Compositions of linear transformations 1 | Matrix transformations | Linear Algebra | Khan Academy
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Compositions of linear transformations 1 | Matrix transformations | Linear Algebra | Khan Academy

October 15, 2019


Let’s see if we can build a bit
on some of our work with linear transformation. I have two linear
transformations. I have the transformation S,
that’s a mapping, or function, from the set X to the set Y. And let’s just say that X
is a subset of of Rn. Y is a subset of Rm. Then we know S is a linear
transformation. It can be represented by a
matrix vector product. We can write S of X. Let me do it in the same color
as I was doing it before. We can write that S of some
vector X, is equal to some matrix A times X. The matrix A, it’s going to be
X, whatever X we input into the function, although we
take the mapping of. It’s going to be in this set,
right here, is going to be a member of Rn. This is going to
be right here. Let me do it like this. X is going to be
a member of Rn. Well, it’s actually going to be
a member of X, which is a subset of Rn. I’m just trying to figure out
what the dimensions of matrix A are going to be. This is going to have n
components right here. Matrix A has to have
n columns. Matrix A is going to be, let’s
just say, is an m by n matrix. Fair enough. Let’s say we have another
linear transformation. Let me draw what I’ve
done so far. We have sum set X, right
here, that is set X. It is a subset of Rn. Rn, I can draw out there. We have this mapping, S, or this
linear transformation, from X to Y. It goes to a new set,
Y, right here. Y is a member of Rm. The mapping X, right here. You take some element here,
and you apply the transformation S. I’ve told you it’s a linear
transformation. You’ll get to some value in
set Y, which is in Rm. I said that the matrix
representation of our linear transformation is going to
be an m by n matrix. You’re going to start with
something that has n entries, or a vector that’s
a member of Rn. You want to end up with
a vector that’s in Rm. Fair enough. Now, let’s say I have another
linear transformation, T. It’s a mapping from the
set Y to the set Z. Let me draw. I have another set here
called set Z. I can map from elements of Y, so
I could map from here, into elements of Z using the linear
transformation T. Similar to what I did before. We know that Y is
a member of Rm. You know that this is a subset,
not a member, more of a subset of Rm. These are just arbitrary
letters. It could be 100 or
5, or whatever. I’m just trying to
stay abstract. Z is a member, I’m running out
of letters, let’s say Z is a member of Rl. Z is a member of Rl. Then, what’s the transformation
T, what’s it’s matrix representation
going to be. You know it’s a linear
transformation. I told you that. We know it can be represented
in this form. We could say that T of X, where
X is a member of Rm, is going to be equal to some
matrix B times X. What are the dimensions of
matrix B going to be. X is going to be a member of Rm,
so B is going to have to have m columns. And then it’s a mapping into a
set that’s a member of Rl. It’s going to map from members
of Rm to members of Rl. It’s going to be
l by m matrix. When you see this, a very
natural question might arise in your head. Can we construct some mapping
that goes all the way, that goes all the way, from set
X all the way to set T. Maybe we’ll call that the
composition of– I mean we can create that mapping using a
combination of S and T. Let’s just make up some word. Let’s just call T, with this
little circle S, let’s just call this a mapping from
X all the way to Z. We’ll call this the composition
of T with S. We’re essentially just combining
the two functions in order to try to create some
mapping that takes us from T, from set X, all the
way to set Z. We still haven’t defined this. How can we actually
construct this. A natural thing might be to
first apply transformation S. Let’s say that this is our X
we’re dealing with right here. Maybe the first thing we want to
do is apply S, and that’ll give us an S of X. That will give us this value,
right here, that’s in set Y. And then what if we were to take
that value and apply the transformation T to it? We would take this value, and
apply the transformation T to it, to maybe get
to this value. This would be the linear
transformation T applied to this value, this member of the
set Y, which is in Rm. We are just going to apply that
transformation to this guy, right here, which was the
transformation S applied to X. This might look fancy, but all
this is, remember this is just a vector, right here,
in the set Y, which is a subset of Rm. This is a vector that is in X. When you apply mapping,
you get another vector that’s in Y. You apply the linear
transformation T to that, then you get another vector
that’s at set Z. Let’s define the composition
of T with S. This is going to be
a definition. Let’s define the composition of
T with S to be– first we apply S to some vector in X. Apply S to some vector
in X to get us here. Then we apply T to that vector
to get us to set Z. To get us to set– so we apply
T to this thing right there. The first question might be,
is this even a linear transformation? Is the composition of two linear
transformations even a linear transformation? Well there are two requirements
to be a linear transformation. The sum of the linear
transformation of the sum of two vectors, should be the
linear transformation of each of them summed together. I know when I just say that
verbally, it probably doesn’t make a lot of sense. Let’s try to take the
composition, the composition of T with S of the sum
of two vectors in X. I’m taking the vectors
x and the vectors y. By definition, what
is this equal to? This is equal to applying to
linear transformation T to the linear transformation S,
applied to our two vectors, x plus y. What is this equal to? I told you at the beginning of
the video, that S is a linear transformation. So by definition, of a linear
transformation, one of our requirements, we know that S of
x plus y is the same thing as S of x plus S of y, because
S is a linear transformation. We know that is true. We know that we can replace this
thing right there with that thing right there. We also know that T is a
linear transformation. Which means that the
transformation applied to the sum of two vectors is equal to
the transformation of each of the vectors summed up. The transformation of S of x, or
the transformation applied to the transformation of S
applied to x, I know the terminology is getting confused,
plus T of S of y. We can do this because we
know that T is a linear transformation. But what is this right here? All this statement right here is
equal to the composition of T with S, applied to x, plus the
composition of T with S, applied to y. Given that both T and S are
linear transformations, we got our first requirement. That the composition applied to
the sum of two vectors is equal to the composition
applied to each of the vectors summed up. That was our first requirement
for linear transformation. Our second one is, we need
to apply this to a scalar multiple of a vector in X. So, T of S, or let me say it
this way, the composition of T with S applied to some scalar
multiple of some vector x, that’s in our set X. This is a vector x,
that’s our set X. This should be a capital X. This is equal to what. Well, by our definition of our
linear, of our composition, this is equal to the
transformation T applied to the transformation S, applied
to c times our vector x. What is this equal to? We know that this is a linear
transformation. Given that this is a linear
transformation, that S is a linear transformation, we know
that this can be rewritten as T times c times S
applied to x. This little replacing that I
did, with S applied to c times x, is the same thing as
c times the linear transformation applied to x. This just comes out of the
fact that S is a linear transformation. We’ve done that multiple
times. Now we have T applied
to some scalar multiple of some vector. We can do the same thing. We know that T is a linear
transformation. We know that this is equal to,
I’ll do it down here, this is equal to c times T applied to
S applied to some vector x that’s in there. What is this equal=? This is equal to the constant c
times the composition T with S of our vector x right there. We’ve met our second requirement
for linear transformation. The composition as we’ve defined
it is definitely a linear transformation. This means that the composition
of T with S can be written as some matrix– let
me write it this way– the composition of T with
S applied to, or the transformation of, which is
a composition of T with S, applied to some vector x, can
be written as some matrix times our vector x. And what will be the dimensions
of our matrix? We’re going from a n dimension
space, so this is going to have n columns, to a
l dimension space. So this is going
to have l rows. This is going to be
an l by n matrix. I’ll leave you there
in this video. I realize I’ve been making too
many 20 minutes plus videos. The next video, now that we
know this is a linear transformation, and that we know
that we can represent it as a matrix vector product. We’ll actually figure out how
to represent this matrix, especially in relation to the
two matrices that define our transformations, S and T.

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