you have?

00:16:00.820 --> 00:16:05.430
You would have S2 times S1,
times the identity matrix.

00:16:05.430 --> 00:16:08.300
Now, our last row operation we
represented with the matrix

00:16:08.300 --> 00:16:09.800
product S3.

00:16:09.800 --> 00:16:12.690
We're multiplying it by the
transformation matrix S3.

00:16:12.690 --> 00:16:16.990
So if you did that, you
have S3, S2, S1 A.

00:16:16.990 --> 00:16:19.550
But if you perform the same
exact row operations on this

00:16:19.550 --> 00:16:24.940
guy right here, you have
S3, S2, S1, times

00:16:24.940 --> 00:16:26.360
the identity matrix.

00:16:26.360 --> 00:16:28.510
Now when you did this, when
you performed these row

00:16:28.510 --> 00:16:32.690
operations here, this got you
to the identity matrix.

00:16:32.690 --> 00:16:35.310
Well, what are these going
to get you to?

00:16:35.310 --> 00:16:37.800
When you just performed the same
exact row operations you

00:16:37.800 --> 00:16:40.270
performed on A to get to the
identity matrix, if you

00:16:40.270 --> 00:16:43.110
performed those same exact row
operations on the identity

00:16:43.110 --> 00:16:44.630
matrix, what do you get?

00:16:44.630 --> 00:16:46.990
You get this guy right here.

00:16:46.990 --> 00:16:48.790
Anything times that identity
matrix is going

00:16:48.790 --> 00:16:50.930
to be equal to itself.

00:16:50.930 --> 00:16:52.350
So what is that right there?

00:16:52.350 --> 00:16:53.600
That is A inverse.

00:16:56.370 --> 00:17:00.850
So we have a generalized way
of figuring out the inverse

00:17:00.850 --> 00:17:02.630
for transformation matrix.

00:17:02.630 --> 00:17:04.819
What I can do is, let's
say I have some

00:17:04.819 --> 00:17:07.160
transformation matrix A.

00:17:07.160 --> 00:17:09.420
I can set up an augmented
matrix where I put the

00:17:09.420 --> 00:17:13.750
identity matrix right there,
just like that, and I perform

00:17:13.750 --> 00:17:15.000
a bunch of row operations.

00:17:17.670 --> 00:17:20.060
And you could represent them
as matrix products.

00:17:20.060 --> 00:17:23.069
But you perform a bunch of row
operations on all of them.

00:17:23.069 --> 00:17:25.180
You perform the same operations
you perform on A as

00:17:25.180 --> 00:17:27.119
you would do on the
identity matrix.

00:17:27.119 --> 00:17:31.340
By the time you have A as an
identity matrix, you have A in

00:17:31.340 --> 00:17:33.250
reduced row echelon form.

00:17:33.250 --> 00:17:38.950
By the time A is like that, your
identity matrix, having

00:17:38.950 --> 00:17:42.290
performed the same exact
operations on it, it is going

00:17:42.290 --> 00:17:46.300
to be transformed into
A's inverse.

00:17:46.300 --> 00:17:50.340
This is a very useful tool for
solving actual inverses.

00:17:50.340 --> 00:17:52.150
Now, I've explained
the theoretical

00:17:52.150 --> 00:17:53.180
reason why this works.

00:17:53.180 --> 00:17:54.740
In the next video we'll
actually solve this.

00:17:54.740 --> 00:17:57.610
Maybe we'll do it for the
example that I started off

00:17:57.610 --> 00:17:59.740
with in this video.