Current time:0:00Total duration:18:02

0 energy points

# Transformation matrix with respect to a basis

Video transcript

Let's say I've got some linear
transformation T that is a mapping from Rn to Rn. So if this is its domain, which
is just Rn, then its codomain is also Rn. If you give me some vector in
our domain, let's call that vector x, then T will map it
to some other member of Rn, which is also the codomain. So it'll map it over here. We could call that the mapping
of T, or the mapping of x, or T of x. Since T is a linear
transformation, we know that the mapping of x to its codomain
is equivalent to x being multiplied by
some matrix A. So we know that this thing right
here is equal to some matrix A times x. You've seen all of this
multiple, multiple times. Just to make sure we understand
the wording properly, we said we've used
the word that A is the-- we could either call it the matrix
for T, or let's say it's the transformation
matrix for T. Now, in the last couple of
videos, we've learned that the same vector can be represented
in different ways. It can be represented in
different coordinate systems. When I just write the vector x
like that, we just assume that it's being represented in
standard coordinates, or it's being represented with respect
to the standard basis. So let's be a little bit
more particular. This A is the transformation
for T only when x is represented in standard
coordinates, or only when x is written in coordinates with
respect to the standard basis. So let me write a little
qualifier here. A is the transformation matrix
for T with respect to the standard basis. I never wrote this
blue part before. I never even said this blue part
before, because the only coordinate system we were
dealing with was the standard coordinate system or the
coordinates with respect to the standard basis. But now we know that there are
multiple coordinate systems. There are multiple ways to
represent this vector. There are multiple ways to
represent that vector, because Rn has multiple spanning
bases. There are multiple bases that
can represent Rn, and each of those bases can generate a
coordinate system where you can represent any vector in Rn
with coordinates with respect to any of those bases. So that last part I said was a
bit of a mouthful, so let me make it a little bit
more concrete. Let's say that I have some basis
B that's made up of n-- it has to be linearly
independent. That's the definition of a
basis-- of n vectors v1, v2, all the way to vn. Now, these are n linearly
independent vectors. Each of these are
members of Rn. So B is a basis for Rn, which is
just another way of saying that all of these vectors are
linearly independent and any vector in Rn can be represented
as a linear combination of these guys, which
is another way of saying that any vector in Rn can be
represented with coordinates with respect to this
basis right there. So the same vector x,
I'm going to put the same dot here. When we represent it in standard
coordinates, it's just going to be that right
there, that vector x. But what if we want to represent
it in coordinates with respect to this
new basis? Well, then that same vector
x will look like this. We would denote it by this. The same vector can be
represented with respect to this basis. This could be some set
of coordinates. This would be some other set of
coordinates, but it's still representing the same basis. Likewise, this vector right
here, that vector right there, is also in Rn. So it can be represented by
some linear combination of these guys, or you can represent
it with coordinates with respect to this basis. So that same point right there,
I could represent it. So that point is this. But I could represent it with
coordinates with respect to my basis just like that. So this is an interesting
question. This should maybe bring
an interesting question into your brain. If I start off with something
that's in standard coordinates, and I apply the
transformation T-- that's like applying this matrix A to it or
multiplying that thing in standard coordinates times the
matrix A-- I then get the mapping of T in standard
coordinates. Now, what if I start off with
that thing in nonstandard coordinates if I have
coordinates with respect to this other basis here? Well, T should still
map it to this guy. The transformation, no matter
what, should always map from that dot to that dot. It shouldn't care what
your coordinates are. So T should still map to
that same exact point. T should still be a linear
transformation. It can map from x to T of x, but
that's the same thing as mapping from this kind of way
of labeling x to this way of labeling x. So we could say maybe this guy
right here could be some other matrix times this
guy over here. So let me write this
over here. These are just different
coordinate systems. I shouldn't just even say maybe. This guy should be able
to be represented. So if I represent the mapping
of x in our codomain in coordinates with respect to B--
so that's what that guy is right there-- so if I want to
represent that dot with this other coordinate system,
coordinates with respect to this basis, it should be
equal to the product of some other matrix. Let me call that
other matrix D. Some other matrix D times this
representation of x times the coordinates of x with respect
to my alternate nonstandard coordinate system. I should be able to find some
matrix D that does this. Then we would say that D is the transformation matrix for T. A assumes that you have x in
terms of standard coordinates. Now D assumes that you have x in
coordinates with respect to this basis, so with respect
to the basis B. There's no reason why we
shouldn't be able to do this. These things are just different
ways of representing the exact same vector,
the exact same dot in our sets here. So if I represent it one way,
the standard way, I multiply by A, and I get Ax. If I represent it in nonstandard
coordinates, I should be able to multiply it
by some other matrix and get another nonstandard coordinate
representation of what it gets mapped to. So let's see if we can find
some relation between D and between A. So we learned a couple of videos
ago that there's a change of basis matrix that we
can generate from this basis. It's pretty easy to generate. The change of basis matrix is
just a matrix whose columns are these basis vectors, so
v1, v2-- I shouldn't put a comma there. These are just the columns--
v2 all the way to vn. This is an n-by-n matrix. Each of these guys are members
of Rn and we have n of them. This is an n-by-n matrix where
all of the columns are linearly independent, so we
know that C is invertible. These are column vectors
right here. So we know that C
is invertible. We learned in the last two or
three videos that if we have some vector x, and it's being
represented by coordinates with respect to our basis B, we
can just multiply that by C, and we'll get our vector x. This essentially will tell us
the linear combination of these guys that'll get us x. Since C is invertible, we also
saw that if we have the standard format for x, or the
standard coordinates for x, we can multiply that
by C inverse. That will get us the coordinates
for x with respect to the basis B. These two things, if you just
multiply both sides of this equation-- let me do it in a
different color-- if you just multiply both sides of this
equation by C inverse on the left-hand side, you're
going to get this equation right there. Now given that, let's
see if we can find some type of relation. Let's see what D times
xB is equal to. So let's say if we take D times
xB, so this thing right here should be equal to D times
the representation or the coordinates of x with
respect to the basis B. That's what we're claiming. We're saying that this guy
is equal to D times the representation of x with respect
to the coordinates with respect to the basis B. Let me write all of this down. I'll do it right here, because I
think it's nice to have this graphic up here. So we can say that D times
xB is equal to this thing right here. It's the same thing as the
transformation of x represented in coordinates with
respect to B, or in these nonstandard coordinates. So it's equal to the
transformation of x represented in this coordinate
system, represented in coordinates with respect to B. We see that right there. But what is the transformation
of x? That's the same thing
as A times x. That's kind of the standard
transformation if x was represented in standard
coordinates. So this is equal to x in
standard coordinates times the matrix A. Then that will get us to this
dot in standard coordinates, but then we want to convert
it to these nonstandard coordinates just like that. Now, if we have this, how can
we just figure out what the vector Ax should look like? What this vector should
look like? Well, we can look at this
equation right here. We have this. This is the same
thing as this. Actually. we want to go the other way. We have this. We have that right there. That's this right there. We want to get just this dot
represented in regular standard coordinates. So what do we do? We multiply it by C. Let me write it this way. If we multiply both sides
of this equation times C, what do we get? We get this right here. Actually, no. I was looking at the right
equation the first time. We have this right here, which
is the same-- first intuition is always right. We have this, which is the same
thing as this right here. So this can be rewritten. This thing can be rewritten
as C inverse-- we don't have an x here. We have an Ax here, so
C inverse times Ax. The vector Ax represented in
these nonstandard coordinates is the same thing as multiplying
the inverse of our change of basis matrix
times the vector Ax. If I have my vector Ax and I
multiply it times the inverse of the change of basis matrix,
I will then have a representation of the vector
Ax in my nonstandard basis. Now, what is the vector
x equal to? Well, the vector x is equal to
our change of basis matrix times x represented in these
nonstandard coordinates. So this is going to be equal to
C inverse A times x. x is just the same thing as C. x is just C times our
nonstandard coordinates for x, just like that. Let me summarize it, just
because I waffled a little bit on this point right there
just because I got a little bit confused. If I start off with the
nonstandard representation of x, or x in coordinates
with respect to B, I multiply them times D. So if I start with this, I
multiply them times D, I get to that point right there. So this right there is
the same thing as this point right there. That point right there should
be the nonstandard representation of the
transformation of x, or the coordinates of the
transformation of x with respect to B. Now, the transformation of
x, if x is in standard coordinates, is just
A times x. So this is just A times x. But I want to represent it in
these nonstandard coordinates. Now, A times x in nonstandard
coordinates is the same thing as C inverse times A times x, if
you think this is the same thing as this. So if you have this and you
want to represent it in nonstandard coordinates, you
multiply it by C inverse, so then you'll get that
representation in nonstandard coordinates. Then finally, we say look, x is
the same thing as C times the nonstandard coordinate
representation of x. So we can replace x with
that right there. So the big takeaway here is that
D times the coordinates of x with respect to the basis
B is equal to C inverse A times C times the coordinates
of x with respect to the basis B. So D must be equal
to C inverse AC. So if D is the transformation
matrix for T with respect to the basis B-- and let me write
here-- and C is the change of basis matrix for B-- let me
write that down, might as well because this is our big
takeaway-- and A is the transformation-- I'll write it
in shorthand-- matrix for T with respect to the standard
basis, then we can say-- this is the big takeaway-- that D,
our matrix D, is equal to C inverse times A times C. That's our big takeaway from
this video, which is really interesting. I don't want you to
lose this point. We now understand that A is
just for a certain set of coordinates. But there's arbitrary different
bases that we can use to represent Rn, so we can
have different matrices that represent the linear
transformation under different coordinate systems. If we want
to figure out those different matrices for different
coordinate systems, we can essentially just construct the
change of basis matrix for the coordinate system we care about,
and then generate our new transformation matrix with
respect to the new basis by just applying this result.