Changing coordinate systems to help find a transformation matrix

WEBVTT
Kind: captions
Language: en

00:00:00.750 --> 00:00:02.220
Let's say we're in R2.

00:00:02.220 --> 00:00:03.130
Let me graph it.

00:00:03.130 --> 00:00:06.085
Let me draw my vertical
axis, just like that.

00:00:06.085 --> 00:00:09.220
And then let me draw
my horizontal

00:00:09.220 --> 00:00:10.870
axis, just like that.

00:00:10.870 --> 00:00:20.890
And let's say I have
a vector 1, 2.

00:00:20.890 --> 00:00:24.610
So my vector looks like this,
1, and then we go up 2.

00:00:24.610 --> 00:00:27.230
So my vector looks like this.

00:00:27.230 --> 00:00:29.110
Let me draw it.

00:00:29.110 --> 00:00:30.370
That is my vector.

00:00:30.370 --> 00:00:32.080
It's the vector 1, 2.

00:00:32.080 --> 00:00:34.650
And I want to look at
the line that is

00:00:34.650 --> 00:00:36.200
spanned by this vector.

00:00:36.200 --> 00:00:39.340
So let me just define some line
L, and it's equal to the

00:00:39.340 --> 00:00:47.850
set of t times 1, 2, where
t is any real number.

00:00:47.850 --> 00:00:50.930
So this is just a line with
a slope of 2, right?

00:00:50.930 --> 00:00:53.185
You rise 2 for every
1 you go over.

00:00:53.185 --> 00:00:57.270
So let me draw my line
just like this.

00:00:57.270 --> 00:01:00.520
There's nothing new
at this point.

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Let me draw it a little
bit better than that.

00:01:03.220 --> 00:01:04.569
There you go.

00:01:04.569 --> 00:01:06.230
And it keeps going, because
obviously, you could get

00:01:06.230 --> 00:01:09.150
negative multiples of my vector,
and it also goes here.

00:01:09.150 --> 00:01:12.230
If you look at all the points
that are specified by all of

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the scaled up or scaled down or
negative scaled versions of

00:01:15.750 --> 00:01:17.880
this vector, you're going to
get all the points on this

00:01:17.880 --> 00:01:20.640
line if you draw those vectors
in standard position.

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So this is my line
right there.

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Now, I want to construct a
linear transformation that

00:01:27.050 --> 00:01:29.730
reflects around this line.

00:01:29.730 --> 00:01:33.660
For example, if I have
this vector-- let me

00:01:33.660 --> 00:01:34.570
draw it like this.

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Let's say I have that vector
that specifies that point

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right there.

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Let me call that vector x.

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I want the transformation of
x to be essentially the

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reflection of x around
this line.

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So if I go in this direction
right here, I want this to be

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the transformation.

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I want that to be the
transformation of x, so it

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should look just like that.

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I'll do another example.

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Let's say I have this vector.

00:02:03.570 --> 00:02:05.250
Let me give you a particular
vector.

00:02:05.250 --> 00:02:08.229
Let's say I have a vector that's
orthogonal to my line.

00:02:08.229 --> 00:02:11.039
So let's say I have the vector,
I'll call it v1.

00:02:11.039 --> 00:02:13.150
And let's see, something that's
orthogonal to it, I can

00:02:13.150 --> 00:02:15.750
switch these two guys and make
one of them negative.

00:02:15.750 --> 00:02:22.040
So if I go to the vector 2,
minus 1, so if I have this

00:02:22.040 --> 00:02:26.850
vector right here, visually
it looks orthogonal.

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So if I have v2-- let
me call it v1.

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v1 is equal to 2 minus 1.

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Visually, it looks orthogonal,
but then even if you actually

00:02:39.700 --> 00:02:43.670
take the dot product 2 times 1
plus minus 1 times 2, that's

00:02:43.670 --> 00:02:44.270
going to be equal to 0.

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So this is definitely orthogonal
to my spanning

00:02:47.450 --> 00:02:49.290
vector, so it's definitely
orthogonal to every

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point on that line.

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But if I apply the
transformation to this guy

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right here, I'm just going
to flip it over the line.

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So let me draw it like this.

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So it's going to be equal to--
it's just going to look like--

00:03:00.160 --> 00:03:02.300
let me do it in a green color.

00:03:02.300 --> 00:03:05.360
So the transformation of this
guy, I'm just flipping him

00:03:05.360 --> 00:03:06.030
over this line.

00:03:06.030 --> 00:03:08.060
I'm finding the reflection
across line L.

00:03:08.060 --> 00:03:13.360
So this is going to be the
transformation of v1, and it

00:03:13.360 --> 00:03:18.430
would be-- we don't know what
the actual transformation

00:03:18.430 --> 00:03:20.340
matrix is and that's actually
going to be the subject of

00:03:20.340 --> 00:03:23.000
this video, but if you just
look at the transformation

00:03:23.000 --> 00:03:25.570
just for v1, it's going to be
the negative of that, so it's

00:03:25.570 --> 00:03:30.770
going to be minus 2, 1.

00:03:30.770 --> 00:03:33.550
So I know I want to define
this transformation.

00:03:33.550 --> 00:03:39.240
I know I want to define this
transformation from R2 to R2.

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That is a reflection.

00:03:41.240 --> 00:03:53.070
So the transformation of some
vector x is the reflection of

00:03:53.070 --> 00:03:58.770
x around or across, or however
you want to describe it,

00:03:58.770 --> 00:04:02.660
around line L, around L.

00:04:02.660 --> 00:04:05.470
Now, in the past, if we wanted
to find the transformation

00:04:05.470 --> 00:04:07.033
matrix-- we know this is a
linear transformation.

00:04:07.033 --> 00:04:11.430
I don't have to go through all
of the exercise of this is a

00:04:11.430 --> 00:04:12.580
linear transformation.

00:04:12.580 --> 00:04:15.380
But in the past, if we wanted
to find the transformation

00:04:15.380 --> 00:04:17.370
matrix for a linear
transformation, let's say we

00:04:17.370 --> 00:04:22.130
say T of x is equal to some
2-by-2 matrix, because it's a

00:04:22.130 --> 00:04:25.590
mapping from R2 to R2 times x.

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Before in the past to find A, we
would say A is equal to the

00:04:30.110 --> 00:04:36.180
transformation applied to our
first standard basis vector,

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so the transformation
applied to 1, 0.

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And then our second column in A
would be the transformation

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applied to 0, 1.

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And if we were dealing in Rn,
we would do this n times and

00:04:49.690 --> 00:04:51.420
we would have the
n basis vectors.

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We've seen this multiple,
multiple times.

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Now, in the past, when we
were constructing these

00:04:56.290 --> 00:04:59.440
transformation matrices, finding
these things were

00:04:59.440 --> 00:05:02.300
pretty straightforward, but
you'll see now it's not that

00:05:02.300 --> 00:05:03.010
straightforward.

00:05:03.010 --> 00:05:08.620
If I have 1, 0, if I have the
vector 1, 0, finding its

00:05:08.620 --> 00:05:11.380
transformation, if I were to
flip it over, it's going to

00:05:11.380 --> 00:05:13.310
look something like this.

00:05:13.310 --> 00:05:17.440
And we could do it if we were
creative with our geometry and

00:05:17.440 --> 00:05:18.260
our trigonometry.

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We could actually figure this
number out, and we could

00:05:20.860 --> 00:05:21.910
figure this number out.

00:05:21.910 --> 00:05:28.130
If I were to flip 0, 1
over, it might look

00:05:28.130 --> 00:05:29.110
something like that.

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We could figure it out,
but it's not easy.

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So let me just write
this down.

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It's not easy.

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And we want to at least-- I
mean, you know, we could do

00:05:38.220 --> 00:05:40.340
this if we had to, but if
there's an easier way, we

00:05:40.340 --> 00:05:43.570
should at least be exposed
to that easier way.

00:05:43.570 --> 00:05:45.500
So you might have
an insight here.

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We've been talking about
alternate coordinate systems,

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and this transformation, it
looks difficult in our

00:05:51.690 --> 00:05:54.870
standard coordinate system,
because we're reflecting

00:05:54.870 --> 00:05:58.010
around this line right there.

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But what if we defined a
coordinate system where

00:06:00.410 --> 00:06:03.490
reflection around this line
was more natural?

00:06:03.490 --> 00:06:05.610
And you might already see an
interesting coordinate system.

00:06:05.610 --> 00:06:10.430
What if we had a coordinate
system with v1 as one of the

00:06:10.430 --> 00:06:15.040
basis vectors and then 1, 2
as the other basis vector?

00:06:18.070 --> 00:06:20.180
Then, all of a sudden, at
least in that coordinate

00:06:20.180 --> 00:06:23.340
system, the reflection wouldn't
be around this

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inclined line.

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The reflection would be around
the second coordinate's axis.

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It seems like it would be a more
natural transformation.

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So this leads us to a very
interesting thing.

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Let me review what we've done in
the last couple of videos.

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In the last couple of videos,
we could say, look, if we're

00:06:38.020 --> 00:06:44.150
in standard coordinates, you
multiply by some matrix A and

00:06:44.150 --> 00:06:45.920
you get the transformation
of x.

00:06:45.920 --> 00:06:47.390
We've seen that multiple
times.

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This is in standard
coordinates.

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And we've also seen that we
could change our coordinates.

00:06:52.640 --> 00:06:55.480
We can get coordinates with
respect to some other basis.

00:06:55.480 --> 00:06:58.740
We can multiply x times C
inverse and then we'll get the

00:06:58.740 --> 00:07:01.980
representation of x or in
coordinates with respect to

00:07:01.980 --> 00:07:05.750
some other basis, and then we
can multiply that times some

00:07:05.750 --> 00:07:07.800
other matrix D.

00:07:07.800 --> 00:07:14.130
And we've seen-- let me write
this down-- that D is equal to

00:07:14.130 --> 00:07:18.120
C inverse times A times C.

00:07:18.120 --> 00:07:21.760
We showed this I think it was
two or three videos ago, where

00:07:21.760 --> 00:07:24.395
C is just the change
of basis matrix.

00:07:24.395 --> 00:07:27.330
C is just the matrix
that has our new

00:07:27.330 --> 00:07:29.040
basis vectors as columns.

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And C inverse is obviously
its inverse.

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So we can apply D.

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And then if we multiply D
times this B coordinate

00:07:37.300 --> 00:07:43.750
version of x, we will get the
B coordinate version of the

00:07:43.750 --> 00:07:46.970
transformation of x, so the
transformation of x

00:07:46.970 --> 00:07:49.640
represented in B coordinates.

00:07:49.640 --> 00:07:52.440
And then, you know, you could
convert back and forth between

00:07:52.440 --> 00:07:52.970
these guys.

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You could go this way if you
multiply by C inverse.

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You could go that way
if you multiply C.

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We've seen this in
multiple videos.

00:08:00.030 --> 00:08:03.140
Now, in this particular case
that we're dealing with, I

00:08:03.140 --> 00:08:05.680
want to find A.

00:08:05.680 --> 00:08:08.330
I've kind of defined my
transformation in kind of

00:08:08.330 --> 00:08:10.850
touchy-feely terms. I've just
verbally defined it, but I

00:08:10.850 --> 00:08:14.170
would like to have a
transformation matrix with

00:08:14.170 --> 00:08:16.500
respect to my standard basis.

00:08:16.500 --> 00:08:18.190
But I just showed you
that it's not easy.

00:08:18.190 --> 00:08:20.960
I'll have to break out some
serious geometry and

00:08:20.960 --> 00:08:24.480
trigonometry to figure out what
happens to 1, 0 when I

00:08:24.480 --> 00:08:27.540
reflect around the line
spanned by 1, 2.

00:08:27.540 --> 00:08:28.640
It's not easy.

00:08:28.640 --> 00:08:30.180
But what if we change basis?

00:08:30.180 --> 00:08:33.030
What if we change basis
to some new basis?

00:08:33.030 --> 00:08:36.760
Let's say we define some new
basis, where our spanning

00:08:36.760 --> 00:08:40.940
vectors or our basis vectors are
this guy, are the vectors

00:08:40.940 --> 00:08:44.150
2, 1, and the vectors 1, 2.

00:08:46.890 --> 00:08:50.240
So you can almost view that
our new horizontal axis--

00:08:50.240 --> 00:08:51.990
obviously, it's not horizontal,
but if you want to

00:08:51.990 --> 00:08:54.450
think of it this way-- is going
to be like that, and our

00:08:54.450 --> 00:08:58.360
new vertical axis
is like this.

00:08:58.360 --> 00:09:01.880
And now in this world, our
transformation is just a

00:09:01.880 --> 00:09:04.520
reflection around our
new vertical axis.

00:09:04.520 --> 00:09:09.720
So maybe it'll be easier to find
D, because this might be

00:09:09.720 --> 00:09:12.570
a simpler transformation in this
other coordinate system.

00:09:12.570 --> 00:09:16.020
And then if we can find D,
then we can solve for A.

00:09:16.020 --> 00:09:16.810
We've seen this.

00:09:16.810 --> 00:09:22.310
A is just equal to C times
D times C inverse.

00:09:22.310 --> 00:09:24.950
I showed that to you a
couple of videos ago.

00:09:24.950 --> 00:09:28.455
So let's see if it's
easier to find D.

00:09:28.455 --> 00:09:34.030
So let's just experiment
a little bit.

00:09:34.030 --> 00:09:36.260
Let's label our transformation
vectors.

00:09:36.260 --> 00:09:38.770
Let's say this is-- or
our basis vectors.

00:09:38.770 --> 00:09:39.820
Let's say that's v1.

00:09:39.820 --> 00:09:41.990
That's the same label I applied
to it over here.

00:09:41.990 --> 00:09:45.030
Let's say that this spanning
vector right here of our line

00:09:45.030 --> 00:09:47.620
that we're reflecting around,
let's call that v2.

00:09:47.620 --> 00:09:50.560
So our new coordinate
system has the basis

00:09:50.560 --> 00:09:52.910
vectors v1 and v2.

00:09:52.910 --> 00:10:05.930
Now, what is v1 represented
in our new basis?

00:10:05.930 --> 00:10:09.430
Remember, when you transfer to
a new basis, all it means is

00:10:09.430 --> 00:10:12.560
how much do your coordinates--
your first coordinate now is

00:10:12.560 --> 00:10:16.700
going to be-- how much of your
first basis vector do you need

00:10:16.700 --> 00:10:18.990
to get a linear combination
of v1 and how much

00:10:18.990 --> 00:10:19.860
of your second one?

00:10:19.860 --> 00:10:30.760
Well, v1-- is just equal to 1
times v1 plus 0 times v2.

00:10:30.760 --> 00:10:33.050
I mean, if I wanted to, I
could've found C inverse and

00:10:33.050 --> 00:10:36.280
multiplied by it, but it's
almost trivial to find the

00:10:36.280 --> 00:10:39.430
linear combination of your basis
vectors that gets v1.

00:10:39.430 --> 00:10:41.560
In fact, it's not almost
trivial, it is trivial.

00:10:41.560 --> 00:10:44.360
v1 is 1 times v1 plus
0 times v2.

00:10:44.360 --> 00:10:47.940
So its coordinates with respect
to the basis B are

00:10:47.940 --> 00:10:53.940
going to be 1 times v1
plus 0 times v2.

00:10:53.940 --> 00:10:54.910
Fair enough.

00:10:54.910 --> 00:11:01.520
Now, what is D going to be?

00:11:01.520 --> 00:11:02.590
Let's see.

00:11:02.590 --> 00:11:05.000
Let's say we want to find D
first, because if we can find

00:11:05.000 --> 00:11:10.510
D, then we can just apply this
formula here to figure out A.

00:11:10.510 --> 00:11:17.030
Now, let's just say D is
equal to-- it's going

00:11:17.030 --> 00:11:18.210
to be a 2-by-2 vector.

00:11:18.210 --> 00:11:20.890
It's still mapping
from R2 to R2.

00:11:20.890 --> 00:11:22.540
This is still a two-dimensional
space.

00:11:22.540 --> 00:11:25.700
This is going to map within this
two-dimensional space.

00:11:25.700 --> 00:11:30.730
So let's say D has just two
column vectors d1 and

00:11:30.730 --> 00:11:32.740
d2, just like that.

00:11:32.740 --> 00:11:35.690
So what is D times this
going to give us?

00:11:35.690 --> 00:11:40.040
So let me write it this way.

00:11:40.040 --> 00:11:42.380
Let me scroll down
a little bit.

00:11:42.380 --> 00:11:45.400
I'd like to show that little
chart right there.

00:11:45.400 --> 00:11:51.830
But D times the B coordinate
version of our first basis

00:11:51.830 --> 00:11:57.030
vector is going to be equal to
d1, d2-- those are the two

00:11:57.030 --> 00:12:05.420
columns of D-- d1, d2 times v1
in B coordinates, right?

00:12:05.420 --> 00:12:08.800
So it's times 1, 0.

00:12:08.800 --> 00:12:12.770
Now, what is this going
to be equal to?

00:12:12.770 --> 00:12:14.870
Well, this is just going
to be a linear

00:12:14.870 --> 00:12:16.155
combination of these columns.

00:12:16.155 --> 00:12:23.180
So it's going to be 1 times d1
plus 0 times d2, which is just

00:12:23.180 --> 00:12:26.720
d1, so it's the first
column of this.

00:12:26.720 --> 00:12:28.160
But what is that also
going to equal?

00:12:28.160 --> 00:12:32.750
If I multiply D times the B
coordinate version of x, I've

00:12:32.750 --> 00:12:34.600
just gotten the B coordinate
version of the

00:12:34.600 --> 00:12:36.550
transformation of x.

00:12:36.550 --> 00:12:46.260
So this is going to be equal to
the B coordinate version of

00:12:46.260 --> 00:12:52.730
the transformation of v1.

00:12:52.730 --> 00:12:56.540
If you just put a v1 here
instead of an x, v1's B

00:12:56.540 --> 00:13:00.480
version, right there, times D
is going to be equal to the

00:13:00.480 --> 00:13:04.940
transformation of v1's B version
Or its coordinates

00:13:04.940 --> 00:13:06.040
with respect to B.

00:13:06.040 --> 00:13:08.430
So it's just that thing
right there.

00:13:11.520 --> 00:13:13.510
So there's a couple of
interesting things here.

00:13:13.510 --> 00:13:17.540
We just got that the first
column, just by applying D

00:13:17.540 --> 00:13:20.280
times this has got to be equal
to this, we just figured out

00:13:20.280 --> 00:13:24.290
that the first column of D,
because if you take this times

00:13:24.290 --> 00:13:30.260
1, 0, which is the first basis
vector's representation in its

00:13:30.260 --> 00:13:32.880
own basis, and you're going to
see in general that when you

00:13:32.880 --> 00:13:35.670
represent these basis vectors
in their own basis, they're

00:13:35.670 --> 00:13:38.660
going to look just like your
standard basis vectors in this

00:13:38.660 --> 00:13:40.030
new coordinate system.

00:13:40.030 --> 00:13:43.240
So when you represented 2, 1 in
your new basis with respect

00:13:43.240 --> 00:13:45.850
to this, of course, it's 1 times
this plus 0 times this.

00:13:45.850 --> 00:13:47.330
It's going to be 1, 0.

00:13:47.330 --> 00:13:52.760
If you want to represent
v2 in your new basis,

00:13:52.760 --> 00:13:53.550
well, what is it?

00:13:53.550 --> 00:13:56.410
It's 0 times this guy plus
1 times this guy.

00:13:56.410 --> 00:13:59.790
It's 0 times v1 plus
1 times v2.

00:13:59.790 --> 00:14:01.690
So it's going to be 0, 1.

00:14:01.690 --> 00:14:03.510
And you're going to see this
is generally the case.

00:14:03.510 --> 00:14:06.050
And when you think about it,
it's almost ridiculously

00:14:06.050 --> 00:14:09.550
obvious that the first basis
vector is going be 1 times the

00:14:09.550 --> 00:14:12.510
first basis vector plus 0 times
the second basis vector.

00:14:12.510 --> 00:14:15.050
And if you had n basis vectors,
it would be 1 times

00:14:15.050 --> 00:14:18.360
the first basis vector plus 0
times all of the other ones.

00:14:18.360 --> 00:14:24.990
And in general, if you're
dealing with the nth basis

00:14:24.990 --> 00:14:29.340
vector and you want to express
it in your basis, you're

00:14:29.340 --> 00:14:32.870
essentially going to have a
bunch of the zeroes all the

00:14:32.870 --> 00:14:36.460
way-- because it's 1 times your
nth basis vector, and

00:14:36.460 --> 00:14:37.790
then everything else
is zeroes, so

00:14:37.790 --> 00:14:39.600
this'll be the nth term.

00:14:39.600 --> 00:14:43.290
So this applies generally, or
you could called this en, if

00:14:43.290 --> 00:14:45.310
you want to call your standard
basis vectors that way.

00:14:45.310 --> 00:14:46.440
This is a bit of an aside.

00:14:46.440 --> 00:14:50.800
I just want to show you that
this idea is generalizable.

00:14:50.800 --> 00:14:53.140
Now, what's useful about this?

00:14:53.140 --> 00:14:57.725
Well, just applying this little
logic, D times this guy

00:14:57.725 --> 00:15:01.480
is that guy, we figured out that
the first column of D is

00:15:01.480 --> 00:15:05.580
equal to the transformation
applied to our first basis

00:15:05.580 --> 00:15:08.510
vector in B coordinates.

00:15:08.510 --> 00:15:12.230
So if I wanted to rewrite D, I
could rewrite D like this.

00:15:12.230 --> 00:15:16.610
The first column is d1, which
is the same thing as the

00:15:16.610 --> 00:15:24.250
transformation applied to v1,
our first basis vector, in B

00:15:24.250 --> 00:15:25.340
coordinates.

00:15:25.340 --> 00:15:26.770
Now, what is d2?

00:15:26.770 --> 00:15:29.480
Well, let's do the same
thing to v2, to our

00:15:29.480 --> 00:15:31.690
second basis vector.

00:15:31.690 --> 00:15:40.940
So if I multiply D times the
second basis vector v2 in B

00:15:40.940 --> 00:15:44.150
coordinates, in our new
coordinate system, that is

00:15:44.150 --> 00:15:50.425
equal to d1, d2 times-- what is
this guy's representation

00:15:50.425 --> 00:15:51.640
in our new coordinate system?

00:15:51.640 --> 00:15:55.140
It's 0 times v1, plus
1 times v2.

00:15:55.140 --> 00:15:56.810
It's 0, 1.

00:15:56.810 --> 00:15:57.640
I wrote it up here.

00:15:57.640 --> 00:15:58.470
It's 0, 1.

00:15:58.470 --> 00:16:05.920
So this is going to be equal to
0 times d1 plus 1 times d2,

00:16:05.920 --> 00:16:08.320
which is equal to d2.

00:16:08.320 --> 00:16:09.450
And what is that going
to be equal to?

00:16:09.450 --> 00:16:13.260
If you multiply D times some B
coordinate representation of

00:16:13.260 --> 00:16:16.240
x, you're going to get the B
coordinate representation of

00:16:16.240 --> 00:16:18.680
the transformation of x.

00:16:18.680 --> 00:16:22.340
So this is going to be equal
to the B coordinate

00:16:22.340 --> 00:16:25.950
representation of the
transformation of v2.

00:16:28.480 --> 00:16:32.460
And just like that, we see that
column 2 in D, the second

00:16:32.460 --> 00:16:36.320
column, so this was the first
column, the second column in D

00:16:36.320 --> 00:16:37.580
is just this guy.

00:16:37.580 --> 00:16:45.985
It's the transformation of
v2 in B coordinates.

00:16:49.250 --> 00:16:52.810
Now, is this any easier to find
than this stuff up here?

00:16:52.810 --> 00:16:54.670
Remember, before, we said,
hey, if we just apply the

00:16:54.670 --> 00:16:57.550
transformation to the standard
basis vectors, we can get the

00:16:57.550 --> 00:17:00.400
standard transformation or we
can find the transformation

00:17:00.400 --> 00:17:02.630
matrix with respect to the
standard basis, and that's all

00:17:02.630 --> 00:17:03.370
we wanted to do.

00:17:03.370 --> 00:17:05.060
And we said that's not easy.

00:17:05.060 --> 00:17:07.490
Is this any easier?

00:17:07.490 --> 00:17:13.020
Well, what is the transformation
of v1?

00:17:13.020 --> 00:17:15.890
Let's go back to our original
transformation definition.

00:17:15.890 --> 00:17:18.630
This is v1 right here.

00:17:18.630 --> 00:17:25.400
The transformation of v1 ended
up being minus v1.

00:17:25.400 --> 00:17:30.710
It equals minus v1.

00:17:30.710 --> 00:17:34.870
So if we wanted to write the
transformation of v1 in B

00:17:34.870 --> 00:17:39.130
coordinates-- so let me
write this right here.

00:17:39.130 --> 00:17:44.890
So the transformation of v1,
if I wanted to reflect it

00:17:44.890 --> 00:17:47.820
around the line L, it just
was equal to minus v1.

00:17:47.820 --> 00:17:50.530
That's because v1 was orthogonal
to the line, right?

00:17:50.530 --> 00:17:51.700
That's why we picked it.

00:17:51.700 --> 00:17:53.455
If you just take the
transformation of this guy, it

00:17:53.455 --> 00:17:56.610
just flips over and you get
the minus of the vector.

00:17:56.610 --> 00:18:00.220
So what is this guy, if we want
to apply it, if we wanted

00:18:00.220 --> 00:18:02.266
to write this in
B coordinates?

00:18:06.580 --> 00:18:13.940
Well, minus v1 is just equal
to minus 1 times v1

00:18:13.940 --> 00:18:16.380
plus 0 times v2.

00:18:16.380 --> 00:18:20.540
So minus v1's coordinates, in
B coordinates, they're just

00:18:20.540 --> 00:18:25.260
going to be equal to minus 1
times v1 plus 0 times v2.

00:18:25.260 --> 00:18:28.130
Remember, these are just the
weights on your basis vector.

00:18:28.130 --> 00:18:30.950
So this thing right here is
pretty straightforward.

00:18:30.950 --> 00:18:35.810
This is just going to be equal
to minus 1 and 0, the first

00:18:35.810 --> 00:18:36.960
column of D.

00:18:36.960 --> 00:18:38.340
Now, what's the second
column of D?

00:18:38.340 --> 00:18:40.660
It's the transformation of v2.

00:18:40.660 --> 00:18:44.170
v2 was the vector that
spanned our line.

00:18:44.170 --> 00:18:46.220
This is v2 right here.

00:18:46.220 --> 00:18:49.140
So what happens to v2 when
you transform it?

00:18:49.140 --> 00:18:50.790
Well, you take the reflection
of something that's on the

00:18:50.790 --> 00:18:52.990
line, around that line, it's
just going to be the same.

00:18:52.990 --> 00:18:53.980
Nothing's going to change.

00:18:53.980 --> 00:18:55.390
You're taking a reflection
around this.

00:18:55.390 --> 00:18:57.330
You can kind of imagine spinning
that vector, but it's

00:18:57.330 --> 00:18:59.600
not going to change the
vector at all, right?

00:18:59.600 --> 00:19:04.430
So the transformation of this
guy, of 1, 2, is just

00:19:04.430 --> 00:19:05.830
going to be 1, 2.

00:19:05.830 --> 00:19:11.550
The transformation of v2 is just
going to be equal to v2.

00:19:11.550 --> 00:19:13.030
Let me write that down.

00:19:13.030 --> 00:19:19.060
The transformation of v2 is just
going to be equal to v2.

00:19:19.060 --> 00:19:21.910
As you can see, we've picked
these basis vectors because

00:19:21.910 --> 00:19:23.820
transforming them was
very natural.

00:19:23.820 --> 00:19:25.320
It was very easy.

00:19:25.320 --> 00:19:28.060
So what about this guy, if I
wanted to represent this guy

00:19:28.060 --> 00:19:29.970
in B coordinates?

00:19:29.970 --> 00:19:34.050
Well, v2 in B coordinates, we
already figured out is 0, 1.

00:19:34.050 --> 00:19:36.790
So this is just going
to be 0, 1.

00:19:36.790 --> 00:19:41.500
So now we have our
transformation matrix D with

00:19:41.500 --> 00:19:43.340
respect to basis B.

00:19:43.340 --> 00:19:46.440
It is equal to the
transformation of our first

00:19:46.440 --> 00:19:49.640
basis vector with respect to
the B coordinates, which is

00:19:49.640 --> 00:19:51.940
minus 1, 0.

00:19:51.940 --> 00:19:54.210
And then its second column is
the transformation of our

00:19:54.210 --> 00:19:59.630
second basis vector with respect
to B coordinates, so

00:19:59.630 --> 00:20:02.730
it's 0, 1, or the B coordinate
representation of the

00:20:02.730 --> 00:20:05.300
transformation of our
second basis vector.

00:20:05.300 --> 00:20:07.950
The language can get convoluted
sometimes.

00:20:07.950 --> 00:20:09.030
And that was neat.

00:20:09.030 --> 00:20:10.010
That was easy.

00:20:10.010 --> 00:20:11.980
We now have solved for D.

00:20:11.980 --> 00:20:14.470
And now that we've solved for
D, you notice that changing

00:20:14.470 --> 00:20:17.520
the basis was actually very
natural, and it was very easy

00:20:17.520 --> 00:20:19.020
to find what D is.

00:20:19.020 --> 00:20:22.360
And now that we know what D is,
we can now solve for A.

00:20:22.360 --> 00:20:25.060
We can now solve for the
transformation matrix with

00:20:25.060 --> 00:20:27.210
respect to the standard basis.

00:20:27.210 --> 00:20:30.560
So to do that, we have to figure
out C and C inverse.

00:20:30.560 --> 00:20:34.750
So C, remember, C is just the
change of basis matrix.

00:20:38.020 --> 00:20:40.160
And all that is is the
basis vectors.

00:20:40.160 --> 00:20:43.290
It's just a matrix with the
basis vectors in the column.

00:20:43.290 --> 00:20:46.030
So we have our basis
right here.

00:20:46.030 --> 00:20:50.050
So C is just 2, 1 and 1, 2.

00:20:50.050 --> 00:20:52.210
And then what is C inverse?

00:20:52.210 --> 00:20:58.200
C inverse is equal to-- well,
we have to find that this

00:20:58.200 --> 00:21:01.470
guy's determinant, so it's
2 times 2, which is 4.

00:21:01.470 --> 00:21:03.630
4 minus 1 times 1, so it's 3.

00:21:03.630 --> 00:21:08.960
So we do one over the
determinant, 1/3, times-- you

00:21:08.960 --> 00:21:11.170
swap these dudes right here.

00:21:11.170 --> 00:21:14.290
So this becomes a 2 and a 2,
and then you make these two

00:21:14.290 --> 00:21:15.530
guys negative.

00:21:15.530 --> 00:21:17.390
Negative 1, negative 1.

00:21:17.390 --> 00:21:18.360
That's C inverse.

00:21:18.360 --> 00:21:20.010
So now we're ready
to solve for A.

00:21:22.910 --> 00:21:27.410
So A is going to be equal-- let
me pick a nice color here.

00:21:27.410 --> 00:21:39.310
A is going to be equal to C,
which is 2, 1; 1, 2 times D.

00:21:39.310 --> 00:21:50.320
D is minus 1, 0; 0, 1 times C
inverse, which is 1/3, 2 minus

00:21:50.320 --> 00:21:54.030
1; minus 1, 2.

00:21:54.030 --> 00:21:55.510
So let's solve this.

00:21:55.510 --> 00:21:58.410
Let's just do some matrix-matrix
products.

00:21:58.410 --> 00:22:02.620
So let's do these two guys
first. And so what do we get?

00:22:02.620 --> 00:22:04.850
We're going to get
a 2-by-2 matrix.

00:22:04.850 --> 00:22:09.680
We get 2 times minus 1 plus 1
times 0, so this is just going

00:22:09.680 --> 00:22:11.090
to be minus 2.

00:22:11.090 --> 00:22:16.790
And then here, you get 2 times
0 plus 1 times 1, so you're

00:22:16.790 --> 00:22:18.250
just going to get 1, right?

00:22:18.250 --> 00:22:20.590
2 times 0 plus 1 times 1.

00:22:20.590 --> 00:22:23.920
And then here, 1 times minus
1 plus 2 times 0.

00:22:23.920 --> 00:22:25.500
That's just minus 1.

00:22:25.500 --> 00:22:29.175
And then you get 1 times
0 plus 2 times 1.

00:22:29.175 --> 00:22:31.030
That's just 2.

00:22:31.030 --> 00:22:32.460
And now we're going to have
to multiply this times

00:22:32.460 --> 00:22:33.870
this guy over here.

00:22:33.870 --> 00:22:35.970
Let me put the 1/3 out
front, so we can

00:22:35.970 --> 00:22:38.020
worry about that later.

00:22:38.020 --> 00:22:42.040
And then let me multiply times
this, so times 2, minus 1;

00:22:42.040 --> 00:22:44.120
minus 1, 2.

00:22:44.120 --> 00:22:47.110
And all of this is going
to be equal to A.

00:22:47.110 --> 00:22:49.060
I'll do it in yellow.

00:22:49.060 --> 00:22:52.350
So this is going to be another
2-by-2 matrix.

00:22:52.350 --> 00:22:57.830
And so we get negative 2 times
2, which is minus 4.

00:22:57.830 --> 00:22:59.080
Let me write out, so I
don't make careless

00:22:59.080 --> 00:23:01.420
mistakes Minus 4.

00:23:01.420 --> 00:23:06.560
Minus 2 times 2 plus 1 times
minus 1, so minus 1.

00:23:06.560 --> 00:23:08.320
That's that term right there.

00:23:08.320 --> 00:23:12.470
The next term is minus 2 times
minus 1, which is 2 plus 1

00:23:12.470 --> 00:23:15.420
times 2, so plus 2.

00:23:15.420 --> 00:23:16.550
And then let's do this term.

00:23:16.550 --> 00:23:22.380
Minus 1 times 2 is minus
2 plus 2 times minus 1,

00:23:22.380 --> 00:23:24.000
which is minus 2.

00:23:24.000 --> 00:23:30.040
And then finally we have minus
1 times minus 1, which is 1,

00:23:30.040 --> 00:23:31.840
plus 2 times 2.

00:23:31.840 --> 00:23:33.955
1 plus 4.

00:23:33.955 --> 00:23:35.440
Let me make sure I
did that right.

00:23:35.440 --> 00:23:38.840
Minus 2 times 2 as minus 4,
1 times minus 1-- yeah,

00:23:38.840 --> 00:23:40.000
I think I got it.

00:23:40.000 --> 00:23:42.370
And, of course, we have a 1/3
out front, so this is going to

00:23:42.370 --> 00:23:46.460
be equal to 1/3 times
the matrix.

00:23:46.460 --> 00:23:53.960
Minus 4 minus 1 is minus
5, 4; minus 4, and 5.

00:23:53.960 --> 00:24:04.560
Or it is equal to minus 5/3,
4/3, minus 4/3, and 5/3.

00:24:04.560 --> 00:24:09.470
That is our transformation
matrix A with respect to the

00:24:09.470 --> 00:24:11.950
standard basis.

00:24:11.950 --> 00:24:14.110
Now, let me check something,
because I have a suspicion

00:24:14.110 --> 00:24:16.720
because I did this before, and I
don't have what I did before

00:24:16.720 --> 00:24:19.100
in front of me, but I think
I got a different answer.

00:24:19.100 --> 00:24:20.870
So let me just make sure
I did this correctly.

00:24:20.870 --> 00:24:24.880
So C, these are our basis
vectors right there.

00:24:24.880 --> 00:24:28.340
Let's see, our basis vectors
were the vectors-- oh, well,

00:24:28.340 --> 00:24:31.330
our first basis vector was
the vector 2 minus 1.

00:24:31.330 --> 00:24:32.820
So that's where my
error came from.

00:24:32.820 --> 00:24:37.220
My first basis vector was the
vector 2 minus 1 and my second

00:24:37.220 --> 00:24:39.670
basis vector is the
vector 2, 1.

00:24:39.670 --> 00:24:44.640
So that obviously is going
to change my C.

00:24:44.640 --> 00:24:47.270
It won't change anything else,
because we really didn't use

00:24:47.270 --> 00:24:48.580
this information
anywhere else.

00:24:48.580 --> 00:24:53.380
So it's going to be 2, minus
1, so C is 2, minus 1.

00:24:53.380 --> 00:24:57.480
So C inverse, let's see, the
determinant is going to be 2

00:24:57.480 --> 00:25:03.075
times 2, which is 4 minus
1 times 1, so it's going

00:25:03.075 --> 00:25:04.400
to be 4 plus 1.

00:25:04.400 --> 00:25:06.210
So this is going to be a 5.

00:25:06.210 --> 00:25:07.235
Sorry for the error.

00:25:07.235 --> 00:25:09.200
And then we're going to
switch these two guys.

00:25:09.200 --> 00:25:12.560
And then these two guys are
going to become negative.

00:25:12.560 --> 00:25:14.080
So this guy's going to
become negative.

00:25:14.080 --> 00:25:15.630
This guy's going to be a plus.

00:25:15.630 --> 00:25:18.730
Let me rewrite that.

00:25:18.730 --> 00:25:20.590
I apologize for the error.

00:25:20.590 --> 00:25:27.000
C inverse is going to be 2, 1;
minus 1, 2 times 1/5, not 1/3.

00:25:27.000 --> 00:25:28.930
That's what that negative
1 changes.

00:25:28.930 --> 00:25:29.920
And so what are we
going to get?

00:25:29.920 --> 00:25:33.550
There's going to be 2 minus--
this is C inverse right here,

00:25:33.550 --> 00:25:34.410
so it's going to be 2 minus 1.

00:25:34.410 --> 00:25:37.810
This is a plus 1 and this
is going to be a 5.

00:25:37.810 --> 00:25:40.790
And then so this is going to
be a 5 and this is going

00:25:40.790 --> 00:25:43.600
to be a plus 1.

00:25:43.600 --> 00:25:45.850
This thing wouldn't change, or
actually, this will change,

00:25:45.850 --> 00:25:48.110
because C is now 2 minus 1.

00:25:48.110 --> 00:25:50.490
Actually, let me just redo this
part, because I realize

00:25:50.490 --> 00:25:53.080
that I made it a bit of a mess,
and just to go through

00:25:53.080 --> 00:25:55.570
this and correct this
is a bit hairy.

00:25:55.570 --> 00:25:59.820
It's easier just to redo the
whole multiplication.

00:25:59.820 --> 00:26:00.920
I apologize about that.

00:26:00.920 --> 00:26:04.460
I apologize for wasting your
time with my error.

00:26:04.460 --> 00:26:07.500
So A is equal to-- let
me rewrite them.

00:26:07.500 --> 00:26:11.460
2, 1; minus 1, 2, right?

00:26:11.460 --> 00:26:16.480
2, minus 1; 1, 2-- these
are our basis vectors.

00:26:16.480 --> 00:26:22.190
I forgot that minus sign there--
times our vector D,

00:26:22.190 --> 00:26:27.300
minus 1, 0; 0, 1-- our matrix
D-- times 1/5-- let me just

00:26:27.300 --> 00:26:30.030
write the 1/5 out here--
times 1/5.

00:26:30.030 --> 00:26:31.500
This is C inverse, so
we're going to have

00:26:31.500 --> 00:26:35.220
2, 1; minus 1, 2.

00:26:35.220 --> 00:26:37.850
And what is this going
to be equal to?

00:26:37.850 --> 00:26:40.610
So first we can do these
two guys right there.

00:26:40.610 --> 00:26:45.790
So it's going to be 1/5 times--
we have 2 times minus

00:26:45.790 --> 00:26:48.770
1 plus 1 times 0,
so it's minus 2.

00:26:48.770 --> 00:26:52.720
2 times 0 plus 1 times
1 is 1, right?

00:26:52.720 --> 00:26:55.510
You have minus 1 times minus
1 plus 2 times 0,

00:26:55.510 --> 00:26:56.940
so that's just 1.

00:26:56.940 --> 00:27:00.300
And then minus 1 times 0, which
is 0, plus 2 times 1,

00:27:00.300 --> 00:27:01.780
which is just 2.

00:27:01.780 --> 00:27:06.410
And then we multiply it times
this guy, 2, minus 1; 1, 2.

00:27:06.410 --> 00:27:07.630
And what is this going
to be equal to?

00:27:07.630 --> 00:27:13.510
This is going to be equal to 1/5
times minus 2 times 2 is

00:27:13.510 --> 00:27:18.260
minus 4 plus 1 times
1: plus 1.

00:27:18.260 --> 00:27:20.170
And then your next term is
going to be minus 2 times

00:27:20.170 --> 00:27:24.640
minus 1, which is 2, plus 1
times 2, so it's plus 2.

00:27:24.640 --> 00:27:29.560
And then we have 1 times 2,
which is 2, plus 2 times 1,

00:27:29.560 --> 00:27:30.860
which is 2, right?

00:27:30.860 --> 00:27:32.600
1 times 2, plus 2 times 1.

00:27:32.600 --> 00:27:34.800
And then you have
1 times minus 1.

00:27:34.800 --> 00:27:38.970
You have 1 times minus 1, which
is minus 1, plus 2 times

00:27:38.970 --> 00:27:41.180
2, which is 4.

00:27:41.180 --> 00:27:46.780
So this is going to be equal to
1/5 times-- we have a minus

00:27:46.780 --> 00:27:53.550
3, we have a 4, a 4,
and a minus 3.

00:27:53.550 --> 00:27:56.140
Sorry, a plus 3.

00:27:56.140 --> 00:27:59.350
Minus 4, a minus 3,
and then a plus 3.

00:27:59.350 --> 00:28:02.920
You can imagine when you deal
with matrices, arithmetic is

00:28:02.920 --> 00:28:04.240
what will trip you up.

00:28:04.240 --> 00:28:08.700
So your matrix A, contrary to
what I found out a few minutes

00:28:08.700 --> 00:28:21.180
ago, is equal to minus 3/5,
4/5, and then 4/5,

00:28:21.180 --> 00:28:24.650
3/5, just like that.

00:28:24.650 --> 00:28:28.100
So barring my slight arithmetic
error, which I got

00:28:28.100 --> 00:28:31.230
from writing down the basis
vectors incorrectly,

00:28:31.230 --> 00:28:33.060
forgetting to put that
negative number, now

00:28:33.060 --> 00:28:35.880
hopefully, you realize you can
appreciate that we found an

00:28:35.880 --> 00:28:40.630
easier way to find this
transformation matrix A with

00:28:40.630 --> 00:28:42.780
respect to the standard basis.

00:28:42.780 --> 00:28:46.290
You find the transformation
matrix D first in a more

00:28:46.290 --> 00:28:49.840
natural basis coordinate system,
and then you can solve

00:28:49.840 --> 00:28:51.010
for A from that.

00:28:51.010 --> 00:28:54.150
And you get this result down
there, which is hopefully the

00:28:54.150 --> 00:28:54.690
right answer.

00:28:54.690 --> 00:28:57.590
If my memory serves me right,
this is what I got the first

00:28:57.590 --> 00:28:59.410
time I did the problem.


Alternate basis transformation matrix example part 2

WEBVTT
Kind: captions
Language: en

00:00:00.650 --> 00:00:04.600
We've see now that we can apply
linear transformations

00:00:04.600 --> 00:00:07.460
in different coordinate systems.
The transformations

00:00:07.460 --> 00:00:09.290
that we've been performing
before have all been with

00:00:09.290 --> 00:00:11.130
respect to the standard basis.

00:00:11.130 --> 00:00:13.540
So in the last video I said,
look, in standard coordinates,

00:00:13.540 --> 00:00:17.520
if you have some vector x in
your domain and you apply some

00:00:17.520 --> 00:00:20.930
transformation, then let's say
that A is the transformation

00:00:20.930 --> 00:00:25.290
matrix with respect to the
standard basis, then you're

00:00:25.290 --> 00:00:26.300
just going to have
this mapping.

00:00:26.300 --> 00:00:28.520
You take x, you multiply it by
A, you're going to get the

00:00:28.520 --> 00:00:29.990
transformation of x.

00:00:29.990 --> 00:00:32.280
Now, in the last video and in
a couple of videos before

00:00:32.280 --> 00:00:34.070
that, or actually the one right
before that, we said,

00:00:34.070 --> 00:00:34.700
well, look.

00:00:34.700 --> 00:00:36.940
You can do the same mapping,
but just in an alternate

00:00:36.940 --> 00:00:38.060
coordinate system.

00:00:38.060 --> 00:00:43.960
You could do it in some
coordinate system with respect

00:00:43.960 --> 00:00:46.790
to some basis B, and that should
be the same thing.

00:00:46.790 --> 00:00:49.170
It should just be a different
transformation matrix.

00:00:49.170 --> 00:00:51.520
And in the last video, we
actually figured out what that

00:00:51.520 --> 00:00:53.180
different transformation
matrix is.

00:00:53.180 --> 00:00:56.940
We had a change of basis, so
let's say we had this basis

00:00:56.940 --> 00:00:57.340
right here.

00:00:57.340 --> 00:01:01.090
Let me actually copy and paste
everything so that we

00:01:01.090 --> 00:01:02.370
understand what we did.

00:01:02.370 --> 00:01:03.869
So this was the example.

00:01:03.869 --> 00:01:05.330
Let me copy it.

00:01:05.330 --> 00:01:09.620
Let me paste it up here, put all
of our takeaways from the

00:01:09.620 --> 00:01:14.870
last video up here and
paste it right here.

00:01:14.870 --> 00:01:16.950
For the last video we
said, OK, this is

00:01:16.950 --> 00:01:19.830
my basis right there.

00:01:19.830 --> 00:01:22.025
And then we said-- let
me copy and paste.

00:01:24.920 --> 00:01:30.330
That was my alternate basis, and
then I have my change of

00:01:30.330 --> 00:01:32.300
basis matrix and it's inverse.

00:01:32.300 --> 00:01:33.990
Those will be useful to
deal with, so let me

00:01:33.990 --> 00:01:36.420
copy and paste that.

00:01:36.420 --> 00:01:41.700
OK, copy, And then I'm
going to paste it.

00:01:41.700 --> 00:01:45.150
Edit, paste.

00:01:45.150 --> 00:01:47.340
Maybe I'll just write
it over there.

00:01:47.340 --> 00:01:49.325
Not maybe the best order, maybe
I should have written

00:01:49.325 --> 00:01:51.680
that first, but I think
we get the idea.

00:01:51.680 --> 00:01:56.680
Then we want to write what our
transformation matrix is with

00:01:56.680 --> 00:01:58.390
respect to the standard basis.

00:01:58.390 --> 00:02:02.790
And I wrote that write
over here.

00:02:02.790 --> 00:02:04.770
This was all from the last
problem, if you're wondering

00:02:04.770 --> 00:02:07.070
where I got all this stuff.

00:02:07.070 --> 00:02:11.690
Let me copy and then
paste that.

00:02:11.690 --> 00:02:14.000
Edit, paste.

00:02:14.000 --> 00:02:15.520
So I'll paste that.

00:02:15.520 --> 00:02:18.430
And then the whole point of the
last video is we figured

00:02:18.430 --> 00:02:21.890
out what the transformation
matrix is with respect to this

00:02:21.890 --> 00:02:26.000
basis right here.

00:02:26.000 --> 00:02:29.830
So D, which was the big takeaway
from the last video,

00:02:29.830 --> 00:02:33.280
was equal to this right here.

00:02:33.280 --> 00:02:35.860
Let me copy and paste that.

00:02:35.860 --> 00:02:36.980
Copy and paste.

00:02:36.980 --> 00:02:40.940
And so now we have all of our
takeaways in one place.

00:02:40.940 --> 00:02:42.890
Edit, paste.

00:02:42.890 --> 00:02:47.560
What I want to do in this
video is verify that D

00:02:47.560 --> 00:02:53.460
actually works, that I could
start with some vector x-- let

00:02:53.460 --> 00:02:55.140
me write it up here.

00:02:55.140 --> 00:02:57.570
Let's take some example
vector.

00:02:57.570 --> 00:03:01.900
So this transformation, it's
entire domain is R2, so let's

00:03:01.900 --> 00:03:04.530
start with some vector x.

00:03:04.530 --> 00:03:13.450
Let's say that x is equal
to 1 minus 1.

00:03:13.450 --> 00:03:15.840
Now, we could just apply the
transformation in the

00:03:15.840 --> 00:03:18.520
traditional way and get the
transformation of x.

00:03:18.520 --> 00:03:20.180
Let's just do that.

00:03:20.180 --> 00:03:23.840
The transformation of x is
just this matrix times x.

00:03:23.840 --> 00:03:25.940
And so what is that
going to equal?

00:03:25.940 --> 00:03:26.340
Let me see.

00:03:26.340 --> 00:03:28.140
Maybe I can just do it
right here in this

00:03:28.140 --> 00:03:30.260
corner to save space.

00:03:30.260 --> 00:03:32.330
So it's going to be this
matrix times x.

00:03:32.330 --> 00:03:37.210
So this first term right here is
going to be 3 times 1 plus

00:03:37.210 --> 00:03:41.520
minus 2 times minus 1,
or plus 2, right?

00:03:41.520 --> 00:03:44.810
Minus 2 times minus 1 is just 2,
so it's going to be 3 plus

00:03:44.810 --> 00:03:47.640
2, so it's going to
be equal to 5.

00:03:47.640 --> 00:03:51.560
And then the second term right
here is going to be 2 times 1

00:03:51.560 --> 00:03:53.980
plus minus 2 times minus 1.

00:03:53.980 --> 00:03:57.030
Well, that's just positive 2, so
it's going to be 2 plus 2,

00:03:57.030 --> 00:03:59.090
so that is 4.

00:03:59.090 --> 00:04:00.940
So that's just the
transformation of x.

00:04:00.940 --> 00:04:12.860
Now, what is this vector x
represented in coordinates, or

00:04:12.860 --> 00:04:17.279
I guess you could say, this
alternate basis coordinates?

00:04:17.279 --> 00:04:21.000
So what is that vector x
represented in coordinates

00:04:21.000 --> 00:04:24.000
with respect to this
basis right here?

00:04:24.000 --> 00:04:25.010
Well, you saw it before.

00:04:25.010 --> 00:04:25.910
I wrote them out here.

00:04:25.910 --> 00:04:30.150
Maybe it'll be useful
to do it right here.

00:04:30.150 --> 00:04:31.930
I'll copy this.

00:04:31.930 --> 00:04:33.280
Actually, let me copy
both of these.

00:04:33.280 --> 00:04:35.310
These will both be useful.

00:04:35.310 --> 00:04:36.640
Edit, copy.

00:04:36.640 --> 00:04:40.670
As you can see, if you want to
go from x to the x in an

00:04:40.670 --> 00:04:42.640
alternate basis or the
alternate coordinate

00:04:42.640 --> 00:04:46.080
representations of x, you
essentially multiply x times C

00:04:46.080 --> 00:04:49.580
inverse, so that's why I'm
copying and pasting it.

00:04:49.580 --> 00:04:52.470
Let me copy, and then let me put
it up here so that we can

00:04:52.470 --> 00:04:58.200
apply these, then paste
it right there.

00:04:58.200 --> 00:05:07.940
So if we want to go from x to
the B coordinates of x, I take

00:05:07.940 --> 00:05:11.120
my x and I multiply it
times C inverse.

00:05:11.120 --> 00:05:13.820
C inverse is this thing
right here.

00:05:13.820 --> 00:05:19.690
So if I take x and I multiply
it times C inverse I'l get

00:05:19.690 --> 00:05:21.220
this version of x.

00:05:21.220 --> 00:05:22.680
So let's do that.

00:05:22.680 --> 00:05:23.790
So this times that.

00:05:23.790 --> 00:05:26.430
Let me just put the minus
1/3 out front.

00:05:26.430 --> 00:05:30.610
It's going to be equal to minus
1/3 times-- let's see if

00:05:30.610 --> 00:05:33.090
we can do this one in
our head as well.

00:05:33.090 --> 00:05:38.950
So it's going to be 1 times 1
plus minus 2 times minus 1,

00:05:38.950 --> 00:05:40.430
which is just positive 2.

00:05:40.430 --> 00:05:42.880
So it's going to be 1 plus
2, so it's going

00:05:42.880 --> 00:05:44.780
to be equal to 3.

00:05:44.780 --> 00:05:50.570
And then it's minus 2 times 1,
which is minus 2 plus 1 times

00:05:50.570 --> 00:05:52.730
minus 1, which is
just minus 1.

00:05:52.730 --> 00:05:56.195
So it's minus 2 minus
1, so it's minus 3.

00:05:56.195 --> 00:06:00.480
So if we have minus 1/3 times
this, the B coordinate

00:06:00.480 --> 00:06:10.620
representation of our vector x
is going to be equal to minus

00:06:10.620 --> 00:06:15.550
1 and then 1, just like that,
which is actually interesting

00:06:15.550 --> 00:06:16.260
for this example.

00:06:16.260 --> 00:06:19.510
It just kind of swapped
the first entry

00:06:19.510 --> 00:06:20.800
and the second entry.

00:06:20.800 --> 00:06:25.190
Now, let's see what happens
when we apply D to x.

00:06:25.190 --> 00:06:29.690
So if we apply D to x, D should
be our transformation

00:06:29.690 --> 00:06:32.640
matrix if we're dealing
in the B coordinates.

00:06:32.640 --> 00:06:34.610
So let's see what happens.

00:06:34.610 --> 00:06:40.990
If we apply D to x-- let me
scroll over a little bit, just

00:06:40.990 --> 00:06:43.210
so we get a little bit
more real estate.

00:06:43.210 --> 00:06:45.620
So if we apply D to
x, what do we get?

00:06:45.620 --> 00:06:52.590
And so this is going to be the
transformation, or this should

00:06:52.590 --> 00:06:57.280
be the transformation of
x in B coordinates.

00:06:57.280 --> 00:07:01.290
So what is this going
to be equal to?

00:07:01.290 --> 00:07:04.370
You have to multiply
this times D.

00:07:04.370 --> 00:07:07.540
So it's going to be minus 1
times minus 1, which is 1,

00:07:07.540 --> 00:07:09.980
plus 0 times 1.

00:07:09.980 --> 00:07:14.050
So it's just minus 1 times
minus 1, which is 1.

00:07:14.050 --> 00:07:18.120
And then we're going to get 0
times minus 1 plus 2 times 1,

00:07:18.120 --> 00:07:20.120
so 2 times 1 is just 2.

00:07:23.850 --> 00:07:26.410
Now, for everything to work
together and assuming I

00:07:26.410 --> 00:07:30.280
haven't made any careless
mistakes, this thing, this is

00:07:30.280 --> 00:07:34.400
vector right here should be the
same as this vector if I

00:07:34.400 --> 00:07:37.950
change my basis, so if I go
from the standard basis

00:07:37.950 --> 00:07:39.420
to the basis B.

00:07:39.420 --> 00:07:41.640
When you go in that direction,
you just multiply this guy

00:07:41.640 --> 00:07:44.720
times C inverse.

00:07:44.720 --> 00:07:46.260
And I'm just using this
formula right here.

00:07:46.260 --> 00:07:48.750
If I'm in the standard basis and
I multiply by C inverse,

00:07:48.750 --> 00:07:50.140
I'm going to get the B basis.

00:07:50.140 --> 00:07:52.100
So let's see what I get.

00:07:52.100 --> 00:07:55.530
So this guy, I'm going to
multiply him times C inverse.

00:07:55.530 --> 00:08:00.510
Let me do it up here just
to get some extra space.

00:08:00.510 --> 00:08:04.450
So I'm going to multiply
the vector 5, 4.

00:08:04.450 --> 00:08:06.890
I'm going to multiply
that by C inverse.

00:08:06.890 --> 00:08:14.950
We're going to have minus 1/3
times 1 minus 2 minus 2, 1,

00:08:14.950 --> 00:08:15.660
just like that.

00:08:15.660 --> 00:08:17.550
So this is going to be equal
to-- Ill just write the minus

00:08:17.550 --> 00:08:19.590
1/3 out front.

00:08:19.590 --> 00:08:25.700
We have 1 times 5, which is
5, plus minus 2 times

00:08:25.700 --> 00:08:28.030
4, so 5 minus 8.

00:08:30.710 --> 00:08:36.490
And then we have minus 2 times
5, which is minus 10.

00:08:36.490 --> 00:08:39.230
And then we have
plus 1 times 4.

00:08:39.230 --> 00:08:41.230
Minus 2 times 5, which
is minus 10, plus 1

00:08:41.230 --> 00:08:42.609
times 4: plus 4.

00:08:42.609 --> 00:08:46.530
So this is equal to minus
1/3 times minus

00:08:46.530 --> 00:08:47.930
3, and this is what?

00:08:47.930 --> 00:08:51.360
This is minus 6.

00:08:51.360 --> 00:08:53.920
If you multiply the minus 1/3
times it, all the negatives

00:08:53.920 --> 00:08:58.700
cancel out, and you get 1 and
2, which is exactly what we

00:08:58.700 --> 00:08:59.200
needed to get.

00:08:59.200 --> 00:09:04.220
When you take this guy and you
change it's basis to basis B

00:09:04.220 --> 00:09:06.230
or you change its coordinate
system to the coordinate

00:09:06.230 --> 00:09:09.830
system with respect to B, you
multiply it by C inverse, you

00:09:09.830 --> 00:09:10.900
get that right there.

00:09:10.900 --> 00:09:15.990
So literally is the B coordinate
representation of

00:09:15.990 --> 00:09:17.080
the transformation of x.

00:09:17.080 --> 00:09:19.680
We just did it by multiplying
it by C inverse, which is

00:09:19.680 --> 00:09:23.800
exactly what we got when we took
the B coordinate version

00:09:23.800 --> 00:09:30.440
of x and we applied that matrix
that we found, the

00:09:30.440 --> 00:09:32.960
transformation matrix with
respect to the B coordinates,

00:09:32.960 --> 00:09:35.890
and you multiply it times
this guy right here.

00:09:35.890 --> 00:09:37.190
We got the same answer.

00:09:37.190 --> 00:09:39.470
So it didn't matter whether we
went this way around the

00:09:39.470 --> 00:09:41.680
little cycle or we
went this way.

00:09:41.680 --> 00:09:42.800
We got the same answer.

00:09:42.800 --> 00:09:45.350
This isn't a proof, but it shows
us that what we did in

00:09:45.350 --> 00:09:47.920
the last video at least works
for this case, and I literally

00:09:47.920 --> 00:09:49.770
did pick a random x here.

00:09:49.770 --> 00:09:52.790
And you can verify it, if you
like, for other things.

00:09:52.790 --> 00:09:55.700
Now, you should hopefully be
reasonably convinced that we

00:09:55.700 --> 00:09:58.650
can do this, that you can change
your basis and find a

00:09:58.650 --> 00:09:59.690
transformation matrix.

00:09:59.690 --> 00:10:02.570
We've shown how to do it, but
the obvious question is

00:10:02.570 --> 00:10:03.970
why do you do it?

00:10:07.220 --> 00:10:09.520
And someone actually wrote a
comment on the last video,

00:10:09.520 --> 00:10:14.330
which I think kind of captures
the art of why you do it.

00:10:14.330 --> 00:10:16.520
I'm not looking at the comment
right now, but if I remember

00:10:16.520 --> 00:10:19.750
correctly, they said their
linear algebra teacher said

00:10:19.750 --> 00:10:23.680
that linear algebra is the art
of choosing the right basis.

00:10:23.680 --> 00:10:24.930
Let me write that down.

00:10:33.330 --> 00:10:36.430
Or you could imagine, the
right coordinate system.

00:10:36.430 --> 00:10:38.440
And why is there a right
coordinate system?

00:10:38.440 --> 00:10:41.230
Maybe I'll put little quotes
inside the quotation.

00:10:41.230 --> 00:10:44.650
What does it mean to have the
right coordinate system?

00:10:44.650 --> 00:10:47.510
Well, if you look at your
original transformation matrix

00:10:47.510 --> 00:10:50.750
with respect to the standard
basis, it's fine.

00:10:50.750 --> 00:10:51.750
It's got this 2 by 2.

00:10:51.750 --> 00:10:54.340
But if you performed matrix
operations with this, you've

00:10:54.340 --> 00:10:56.100
go to do some math.

00:10:56.100 --> 00:10:58.460
And if you had to perform it
over and over, if you have to

00:10:58.460 --> 00:11:02.470
perform it on a bunch of
vectors, it is what it is.

00:11:02.470 --> 00:11:05.570
But when you transfer your
bases, when you go to a new

00:11:05.570 --> 00:11:08.950
basis, when you went to this
basis right here, all of a

00:11:08.950 --> 00:11:10.720
sudden, you find that
the transformation

00:11:10.720 --> 00:11:12.610
matrix is much simpler.

00:11:12.610 --> 00:11:14.060
It's a diagonal matrix.

00:11:14.060 --> 00:11:17.290
When you multiply a diagonal
matrix times something, you're

00:11:17.290 --> 00:11:19.870
literally just taking its
scaling factors of the first

00:11:19.870 --> 00:11:22.640
and second terms. When you
multiply this guy times some

00:11:22.640 --> 00:11:24.220
vector-- we did it here.

00:11:24.220 --> 00:11:30.080
When you multiplied this guy
times this vector, you

00:11:30.080 --> 00:11:33.460
literally just scaled the first
term times minus 1 and

00:11:33.460 --> 00:11:36.340
you scaled the second
term by 2.

00:11:36.340 --> 00:11:37.540
So it's a much simpler
operation.

00:11:37.540 --> 00:11:40.010
And you might say, hey, but we
had to do all of that work of

00:11:40.010 --> 00:11:42.470
multiplying by C inverse to get
there, and then once you

00:11:42.470 --> 00:11:45.650
get this answer, you're going
to multiply by C to go back

00:11:45.650 --> 00:11:48.340
into standard coordinates, you
know, that's a lot more work

00:11:48.340 --> 00:11:49.780
than just what you save here.

00:11:49.780 --> 00:11:52.610
But imagine if you had to
apply D multiple times.

00:11:52.610 --> 00:11:57.730
Imagine if you had to apply D
times D times D times D to x.

00:11:57.730 --> 00:11:58.570
Let me say it this way.

00:11:58.570 --> 00:12:00.920
Imagine if you had to apply A
times A times A or you have to

00:12:00.920 --> 00:12:06.210
apply A 100 times to some vector
up here, then applying

00:12:06.210 --> 00:12:09.060
A 100 times to some vector
would be much more

00:12:09.060 --> 00:12:12.870
computationally intensive than
applying D 100 times to this

00:12:12.870 --> 00:12:15.200
vector, even though you had a
little bit of overhead from

00:12:15.200 --> 00:12:18.330
converting in this direction
and then converting back.

00:12:18.330 --> 00:12:20.600
So in a lot of problems,
especially in computer science

00:12:20.600 --> 00:12:23.090
frankly, or some other
applications you might be

00:12:23.090 --> 00:12:27.830
doing, you want to pick
the right basis.

00:12:27.830 --> 00:12:30.880
The computations for many
problems get a lot simpler if

00:12:30.880 --> 00:12:35.400
you pick the right coordinate
system.


Alternate basis transformation matrix example

WEBVTT
Kind: captions
Language: en

00:00:00.390 --> 00:00:02.650
Let's review a bit of what we
learned in the last video.

00:00:02.650 --> 00:00:04.760
If I have some linear
transformation that's a

00:00:04.760 --> 00:00:09.750
mapping from rn to rn, and if
we're dealing with standard

00:00:09.750 --> 00:00:12.750
coordinates, that transformation
-- applied to

00:00:12.750 --> 00:00:15.530
some vector x in standard
cooridenties -- will be equal

00:00:15.530 --> 00:00:18.740
to the matrix a times x.

00:00:18.740 --> 00:00:20.220
So let me write this down.

00:00:20.220 --> 00:00:29.730
If we are dealing with standard
coordinates-- So I

00:00:29.730 --> 00:00:32.350
have x in standard
coordinates .

00:00:32.350 --> 00:00:36.210
If I apply the transformation,
that is equivalent to

00:00:36.210 --> 00:00:39.770
multiplying x by a.

00:00:39.770 --> 00:00:41.730
If I multiply x by a then
I'm going to get the

00:00:41.730 --> 00:00:47.220
transformation of x in
standard corodinates.

00:00:47.220 --> 00:00:50.340
This is a world that we're
very, very familiar with.

00:00:50.340 --> 00:00:54.220
Now, let's say that we have
an alternate basis to rn.

00:00:54.220 --> 00:01:02.860
So let's say that b is equal to
v1, v2, all the way to vn.

00:01:02.860 --> 00:01:05.370
So it has n linearly independent
vectors.

00:01:05.370 --> 00:01:12.310
Let's say b is a basis for rn.

00:01:12.310 --> 00:01:14.840
So it's a basis for rn, but
it's a nonstandard basis.

00:01:14.840 --> 00:01:17.880
These aren't just our standard
basis vectors.

00:01:17.880 --> 00:01:19.760
So b is a basis for rn.

00:01:19.760 --> 00:01:25.980
And let's say that c , which
just has these guys as it's

00:01:25.980 --> 00:01:32.210
column vectors, v1, v2, all the
way to vn , is the change

00:01:32.210 --> 00:01:44.710
of basis matrix for
the basis b.

00:01:44.710 --> 00:01:47.800
Now, we've learned -- we've
seen this several times

00:01:47.800 --> 00:01:53.750
already -- that if I have some
vector x in rn represented in

00:01:53.750 --> 00:01:57.310
b coordinates, or in coordinates
with respect to b,

00:01:57.310 --> 00:02:01.310
I can multiply it by the change
of basis matrix and

00:02:01.310 --> 00:02:04.520
then I'll get just the standard
coordinates for x.

00:02:04.520 --> 00:02:08.270
Or, if you multiply both sides
of this equation by c inverse,

00:02:08.270 --> 00:02:12.760
you can get -- And, if I start
with the standard coordinates

00:02:12.760 --> 00:02:16.710
for x , I can multiply it by c
inverse and then I can get the

00:02:16.710 --> 00:02:20.140
b coordinates for x, or the
alternate nonstandard

00:02:20.140 --> 00:02:21.710
coordinates for x.

00:02:21.710 --> 00:02:23.810
So we've seen both
of these before.

00:02:23.810 --> 00:02:26.430
So let's apply that to this
little diagram here.

00:02:26.430 --> 00:02:29.300
So if I want to get x and if
I wanted to write it in

00:02:29.300 --> 00:02:31.980
nonstandard coordinates,
what do I do?

00:02:31.980 --> 00:02:39.400
Well, if I have x, and if I want
to write it, what do I

00:02:39.400 --> 00:02:42.710
multiply it by if I want to go
to nonstandard coordinates?

00:02:42.710 --> 00:02:45.410
Well, I multiply it
by c inverse.

00:02:45.410 --> 00:02:49.140
If I multiply it by c inverse--
Whatever I write

00:02:49.140 --> 00:02:49.840
next to this line.

00:02:49.840 --> 00:02:54.370
You could say, what matrix do
you have to multiply by to get

00:02:54.370 --> 00:02:56.310
to the other end point
on your line?

00:02:56.310 --> 00:03:03.380
So I multiply x by c inverse
then I get the b

00:03:03.380 --> 00:03:07.490
coordinates for x.

00:03:07.490 --> 00:03:20.920
So these are coordinates
with respect to b.

00:03:20.920 --> 00:03:23.480
I could do the same thing here
with the transformation of x.

00:03:23.480 --> 00:03:25.910
This is just the standard
representation of the

00:03:25.910 --> 00:03:28.000
transformation of x.

00:03:28.000 --> 00:03:32.100
So I could multiply it by c
inverse if we want to go in

00:03:32.100 --> 00:03:33.030
that direction.

00:03:33.030 --> 00:03:37.650
And then we're going to get
the transformation of x

00:03:37.650 --> 00:03:40.490
represented in b coordinates.

00:03:40.490 --> 00:03:44.580
Now, in the last video what we
saw was, hey, why do these

00:03:44.580 --> 00:03:45.250
separately?

00:03:45.250 --> 00:03:48.730
Maybe there is some matrix --
and we found out what it is --

00:03:48.730 --> 00:03:52.850
maybe there's some matrix d that
if we multiply this guy

00:03:52.850 --> 00:03:55.760
times it, I can go straight from
the b coordinates of x to

00:03:55.760 --> 00:03:59.290
the b coordinates of the
transformation of x.

00:03:59.290 --> 00:04:01.600
And we said that is matrix d.

00:04:01.600 --> 00:04:04.400
And in that last video we
showed that d can be

00:04:04.400 --> 00:04:07.090
represented by a-- Actually you
could go around the circle

00:04:07.090 --> 00:04:09.060
and rederive it, if you like.

00:04:09.060 --> 00:04:13.200
But we found out that -- let me
write in another color --

00:04:13.200 --> 00:04:19.810
that d is equal to c inverse
times a times c.

00:04:19.810 --> 00:04:22.270
Now, this is all a review of
everything that we learned in

00:04:22.270 --> 00:04:22.750
the last video.

00:04:22.750 --> 00:04:25.070
Hopefully, it clarified things
up a little bit.

00:04:25.070 --> 00:04:28.750
It's nice to just realize that
these are just alternate ways

00:04:28.750 --> 00:04:29.910
of doing the same thing.

00:04:29.910 --> 00:04:32.440
Both of these are the
transformation.

00:04:32.440 --> 00:04:34.650
When you multiply by a you're
applying the same

00:04:34.650 --> 00:04:36.440
transformation when
you multiply by d.

00:04:36.440 --> 00:04:39.030
You're just doing it in a
different coordinate system.

00:04:39.030 --> 00:04:41.610
Different coordinate systems
are just different ways of

00:04:41.610 --> 00:04:43.350
representing the same vector.

00:04:43.350 --> 00:04:49.020
This and this are different
labels for the same vector.

00:04:49.020 --> 00:04:53.510
This and this are different
labels for the same vector.

00:04:53.510 --> 00:04:56.990
So these are both performing
the transformation d.

00:04:56.990 --> 00:04:59.280
Now, this was a relation we
got in the last video.

00:04:59.280 --> 00:05:03.300
That if we have our change of
basis matrix, we have it's

00:05:03.300 --> 00:05:09.190
inverse, and we have just our
standard basis linear

00:05:09.190 --> 00:05:11.460
transformation matrix, we're
able to get this.

00:05:11.460 --> 00:05:12.970
Let's see if we can
go the other way.

00:05:12.970 --> 00:05:15.440
If we have d, can
we solved for a?

00:05:15.440 --> 00:05:17.900
Well, if you multiply both sides
of this equation on the

00:05:17.900 --> 00:05:24.280
right by c inverse, you get dc
inverse is equal to c inverse

00:05:24.280 --> 00:05:26.150
acc inverse.

00:05:26.150 --> 00:05:28.940
I just put a c inverse on the
right-hand side of both sides

00:05:28.940 --> 00:05:29.890
of this equation.

00:05:29.890 --> 00:05:32.300
This is going to be the
identity matrix, so

00:05:32.300 --> 00:05:33.780
we can ignore it.

00:05:33.780 --> 00:05:36.800
And then let's multiply both
sides on the left by c.

00:05:36.800 --> 00:05:42.690
So then you get cdc inverse
is equal to cc inverse a.

00:05:42.690 --> 00:05:44.830
And this is going to be
the identity matrix.

00:05:44.830 --> 00:05:50.520
And then you're left with a is
equal to c times d c inverse.

00:05:50.520 --> 00:05:52.040
Which is another interesting
result.

00:05:52.040 --> 00:05:55.380
It's another thing to
put in our tool kit.

00:05:55.380 --> 00:05:56.460
Now, everything I've been doing

00:05:56.460 --> 00:05:58.070
has been fairly abstract.

00:05:58.070 --> 00:06:01.880
Let's actually apply some of
these principles with a real

00:06:01.880 --> 00:06:04.590
concrete example.

00:06:04.590 --> 00:06:11.310
So let's say that I have a
transformation T -- I'll keep

00:06:11.310 --> 00:06:13.900
these guys around just because
they might be useful -- that

00:06:13.900 --> 00:06:18.270
is a mapping from R2 to R2.

00:06:18.270 --> 00:06:22.150
And let's say that the
transformation matrix for T,

00:06:22.150 --> 00:06:27.580
so let's say that T of x in
standard coordinates, is equal

00:06:27.580 --> 00:06:38.360
to the matrix 3, 2, minus 2,
minus 2, minus 2 times x.

00:06:38.360 --> 00:06:41.080
In the example we just said,
this would be our

00:06:41.080 --> 00:06:44.230
transformation matrix with
respect to the standard basis.

00:06:44.230 --> 00:06:46.750
And we could call that
a right there.

00:06:46.750 --> 00:06:48.525
Now let's say we have some
alternate basis.

00:06:52.110 --> 00:06:58.720
So alternate R2 basis.

00:06:58.720 --> 00:07:00.680
Let's call that b, because
we've been

00:07:00.680 --> 00:07:02.400
calling it b so far.

00:07:02.400 --> 00:07:05.470
And let's say this alternate
R2 basis is

00:07:05.470 --> 00:07:11.600
vectors 1, 2 and 2, 1.

00:07:11.600 --> 00:07:14.140
So let's see, given this
alternate basis, whether we

00:07:14.140 --> 00:07:17.280
can come up for a transformation
matrix in that

00:07:17.280 --> 00:07:18.660
coordinate world.

00:07:18.660 --> 00:07:31.090
So we're looking for some matrix
d such that, if I apply

00:07:31.090 --> 00:07:40.490
my transformation to x and b
coordinates, or in coordinates

00:07:40.490 --> 00:07:44.410
with respect to this alternate
basis, it should be equal to

00:07:44.410 --> 00:07:46.860
this matrix.

00:07:46.860 --> 00:07:54.390
It should be equal to d times
x in the v coordinates.

00:07:54.390 --> 00:07:55.950
So this is what I'm
looking for.

00:07:55.950 --> 00:07:56.860
I'm looking for that.

00:07:56.860 --> 00:07:59.370
Or, if we go back to our
diagram, I'm looking for that.

00:07:59.370 --> 00:08:02.690
You give me x and b coordinates
and you multiply

00:08:02.690 --> 00:08:05.020
it by d, and I'm going to give
you the transformation of x

00:08:05.020 --> 00:08:06.280
and b coordiniates.

00:08:06.280 --> 00:08:10.830
Now, just applying it to this
concrete example here.

00:08:10.830 --> 00:08:12.230
We have this formula
right here.

00:08:12.230 --> 00:08:14.000
This is the formula
for d, which we

00:08:14.000 --> 00:08:15.320
proved in the last video.

00:08:15.320 --> 00:08:17.650
So we have to figure
out c inverse.

00:08:17.650 --> 00:08:22.610
So what is the change of
basis matrix for b.

00:08:22.610 --> 00:08:25.770
I want to leave this up here.

00:08:25.770 --> 00:08:37.865
So change of basis matrix for b
is just going to be -- let's

00:08:37.865 --> 00:08:38.730
just call it c.

00:08:38.730 --> 00:08:42.539
And it's going to be the basis
vectors for b's within the

00:08:42.539 --> 00:08:45.820
columns so 1, 2 and 2, 1.

00:08:45.820 --> 00:08:47.835
And then we're going to want
to figure out it's inverse.

00:08:47.835 --> 00:08:50.780
So let's figure out it's
determinant first. So the

00:08:50.780 --> 00:08:55.490
determinate of c is equal to
1 times 1 minus 2 times 2.

00:08:55.490 --> 00:08:58.340
So 1 minus 4 is minus 3.

00:08:58.340 --> 00:09:01.950
And so c inverse is going to
be equal to 1 over the

00:09:01.950 --> 00:09:03.070
determinent.

00:09:03.070 --> 00:09:07.410
1 over minus 3 or minus
1/3 times -- We

00:09:07.410 --> 00:09:08.550
switch these two guys.

00:09:08.550 --> 00:09:11.170
So we switched the 1 and the 1,
and then we make these two

00:09:11.170 --> 00:09:14.610
guys negative minus
2, minus 2.

00:09:14.610 --> 00:09:16.230
That is c inverse.

00:09:16.230 --> 00:09:22.160
So this d vector right here is
going to be equal to c inverse

00:09:22.160 --> 00:09:26.830
times a times the transformation
matrix with

00:09:26.830 --> 00:09:29.820
respect to the standard
basis times c.

00:09:29.820 --> 00:09:31.070
Let me write it down here.

00:09:34.410 --> 00:09:39.740
So the d that we're looking for
is going to be equal to c

00:09:39.740 --> 00:09:43.970
inverse times a times c.

00:09:43.970 --> 00:09:51.170
Which is equal to-- c inverse
is minus 1/3 times 1 minus 2

00:09:51.170 --> 00:10:00.570
minus 2, 1 times a -- Let me do
this in a different color,

00:10:00.570 --> 00:10:02.200
I like to switch colors.

00:10:02.200 --> 00:10:07.770
So c inverse times a -- a is
right there -- times 3 minus

00:10:07.770 --> 00:10:12.330
2, 2 minus 2 times c.

00:10:12.330 --> 00:10:13.820
c is right there.

00:10:13.820 --> 00:10:15.300
I'll do it in yellow.

00:10:15.300 --> 00:10:20.410
Times c, which is 1,
2 and then 2, 1.

00:10:20.410 --> 00:10:22.880
Let's do this piece by piece.

00:10:22.880 --> 00:10:23.870
Let's work through this.

00:10:23.870 --> 00:10:26.170
So what is this piece going
to be equal to?

00:10:26.170 --> 00:10:27.850
We have a 2 by 2
times a 2 by 2.

00:10:27.850 --> 00:10:31.730
That's going to give us
another 2 by 2 matrix.

00:10:31.730 --> 00:10:34.980
So this first term right here is
going to be 3 times 1 plus

00:10:34.980 --> 00:10:37.200
minus 2 times 2.

00:10:37.200 --> 00:10:40.025
So 3, 3 minus 4.

00:10:40.025 --> 00:10:42.160
So it's going to be
minus 1, right?

00:10:42.160 --> 00:10:44.490
3 times 1 plus minus
2 times 2.

00:10:44.490 --> 00:10:46.130
Right, it's minus 1.

00:10:46.130 --> 00:10:51.490
Then you have 3 times 2, which
is 6, minus 2 times 1.

00:10:51.490 --> 00:10:53.330
So that is 4.

00:10:53.330 --> 00:10:56.480
3 times 2 minus 2 is 4.

00:10:56.480 --> 00:10:59.130
And then when you go down
here, 2 times 1

00:10:59.130 --> 00:11:00.740
minus 2 times 2.

00:11:00.740 --> 00:11:01.920
That's 2 minus 4.

00:11:01.920 --> 00:11:03.630
That's minus 2.

00:11:03.630 --> 00:11:06.860
And then 2 times 2 is
4 minus 2 times 1.

00:11:06.860 --> 00:11:09.610
So 4 minus 2 is just 2.

00:11:09.610 --> 00:11:14.990
So our matrix d is going to be
equal to minus 1/3 times this

00:11:14.990 --> 00:11:19.560
guy -- 1 minus 2 minus 2, 1 --
times this guy, which was just

00:11:19.560 --> 00:11:22.030
the product of those
two matrices.

00:11:22.030 --> 00:11:23.320
Now let's figure out
what this is.

00:11:23.320 --> 00:11:25.760
If I take the product of these
two guys it's going to be

00:11:25.760 --> 00:11:27.850
another 2 by 2 matrix.

00:11:27.850 --> 00:11:31.210
So 1 times minus 1.

00:11:31.210 --> 00:11:37.560
which is minus 1, plus minus
2 times minus 2--

00:11:37.560 --> 00:11:38.700
So let me be sure.

00:11:38.700 --> 00:11:44.130
So minus 2 times minus 2
is 4, and then 1 times

00:11:44.130 --> 00:11:45.440
minus 1 is minus 1.

00:11:45.440 --> 00:11:48.060
So it's going to be 3.

00:11:48.060 --> 00:11:51.130
And then we go to
the next term.

00:11:51.130 --> 00:11:56.700
We have 1 times 4 plus
minus 2 times 2.

00:11:56.700 --> 00:12:01.130
So that's 4 minus
4, which is 0.

00:12:01.130 --> 00:12:05.200
And then we have minus 2 times
minus 1 which is 2 plus 1

00:12:05.200 --> 00:12:06.750
times minus 2.

00:12:06.750 --> 00:12:08.710
So that is 0.

00:12:08.710 --> 00:12:12.600
And then finally, we have minus
2 times 4, which is

00:12:12.600 --> 00:12:16.230
minus 8, right?

00:12:16.230 --> 00:12:18.175
Plus 1 times 2.

00:12:18.175 --> 00:12:21.470
So minus 8 -- minus 2 times
4 is minus 8 --

00:12:21.470 --> 00:12:24.150
plus 2 is minus 6.

00:12:24.150 --> 00:12:27.230
And all that times minus 1/3.

00:12:27.230 --> 00:12:30.950
So this is going to be equal
to-- 3 times minus 1/3 is

00:12:30.950 --> 00:12:34.030
minus 1 --0 and then 0.

00:12:34.030 --> 00:12:39.460
Minus 6 times minus 1/3 is 2.

00:12:39.460 --> 00:12:47.410
So d is now our transformation
matrix with respect

00:12:47.410 --> 00:12:51.020
to the basis b.

00:12:51.020 --> 00:12:53.550
So we were able to figure
it out just applying

00:12:53.550 --> 00:12:55.490
this formula here.

00:12:55.490 --> 00:12:59.570
Now, what happens -- Actually
I'll save that

00:12:59.570 --> 00:13:00.350
for the next video.

00:13:00.350 --> 00:13:02.155
Where we actually show
that it works.

00:13:04.800 --> 00:13:07.440
We can actually take some
vectors x, apply the

00:13:07.440 --> 00:13:10.420
transformation or apply the
change of coordinates, get to

00:13:10.420 --> 00:13:12.000
this, and then apply d.

00:13:12.000 --> 00:13:15.300
And then maybe we could go up
that way, multiply by c to get

00:13:15.300 --> 00:13:15.715
the transformation.

00:13:15.715 --> 00:13:17.410
It's going to be equivalent
to a.

00:13:17.410 --> 00:13:19.490
I'll do that in the
next video.


Transformation matrix with respect to a basis

WEBVTT
Kind: captions
Language: en

00:00:00.690 --> 00:00:04.420
Let's say I've got some linear
transformation T that is a

00:00:04.420 --> 00:00:08.970
mapping from Rn to Rn.

00:00:08.970 --> 00:00:14.570
So if this is its domain, which
is just Rn, then its

00:00:14.570 --> 00:00:16.670
codomain is also Rn.

00:00:19.170 --> 00:00:23.400
If you give me some vector in
our domain, let's call that

00:00:23.400 --> 00:00:30.650
vector x, then T will map it
to some other member of Rn,

00:00:30.650 --> 00:00:31.770
which is also the codomain.

00:00:31.770 --> 00:00:33.790
So it'll map it over here.

00:00:33.790 --> 00:00:37.690
We could call that the mapping
of T, or the mapping

00:00:37.690 --> 00:00:39.700
of x, or T of x.

00:00:39.700 --> 00:00:42.850
Since T is a linear
transformation, we know that

00:00:42.850 --> 00:00:46.740
the mapping of x to its codomain
is equivalent to x

00:00:46.740 --> 00:00:50.980
being multiplied by
some matrix A.

00:00:50.980 --> 00:00:53.470
So we know that this thing right
here is equal to some

00:00:53.470 --> 00:00:55.600
matrix A times x.

00:00:55.600 --> 00:00:58.880
You've seen all of this
multiple, multiple times.

00:00:58.880 --> 00:01:00.720
Just to make sure we understand
the wording

00:01:00.720 --> 00:01:06.960
properly, we said we've used
the word that A is the-- we

00:01:06.960 --> 00:01:08.750
could either call it the matrix
for T, or let's say

00:01:08.750 --> 00:01:21.710
it's the transformation
matrix for T.

00:01:21.710 --> 00:01:24.950
Now, in the last couple of
videos, we've learned that the

00:01:24.950 --> 00:01:27.860
same vector can be represented
in different ways.

00:01:27.860 --> 00:01:30.810
It can be represented in
different coordinate systems.

00:01:30.810 --> 00:01:35.180
When I just write the vector x
like that, we just assume that

00:01:35.180 --> 00:01:37.980
it's being represented in
standard coordinates, or it's

00:01:37.980 --> 00:01:41.690
being represented with respect
to the standard basis.

00:01:41.690 --> 00:01:45.050
So let's be a little bit
more particular.

00:01:45.050 --> 00:01:50.530
This A is the transformation
for T only when x is

00:01:50.530 --> 00:01:54.180
represented in standard
coordinates, or only when x is

00:01:54.180 --> 00:01:56.930
written in coordinates with
respect to the standard basis.

00:01:56.930 --> 00:01:58.875
So let me write a little
qualifier here.

00:02:01.820 --> 00:02:13.590
A is the transformation matrix
for T with respect to the

00:02:13.590 --> 00:02:14.840
standard basis.

00:02:17.980 --> 00:02:20.120
I never wrote this
blue part before.

00:02:20.120 --> 00:02:24.270
I never even said this blue part
before, because the only

00:02:24.270 --> 00:02:27.560
coordinate system we were
dealing with was the standard

00:02:27.560 --> 00:02:30.000
coordinate system or the
coordinates with respect to

00:02:30.000 --> 00:02:30.950
the standard basis.

00:02:30.950 --> 00:02:33.580
But now we know that there are
multiple coordinate systems.

00:02:33.580 --> 00:02:35.810
There are multiple ways to
represent this vector.

00:02:35.810 --> 00:02:38.690
There are multiple ways to
represent that vector, because

00:02:38.690 --> 00:02:42.060
Rn has multiple spanning
bases.

00:02:42.060 --> 00:02:45.010
There are multiple bases that
can represent Rn, and each of

00:02:45.010 --> 00:02:48.630
those bases can generate a
coordinate system where you

00:02:48.630 --> 00:02:54.450
can represent any vector in Rn
with coordinates with respect

00:02:54.450 --> 00:02:55.740
to any of those bases.

00:02:55.740 --> 00:02:58.470
So that last part I said was a
bit of a mouthful, so let me

00:02:58.470 --> 00:03:00.880
make it a little bit
more concrete.

00:03:00.880 --> 00:03:05.980
Let's say that I have some basis
B that's made up of n--

00:03:05.980 --> 00:03:07.840
it has to be linearly
independent.

00:03:07.840 --> 00:03:14.010
That's the definition of a
basis-- of n vectors v1, v2,

00:03:14.010 --> 00:03:17.480
all the way to vn.

00:03:17.480 --> 00:03:20.150
Now, these are n linearly
independent vectors.

00:03:20.150 --> 00:03:22.480
Each of these are
members of Rn.

00:03:22.480 --> 00:03:35.030
So B is a basis for Rn, which is
just another way of saying

00:03:35.030 --> 00:03:38.060
that all of these vectors are
linearly independent and any

00:03:38.060 --> 00:03:40.690
vector in Rn can be represented
as a linear

00:03:40.690 --> 00:03:43.440
combination of these guys, which
is another way of saying

00:03:43.440 --> 00:03:47.220
that any vector in Rn can be
represented with coordinates

00:03:47.220 --> 00:03:50.410
with respect to this
basis right there.

00:03:50.410 --> 00:03:52.580
So the same vector x,
I'm going to put

00:03:52.580 --> 00:03:53.770
the same dot here.

00:03:53.770 --> 00:03:56.630
When we represent it in standard
coordinates, it's

00:03:56.630 --> 00:03:59.730
just going to be that right
there, that vector x.

00:03:59.730 --> 00:04:01.540
But what if we want to represent
it in coordinates

00:04:01.540 --> 00:04:03.090
with respect to this
new basis?

00:04:03.090 --> 00:04:08.990
Well, then that same vector
x will look like this.

00:04:08.990 --> 00:04:13.240
We would denote it by this.

00:04:13.240 --> 00:04:16.660
The same vector can be
represented with respect to

00:04:16.660 --> 00:04:19.220
this basis.

00:04:19.220 --> 00:04:20.519
This could be some set
of coordinates.

00:04:20.519 --> 00:04:22.270
This would be some other set of
coordinates, but it's still

00:04:22.270 --> 00:04:24.210
representing the same basis.

00:04:24.210 --> 00:04:29.720
Likewise, this vector right
here, that vector right there,

00:04:29.720 --> 00:04:30.980
is also in Rn.

00:04:30.980 --> 00:04:34.300
So it can be represented by
some linear combination of

00:04:34.300 --> 00:04:38.100
these guys, or you can represent
it with coordinates

00:04:38.100 --> 00:04:39.920
with respect to this basis.

00:04:39.920 --> 00:04:42.940
So that same point right there,
I could represent it.

00:04:42.940 --> 00:04:44.680
So that point is this.

00:04:44.680 --> 00:04:47.700
But I could represent it with
coordinates with respect to my

00:04:47.700 --> 00:04:50.380
basis just like that.

00:04:50.380 --> 00:04:52.390
So this is an interesting
question.

00:04:52.390 --> 00:04:55.210
This should maybe bring
an interesting

00:04:55.210 --> 00:04:57.140
question into your brain.

00:04:57.140 --> 00:05:00.270
If I start off with something
that's in standard

00:05:00.270 --> 00:05:03.360
coordinates, and I apply the
transformation T-- that's like

00:05:03.360 --> 00:05:07.010
applying this matrix A to it or
multiplying that thing in

00:05:07.010 --> 00:05:09.780
standard coordinates times the
matrix A-- I then get the

00:05:09.780 --> 00:05:12.770
mapping of T in standard
coordinates.

00:05:12.770 --> 00:05:16.300
Now, what if I start off with
that thing in nonstandard

00:05:16.300 --> 00:05:18.610
coordinates if I have
coordinates with respect to

00:05:18.610 --> 00:05:20.590
this other basis here?

00:05:20.590 --> 00:05:23.365
Well, T should still
map it to this guy.

00:05:27.380 --> 00:05:29.520
The transformation, no matter
what, should always map from

00:05:29.520 --> 00:05:30.700
that dot to that dot.

00:05:30.700 --> 00:05:33.640
It shouldn't care what
your coordinates are.

00:05:33.640 --> 00:05:39.750
So T should still map to
that same exact point.

00:05:39.750 --> 00:05:42.940
T should still be a linear
transformation.

00:05:42.940 --> 00:05:45.850
It can map from x to T of x, but
that's the same thing as

00:05:45.850 --> 00:05:49.940
mapping from this kind of way
of labeling x to this way of

00:05:49.940 --> 00:05:51.100
labeling x.

00:05:51.100 --> 00:05:58.720
So we could say maybe this guy
right here could be some other

00:05:58.720 --> 00:06:01.285
matrix times this
guy over here.

00:06:01.285 --> 00:06:02.810
So let me write this
over here.

00:06:08.480 --> 00:06:10.460
These are just different
coordinate systems. I

00:06:10.460 --> 00:06:11.650
shouldn't just even say maybe.

00:06:11.650 --> 00:06:13.146
This guy should be able
to be represented.

00:06:13.146 --> 00:06:20.830
So if I represent the mapping
of x in our codomain in

00:06:20.830 --> 00:06:24.300
coordinates with respect to B--
so that's what that guy is

00:06:24.300 --> 00:06:26.980
right there-- so if I want to
represent that dot with this

00:06:26.980 --> 00:06:30.120
other coordinate system,
coordinates with respect to

00:06:30.120 --> 00:06:33.970
this basis, it should be
equal to the product

00:06:33.970 --> 00:06:35.170
of some other matrix.

00:06:35.170 --> 00:06:37.650
Let me call that
other matrix D.

00:06:37.650 --> 00:06:44.740
Some other matrix D times this
representation of x times the

00:06:44.740 --> 00:06:50.230
coordinates of x with respect
to my alternate nonstandard

00:06:50.230 --> 00:06:51.400
coordinate system.

00:06:51.400 --> 00:06:54.290
I should be able to find some
matrix D that does this.

00:06:54.290 --> 00:07:02.140
Then we would say that D is the

00:07:02.140 --> 00:07:12.270
transformation matrix for T.

00:07:12.270 --> 00:07:17.120
A assumes that you have x in
terms of standard coordinates.

00:07:17.120 --> 00:07:22.720
Now D assumes that you have x in
coordinates with respect to

00:07:22.720 --> 00:07:35.440
this basis, so with respect
to the basis B.

00:07:35.440 --> 00:07:37.550
There's no reason why we
shouldn't be able to do this.

00:07:37.550 --> 00:07:40.090
These things are just different
ways of representing

00:07:40.090 --> 00:07:42.910
the exact same vector,
the exact same

00:07:42.910 --> 00:07:44.860
dot in our sets here.

00:07:44.860 --> 00:07:47.820
So if I represent it one way,
the standard way, I multiply

00:07:47.820 --> 00:07:50.180
by A, and I get Ax.

00:07:50.180 --> 00:07:52.760
If I represent it in nonstandard
coordinates, I

00:07:52.760 --> 00:07:54.970
should be able to multiply it
by some other matrix and get

00:07:54.970 --> 00:07:58.500
another nonstandard coordinate
representation of what

00:07:58.500 --> 00:08:00.270
it gets mapped to.

00:08:00.270 --> 00:08:05.420
So let's see if we can find
some relation between

00:08:05.420 --> 00:08:08.510
D and between A.

00:08:08.510 --> 00:08:13.200
So we learned a couple of videos
ago that there's a

00:08:13.200 --> 00:08:17.900
change of basis matrix that we
can generate from this basis.

00:08:17.900 --> 00:08:19.420
It's pretty easy to generate.

00:08:19.420 --> 00:08:22.690
The change of basis matrix is
just a matrix whose columns

00:08:22.690 --> 00:08:27.865
are these basis vectors, so
v1, v2-- I shouldn't put a

00:08:27.865 --> 00:08:28.160
comma there.

00:08:28.160 --> 00:08:33.840
These are just the columns--
v2 all the way to vn.

00:08:33.840 --> 00:08:35.530
This is an n-by-n matrix.

00:08:35.530 --> 00:08:39.220
Each of these guys are members
of Rn and we have n of them.

00:08:39.220 --> 00:08:42.760
This is an n-by-n matrix where
all of the columns are

00:08:42.760 --> 00:08:45.570
linearly independent, so we
know that C is invertible.

00:08:45.570 --> 00:08:48.010
These are column vectors
right here.

00:08:48.010 --> 00:08:49.650
So we know that C
is invertible.

00:08:52.450 --> 00:08:59.040
We learned in the last two or
three videos that if we have

00:08:59.040 --> 00:09:04.860
some vector x, and it's being
represented by coordinates

00:09:04.860 --> 00:09:08.600
with respect to our basis B, we
can just multiply that by

00:09:08.600 --> 00:09:13.950
C, and we'll get our vector x.

00:09:13.950 --> 00:09:16.350
This essentially will tell us
the linear combination of

00:09:16.350 --> 00:09:18.160
these guys that'll get us x.

00:09:18.160 --> 00:09:25.980
Since C is invertible, we also
saw that if we have the

00:09:25.980 --> 00:09:30.380
standard format for x, or the
standard coordinates for x, we

00:09:30.380 --> 00:09:33.970
can multiply that
by C inverse.

00:09:33.970 --> 00:09:43.020
That will get us the coordinates
for x with respect

00:09:43.020 --> 00:09:45.080
to the basis B.

00:09:45.080 --> 00:09:48.310
These two things, if you just
multiply both sides of this

00:09:48.310 --> 00:09:51.460
equation-- let me do it in a
different color-- if you just

00:09:51.460 --> 00:09:55.310
multiply both sides of this
equation by C inverse on the

00:09:55.310 --> 00:09:57.290
left-hand side, you're
going to get this

00:09:57.290 --> 00:09:59.310
equation right there.

00:09:59.310 --> 00:10:01.450
Now given that, let's
see if we can find

00:10:01.450 --> 00:10:05.170
some type of relation.

00:10:05.170 --> 00:10:08.740
Let's see what D times
xB is equal to.

00:10:08.740 --> 00:10:13.420
So let's say if we take D times
xB, so this thing right

00:10:13.420 --> 00:10:22.740
here should be equal to D times
the representation or

00:10:22.740 --> 00:10:25.130
the coordinates of x with
respect to the basis B.

00:10:25.130 --> 00:10:26.510
That's what we're claiming.

00:10:26.510 --> 00:10:29.550
We're saying that this guy
is equal to D times the

00:10:29.550 --> 00:10:32.200
representation of x with respect
to the coordinates

00:10:32.200 --> 00:10:34.440
with respect to the basis B.

00:10:34.440 --> 00:10:37.060
Let me write all of this down.

00:10:37.060 --> 00:10:38.940
I'll do it right here, because I
think it's nice to have this

00:10:38.940 --> 00:10:40.660
graphic up here.

00:10:40.660 --> 00:10:52.400
So we can say that D times
xB is equal to

00:10:52.400 --> 00:10:53.660
this thing right here.

00:10:53.660 --> 00:10:56.960
It's the same thing as the
transformation of x

00:10:56.960 --> 00:10:59.990
represented in coordinates with
respect to B, or in these

00:10:59.990 --> 00:11:01.230
nonstandard coordinates.

00:11:01.230 --> 00:11:04.890
So it's equal to the
transformation of x

00:11:04.890 --> 00:11:08.420
represented in this coordinate
system, represented in

00:11:08.420 --> 00:11:10.460
coordinates with respect to B.

00:11:10.460 --> 00:11:12.190
We see that right there.

00:11:12.190 --> 00:11:14.370
But what is the transformation
of x?

00:11:17.710 --> 00:11:20.950
That's the same thing
as A times x.

00:11:20.950 --> 00:11:25.790
That's kind of the standard
transformation if x was

00:11:25.790 --> 00:11:28.050
represented in standard
coordinates.

00:11:28.050 --> 00:11:32.670
So this is equal to x in
standard coordinates

00:11:32.670 --> 00:11:34.900
times the matrix A.

00:11:34.900 --> 00:11:37.740
Then that will get us to this
dot in standard coordinates,

00:11:37.740 --> 00:11:39.830
but then we want to convert
it to these nonstandard

00:11:39.830 --> 00:11:42.970
coordinates just like that.

00:11:42.970 --> 00:11:49.050
Now, if we have this, how can
we just figure out what the

00:11:49.050 --> 00:11:50.550
vector Ax should look like?

00:11:50.550 --> 00:11:52.330
What this vector should
look like?

00:11:52.330 --> 00:11:54.990
Well, we can look at this
equation right here.

00:11:54.990 --> 00:11:57.570
We have this.

00:11:57.570 --> 00:11:58.920
This is the same
thing as this.

00:12:01.560 --> 00:12:02.150
Actually.

00:12:02.150 --> 00:12:03.010
we want to go the other way.

00:12:03.010 --> 00:12:06.670
We have this.

00:12:06.670 --> 00:12:08.180
We have that right there.

00:12:08.180 --> 00:12:10.090
That's this right there.

00:12:10.090 --> 00:12:13.820
We want to get just this dot
represented in regular

00:12:13.820 --> 00:12:14.680
standard coordinates.

00:12:14.680 --> 00:12:15.310
So what do we do?

00:12:15.310 --> 00:12:17.740
We multiply it by C.

00:12:17.740 --> 00:12:20.330
Let me write it this way.

00:12:20.330 --> 00:12:23.350
If we multiply both sides
of this equation times

00:12:23.350 --> 00:12:25.130
C, what do we get?

00:12:25.130 --> 00:12:28.785
We get this right here.

00:12:31.320 --> 00:12:31.780
Actually, no.

00:12:31.780 --> 00:12:34.250
I was looking at the right
equation the first time.

00:12:34.250 --> 00:12:38.490
We have this right here, which
is the same-- first intuition

00:12:38.490 --> 00:12:39.660
is always right.

00:12:39.660 --> 00:12:43.100
We have this, which is the same
thing as this right here.

00:12:43.100 --> 00:12:45.510
So this can be rewritten.

00:12:45.510 --> 00:12:50.690
This thing can be rewritten
as C inverse-- we

00:12:50.690 --> 00:12:51.690
don't have an x here.

00:12:51.690 --> 00:12:57.690
We have an Ax here, so
C inverse times Ax.

00:12:57.690 --> 00:13:03.130
The vector Ax represented in
these nonstandard coordinates

00:13:03.130 --> 00:13:07.930
is the same thing as multiplying
the inverse of our

00:13:07.930 --> 00:13:11.835
change of basis matrix
times the vector Ax.

00:13:11.835 --> 00:13:15.510
If I have my vector Ax and I
multiply it times the inverse

00:13:15.510 --> 00:13:18.160
of the change of basis matrix,
I will then have a

00:13:18.160 --> 00:13:23.460
representation of the vector
Ax in my nonstandard basis.

00:13:23.460 --> 00:13:26.500
Now, what is the vector
x equal to?

00:13:29.300 --> 00:13:33.870
Well, the vector x is equal to
our change of basis matrix

00:13:33.870 --> 00:13:37.230
times x represented in these
nonstandard coordinates.

00:13:37.230 --> 00:13:43.340
So this is going to be equal to
C inverse A times x. x is

00:13:43.340 --> 00:13:46.550
just the same thing as C.

00:13:46.550 --> 00:13:54.200
x is just C times our
nonstandard coordinates for x,

00:13:54.200 --> 00:13:56.210
just like that.

00:13:56.210 --> 00:13:58.360
Let me summarize it, just
because I waffled a little bit

00:13:58.360 --> 00:13:59.736
on this point right there
just because I got

00:13:59.736 --> 00:14:01.080
a little bit confused.

00:14:01.080 --> 00:14:04.710
If I start off with the
nonstandard representation of

00:14:04.710 --> 00:14:06.710
x, or x in coordinates
with respect to B, I

00:14:06.710 --> 00:14:10.000
multiply them times D.

00:14:10.000 --> 00:14:12.490
So if I start with this, I
multiply them times D, I get

00:14:12.490 --> 00:14:13.580
to that point right there.

00:14:13.580 --> 00:14:15.600
So this right there is
the same thing as

00:14:15.600 --> 00:14:17.000
this point right there.

00:14:17.000 --> 00:14:21.500
That point right there should
be the nonstandard

00:14:21.500 --> 00:14:27.070
representation of the
transformation of x, or the

00:14:27.070 --> 00:14:29.310
coordinates of the
transformation of x with

00:14:29.310 --> 00:14:30.730
respect to B.

00:14:30.730 --> 00:14:35.010
Now, the transformation of
x, if x is in standard

00:14:35.010 --> 00:14:37.080
coordinates, is just
A times x.

00:14:37.080 --> 00:14:38.830
So this is just A times x.

00:14:38.830 --> 00:14:41.650
But I want to represent it in
these nonstandard coordinates.

00:14:41.650 --> 00:14:47.750
Now, A times x in nonstandard
coordinates is the same thing

00:14:47.750 --> 00:14:53.290
as C inverse times A times x, if
you think this is the same

00:14:53.290 --> 00:14:54.860
thing as this.

00:14:54.860 --> 00:14:56.730
So if you have this and you
want to represent it in

00:14:56.730 --> 00:14:59.396
nonstandard coordinates, you
multiply it by C inverse, so

00:14:59.396 --> 00:15:01.360
then you'll get that
representation in nonstandard

00:15:01.360 --> 00:15:02.530
coordinates.

00:15:02.530 --> 00:15:07.150
Then finally, we say look, x is
the same thing as C times

00:15:07.150 --> 00:15:10.330
the nonstandard coordinate
representation of x.

00:15:10.330 --> 00:15:14.840
So we can replace x with
that right there.

00:15:14.840 --> 00:15:22.530
So the big takeaway here is that
D times the coordinates

00:15:22.530 --> 00:15:30.280
of x with respect to the basis
B is equal to C inverse A

00:15:30.280 --> 00:15:37.300
times C times the coordinates
of x with respect

00:15:37.300 --> 00:15:38.550
to the basis B.

00:15:41.060 --> 00:15:47.170
So D must be equal
to C inverse AC.

00:15:47.170 --> 00:16:07.240
So if D is the transformation
matrix for T with respect to

00:16:07.240 --> 00:16:26.970
the basis B-- and let me write
here-- and C is the change of

00:16:26.970 --> 00:16:41.830
basis matrix for B-- let me
write that down, might as well

00:16:41.830 --> 00:16:47.030
because this is our big
takeaway-- and A is the

00:16:47.030 --> 00:16:55.800
transformation-- I'll write it
in shorthand-- matrix for T

00:16:55.800 --> 00:17:09.910
with respect to the standard
basis, then we can say-- this

00:17:09.910 --> 00:17:15.240
is the big takeaway-- that D,
our matrix D, is equal to C

00:17:15.240 --> 00:17:20.750
inverse times A times C.

00:17:20.750 --> 00:17:24.099
That's our big takeaway from
this video, which is really

00:17:24.099 --> 00:17:25.089
interesting.

00:17:25.089 --> 00:17:26.920
I don't want you to
lose this point.

00:17:26.920 --> 00:17:31.170
We now understand that A is
just for a certain set of

00:17:31.170 --> 00:17:31.670
coordinates.

00:17:31.670 --> 00:17:34.670
But there's arbitrary different
bases that we can

00:17:34.670 --> 00:17:38.340
use to represent Rn, so we can
have different matrices that

00:17:38.340 --> 00:17:40.580
represent the linear
transformation under different

00:17:40.580 --> 00:17:43.900
coordinate systems. If we want
to figure out those different

00:17:43.900 --> 00:17:47.160
matrices for different
coordinate systems, we can

00:17:47.160 --> 00:17:52.790
essentially just construct the
change of basis matrix for the

00:17:52.790 --> 00:17:55.820
coordinate system we care about,
and then generate our

00:17:55.820 --> 00:17:59.100
new transformation matrix with
respect to the new basis by

00:17:59.100 --> 00:18:01.700
just applying this result.