0:00:00.730,0:00:06.870
For any transformation that maps[br]from Rn to Rn, we've done

0:00:06.870,0:00:09.590
it implicitly, but it's been[br]interesting for us to find the

0:00:09.590,0:00:12.460
vectors that essentially just[br]get scaled up by the

0:00:12.460,0:00:13.880
transformations.

0:00:13.880,0:00:17.230
So the vectors that have the[br]form-- the transformation of

0:00:17.230,0:00:20.950
my vector is just equal[br]to some scaled-up

0:00:20.950,0:00:22.035
version of a vector.

0:00:22.035,0:00:24.290
And if this doesn't look[br]familiar, I can jog your

0:00:24.290,0:00:25.750
memory a little bit.

0:00:25.750,0:00:27.690
When we were looking for[br]basis vectors for the

0:00:27.690,0:00:29.140
transformation--[br]let me draw it.

0:00:29.140,0:00:31.190
This was from R2 to R2.

0:00:33.970,0:00:37.110
So let me draw R2 right here.

0:00:37.110,0:00:44.310
And let's say I had the[br]vector v1 was equal to

0:00:44.310,0:00:45.870
the vector 1, 2.

0:00:45.870,0:00:48.990
And we had the lines spanned[br]by that vector.

0:00:48.990,0:00:52.000
We did this problem several[br]videos ago.

0:00:52.000,0:00:55.350
And I had the transformation[br]that flipped across this line.

0:00:55.350,0:01:01.230
So if we call that line l, T was[br]the transformation from R2

0:01:01.230,0:01:05.410
to R2 that flipped vectors[br]across this line.

0:01:05.410,0:01:13.210
So it flipped vectors[br]across l.

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So if you remember that[br]transformation, if I had some

0:01:15.740,0:01:19.050
random vector that looked like[br]that, let's say that's x,

0:01:19.050,0:01:21.640
that's vector x, then the[br]transformation of x looks

0:01:21.640,0:01:22.410
something like this.

0:01:22.410,0:01:24.640
It's just flipped across[br]that line.

0:01:24.640,0:01:26.770
That was the transformation[br]of x.

0:01:26.770,0:01:28.990
And if you remember that video,[br]we were looking for a

0:01:28.990,0:01:31.670
change of basis that would allow[br]us to at least figure

0:01:31.670,0:01:34.640
out the matrix for the[br]transformation, at least in an

0:01:34.640,0:01:35.500
alternate basis.

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And then we could figure[br]out the matrix for the

0:01:36.900,0:01:38.950
transformation in the[br]standard basis.

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And the basis we picked were[br]basis vectors that didn't get

0:01:42.790,0:01:44.950
changed much by the[br]transformation, or ones that

0:01:44.950,0:01:46.940
only got scaled by the[br]transformation.

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For example, when I took the[br]transformation of v1, it just

0:01:52.750,0:01:54.320
equaled v1.

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Or we could say that the[br]transformation of v1 just

0:01:59.380,0:02:02.800
equaled 1 times v1.

0:02:02.800,0:02:06.780
So if you just follow this[br]little format that I set up

0:02:06.780,0:02:08.860
here, lambda, in this[br]case, would be 1.

0:02:08.860,0:02:11.360
And of course, the vector[br]in this case is v1.

0:02:11.360,0:02:16.395
The transformation just[br]scaled up v1 by 1.

0:02:16.395,0:02:18.860
In that same problem, we had[br]the other vector that

0:02:18.860,0:02:22.450
we also looked at.

0:02:22.450,0:02:28.270
It was the vector minus-- let's[br]say it's the vector v2,

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which is-- let's say[br]it's 2, minus 1.

0:02:32.410,0:02:34.420
And then if you take the[br]transformation of it, since it

0:02:34.420,0:02:36.250
was orthogonal to the[br]line, it just got

0:02:36.250,0:02:37.840
flipped over like that.

0:02:37.840,0:02:39.760
And that was a pretty[br]interesting vector force as

0:02:39.760,0:02:44.960
well, because the transformation[br]of v2 in this

0:02:44.960,0:02:47.050
situation is equal to what?

0:02:47.050,0:02:48.930
Just minus v2.

0:02:48.930,0:02:50.270
It's equal to minus v2.

0:02:50.270,0:02:54.920
Or you could say that the[br]transformation of v2 is equal

0:02:54.920,0:02:58.230
to minus 1 times v2.

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And these were interesting[br]vectors for us because when we

0:03:01.870,0:03:06.390
defined a new basis with these[br]guys as the basis vector, it

0:03:06.390,0:03:09.280
was very easy to figure out[br]our transformation matrix.

0:03:09.280,0:03:12.000
And actually, that basis was[br]very easy to compute with.

0:03:12.000,0:03:14.390
And we'll explore that a little[br]bit more in the future.

0:03:14.390,0:03:16.620
But hopefully you realize that[br]these are interesting vectors.

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There was also the cases where[br]we had the planes spanned by

0:03:21.750,0:03:23.630
some vectors.

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And then we had another vector[br]that was popping out of the

0:03:25.820,0:03:27.040
plane like that.

0:03:27.040,0:03:29.320
And we were transforming things[br]by taking the mirror

0:03:29.320,0:03:31.200
image across this and we're[br]like, well in that

0:03:31.200,0:03:34.360
transformation, these red[br]vectors don't change at all

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and this guy gets[br]flipped over.

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So maybe those would make[br]for good bases.

0:03:38.290,0:03:40.250
Or those would make for[br]good basis vectors.

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And they did.

0:03:41.240,0:03:44.850
So in general, we're always[br]interested with the vectors

0:03:44.850,0:03:47.240
that just get scaled up[br]by a transformation.

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It's not going to be[br]all vectors, right?

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This vector that I drew here,[br]this vector x, it doesn't just

0:03:51.320,0:03:54.650
get scaled up, it actually gets[br]changed, this direction

0:03:54.650,0:03:56.730
gets changed.

0:03:56.730,0:04:00.360
The vectors that get scaled up[br]might switch direct-- might go

0:04:00.360,0:04:03.020
from this direction to that[br]direction, or maybe

0:04:03.020,0:04:04.430
they go from that.

0:04:04.430,0:04:07.270
Maybe that's x and then the[br]transformation of x might be a

0:04:07.270,0:04:08.460
scaled up version of x.

0:04:08.460,0:04:09.710
Maybe it's that.

0:04:12.050,0:04:16.970
The actual, I guess, line that[br]they span will not change.

0:04:16.970,0:04:19.350
And so that's what we're going[br]to concern ourselves with.

0:04:19.350,0:04:21.019
These have a special name.

0:04:21.019,0:04:23.660
And they have a special name and[br]I want to make this very

0:04:23.660,0:04:25.050
clear because they're useful.

0:04:25.050,0:04:27.360
It's not just some mathematical[br]game we're

0:04:27.360,0:04:29.970
playing, although sometimes[br]we do fall into that trap.

0:04:29.970,0:04:31.250
But they're actually useful.

0:04:31.250,0:04:34.140
They're useful for defining[br]bases because in those bases

0:04:34.140,0:04:36.730
it's easier to find[br]transformation matrices.

0:04:36.730,0:04:38.950
They're more natural coordinate[br]systems. And

0:04:38.950,0:04:41.700
oftentimes, the transformation[br]matrices in those bases are

0:04:41.700,0:04:43.620
easier to compute with.

0:04:43.620,0:04:47.060
And so these have[br]special names.

0:04:47.060,0:04:50.040
Any vector that satisfies this[br]right here is called an

0:04:50.040,0:04:57.810
eigenvector for the[br]transformation T.

0:04:57.810,0:05:01.680
And the lambda, the multiple[br]that it becomes-- this is the

0:05:01.680,0:05:12.410
eigenvalue associated with[br]that eigenvector.

0:05:16.870,0:05:19.590
So in the example I just gave[br]where the transformation is

0:05:19.590,0:05:24.020
flipping around this line,[br]v1, the vector 1, 2 is an

0:05:24.020,0:05:27.210
eigenvector of our[br]transformation.

0:05:27.210,0:05:31.080
So 1, 2 is an eigenvector.

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And it's corresponding[br]eigenvalue is 1.

0:05:42.170,0:05:43.820
This guy is also an[br]eigenvector-- the

0:05:43.820,0:05:45.270
vector 2, minus 1.

0:05:45.270,0:05:47.520
He's also an eigenvector.

0:05:47.520,0:05:50.440
A very fancy word, but all it[br]means is a vector that's just

0:05:50.440,0:05:51.920
scaled up by a transformation.

0:05:51.920,0:05:55.030
It doesn't get changed in any[br]more meaningful way than just

0:05:55.030,0:05:56.270
the scaling factor.

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And it's corresponding[br]eigenvalue is minus 1.

0:06:03.860,0:06:05.580
If this transformation--[br]I don't know what its

0:06:05.580,0:06:06.750
transformation matrix is.

0:06:06.750,0:06:07.990
I forgot what it was.

0:06:07.990,0:06:10.820
We actually figured it[br]out a while ago.

0:06:10.820,0:06:16.490
If this transformation matrix[br]can be represented as a matrix

0:06:16.490,0:06:18.180
vector product-- and it should[br]be; it's a linear

0:06:18.180,0:06:22.940
transformation-- then any[br]v that satisfies the

0:06:22.940,0:06:27.610
transformation of-- I'll say[br]transformation of v is equal

0:06:27.610,0:06:32.520
to lambda v, which also would[br]be-- you know, the

0:06:32.520,0:06:33.180
transformation of [? v ?]

0:06:33.180,0:06:36.380
would just be A times v.

0:06:36.380,0:06:39.390
These are also called[br]eigenvectors of A, because A

0:06:39.390,0:06:41.570
is just really the matrix[br]representation of the

0:06:41.570,0:06:43.090
transformation.

0:06:43.090,0:06:51.560
So in this case, this would be[br]an eigenvector of A, and this

0:06:51.560,0:06:53.690
would be the eigenvalue[br]associated with the

0:06:53.690,0:06:54.940
eigenvector.

0:06:58.700,0:07:00.940
So if you give me a matrix that[br]represents some linear

0:07:00.940,0:07:01.880
transformation.

0:07:01.880,0:07:03.880
You can also figure[br]these things out.

0:07:03.880,0:07:05.730
Now the next video we're[br]actually going to figure out a

0:07:05.730,0:07:07.080
way to figure these[br]things out.

0:07:07.080,0:07:10.320
But what I want you to[br]appreciate in this video is

0:07:10.320,0:07:13.920
that it's easy to say,[br]oh, the vectors that

0:07:13.920,0:07:15.130
don't get changed much.

0:07:15.130,0:07:16.620
But I want you to understand[br]what that means.

0:07:16.620,0:07:19.860
It literally just gets scaled up[br]or maybe they get reversed.

0:07:19.860,0:07:22.060
Their direction or the[br]lines they span

0:07:22.060,0:07:23.460
fundamentally don't change.

0:07:23.460,0:07:26.400
And the reason why they're[br]interesting for us is, well,

0:07:26.400,0:07:28.790
one of the reasons why they're[br]interesting for us is that

0:07:28.790,0:07:32.590
they make for interesting basis[br]vectors-- basis vectors

0:07:32.590,0:07:36.530
whose transformation matrices[br]are maybe computationally more

0:07:36.530,0:07:41.610
simpler, or ones that make for[br]better coordinate systems.