Transformations and matrix multiplication
Compositions of Linear Transformations 1 Introduction to compositions of Linear Transformations
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- 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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At 5:31, how is the moon large enough to block the sun? Isn't the sun way larger?
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