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For any transformation that maps
from Rn to Rn, we've done

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it implicitly, but it's been
interesting for us to find the

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vectors that essentially just
get scaled up by the

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transformations.

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So the vectors that have the
form-- the transformation of

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my vector is just equal
to some scaled-up

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version of a vector.

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And if this doesn't look
familiar, I can jog your

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memory a little bit.

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When we were looking for
basis vectors for the

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transformation--
let me draw it.

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This was from R2 to R2.

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So let me draw R2 right here.

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And let's say I had the
vector v1 was equal to

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the vector 1, 2.

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And we had the lines spanned
by that vector.

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We did this problem several
videos ago.

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And I had the transformation
that flipped across this line.

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So if we call that line l, T was
the transformation from R2

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to R2 that flipped vectors
across this line.

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So it flipped vectors
across l.

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So if you remember that
transformation, if I had some

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random vector that looked like
that, let's say that's x,

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that's vector x, then the
transformation of x looks

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something like this.

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It's just flipped across
that line.

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That was the transformation
of x.

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And if you remember that video,
we were looking for a

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change of basis that would allow
us to at least figure

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out the matrix for the
transformation, at least in an

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alternate basis.

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And then we could figure
out the matrix for the

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transformation in the
standard basis.

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

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changed much by the
transformation, or ones that

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only got scaled by the
transformation.

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

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equaled v1.

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Or we could say that the
transformation of v1 just

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equaled 1 times v1.

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So if you just follow this
little format that I set up

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here, lambda, in this
case, would be 1.

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And of course, the vector
in this case is v1.

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The transformation just
scaled up v1 by 1.

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In that same problem, we had
the other vector that

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we also looked at.

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It was the vector minus-- let's
say it's the vector v2,

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

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And then if you take the
transformation of it, since it

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was orthogonal to the
line, it just got

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flipped over like that.

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And that was a pretty
interesting vector force as

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well, because the transformation
of v2 in this

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situation is equal to what?

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Just minus v2.

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It's equal to minus v2.

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Or you could say that the
transformation of v2 is equal

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to minus 1 times v2.

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And these were interesting
vectors for us because when we

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defined a new basis with these
guys as the basis vector, it

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was very easy to figure out
our transformation matrix.

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And actually, that basis was
very easy to compute with.

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And we'll explore that a little
bit more in the future.

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But hopefully you realize that
these are interesting vectors.

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There was also the cases where
we had the planes spanned by

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some vectors.

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

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plane like that.

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And we were transforming things
by taking the mirror

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image across this and we're
like, well in that

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transformation, these red
vectors don't change at all

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and this guy gets
flipped over.

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So maybe those would make
for good bases.

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Or those would make for
good basis vectors.

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And they did.

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So in general, we're always
interested with the vectors

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that just get scaled up
by a transformation.

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

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

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get scaled up, it actually gets
changed, this direction

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gets changed.

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The vectors that get scaled up
might switch direct-- might go

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from this direction to that
direction, or maybe

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they go from that.

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Maybe that's x and then the
transformation of x might be a

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scaled up version of x.

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Maybe it's that.

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The actual, I guess, line that
they span will not change.

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And so that's what we're going
to concern ourselves with.

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These have a special name.

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And they have a special name and
I want to make this very

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clear because they're useful.

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It's not just some mathematical
game we're

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playing, although sometimes
we do fall into that trap.

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But they're actually useful.

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They're useful for defining
bases because in those bases

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it's easier to find
transformation matrices.

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They're more natural coordinate
systems. And

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oftentimes, the transformation
matrices in those bases are

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easier to compute with.

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And so these have
special names.

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Any vector that satisfies this
right here is called an

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eigenvector for the
transformation T.

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And the lambda, the multiple
that it becomes-- this is the

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eigenvalue associated with
that eigenvector.

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So in the example I just gave
where the transformation is

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flipping around this line,
v1, the vector 1, 2 is an

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eigenvector of our
transformation.

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So 1, 2 is an eigenvector.

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And it's corresponding
eigenvalue is 1.

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This guy is also an
eigenvector-- the

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vector 2, minus 1.

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He's also an eigenvector.

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A very fancy word, but all it
means is a vector that's just

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scaled up by a transformation.

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It doesn't get changed in any
more meaningful way than just

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the scaling factor.

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And it's corresponding
eigenvalue is minus 1.

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If this transformation--
I don't know what its

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transformation matrix is.

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I forgot what it was.

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We actually figured it
out a while ago.

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If this transformation matrix
can be represented as a matrix

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vector product-- and it should
be; it's a linear

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transformation-- then any
v that satisfies the

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transformation of-- I'll say
transformation of v is equal

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to lambda v, which also would
be-- you know, the

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transformation of [? v ?]

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would just be A times v.

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These are also called
eigenvectors of A, because A

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is just really the matrix
representation of the

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transformation.

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So in this case, this would be
an eigenvector of A, and this

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would be the eigenvalue
associated with the

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eigenvector.

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So if you give me a matrix that
represents some linear

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transformation.

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You can also figure
these things out.

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Now the next video we're
actually going to figure out a

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way to figure these
things out.

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But what I want you to
appreciate in this video is

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that it's easy to say,
oh, the vectors that

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don't get changed much.

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But I want you to understand
what that means.

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It literally just gets scaled up
or maybe they get reversed.

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Their direction or the
lines they span

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fundamentally don't change.

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And the reason why they're
interesting for us is, well,

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one of the reasons why they're
interesting for us is that

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they make for interesting basis
vectors-- basis vectors

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whose transformation matrices
are maybe computationally more

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simpler, or ones that make for
better coordinate systems.