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.

Thank you Sal

thanks!

You have taught me in 12 minutes what two years of Linear Algebra classes have failed to, thank you so much.

3:22

"I'm running out of letters"

I don't know why this make me laughing too much…..I love you Sal

thank you!! tears falling down right now. 2nd year of engineering.