Transformations and matrix multiplication
Matrix Product Associativity Showing that matrix products are associative
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- We know that if we have some linear transformation, that
- the transformation from x to y -- and these are just sets,
- sets of vectors, and T is a linear transformation from y
- to z-- that we can construct a composition of s with T that
- is a linear transformation from x all the way to z.
- We saw this several videos ago.
- And the definition of our linear transformation, or the
- composition of our linear transformation.
- So the composition of s with t, applied to someone vector x
- in our set x, our domain, is equal to s of t of x.
- This was our definition.
- And then we went on and we said, look, if s of x can be
- represented as the matrix multiplication a x, the matrix
- vector product, and if T of x can be represented -- or the
- transformation T can be represented-- as the product
- of the matrix b with x, we saw that this thing right here --
- which is of course, if we just write this way, this is equal
- to a times T times x, which is just b x -- we saw in multiple
- videos now that this is equivalent to, by our
- definition of matrix products, the matrix a b -- right?
- When you take the product of two matrices you just get
- another matrix -- the product a b times x.
- So you take essentially the first linear transformation in
- your composition, its matrix, which was a, and you take the
- product with the second one.
- Fair enough, all of this is review so far.
- Let's take three linear transformations.
- Let's say that I have the linear transformation h, and
- when I apply that to a vector x, it's equivalent to
- multiplying my vector x by the matrix a.
- Let's say I have the linear transformation g.
- When I applied that to a vector x, it's equivalent to
- multiplying that vectrix -- that vector, there should be a
- new concept called a vetrix -- it's equivalent to multiplying
- that vector times the matrix b.
- And then I have a final linear transformation f.
- When it's applied to some vector x, it's equivalent to
- multiplying that vector x times the matrix z.
- Now what I'm curious about is what happens when I take the
- composition of h with g, and then I take the composition of
- that with f -- these are all linear transformations -- and
- then I apply that to some vector x.
- Which is necessarily going to be in the domain of this guy.
- I haven't actually drawn out their domain and co-domain
- definitions, but I think you get the idea.
- So let's explore what this is a little bit.
- Well by the definition of what a -- let's go back.
- By this definition right here of what composition even
- means, we can just apply that to this right here.
- So we could just imagine this as being our s, and then this
- is our T right there.
- Then what is this going to be equal to?
- If we just do a straight up pattern match right there,
- this is going to be equal to s, the transformation s,
- applied to the transformation f, applied to x.
- So s is h of g.
- So it is h -- or I should say h of g -- the composition of h
- with g, that is our s.
- And then I apply that to f applied to x.
- f is our t.
- I apply that to f applied to x, just like that.
- Now what is this equal to?
- Now we can imagine that this is our x.
- If we just pattern match, according to this definition,
- that this and this guy right here, that this is our t, and
- that this is our s.
- And so if we just pattern match here,
- this is equal to what?
- This is just straight from the definition of a composition.
- So it's equal to s of -- s is our h -- so h of t, which in
- this case is g, g applied to x.
- But instead of an x, we have this vector here, which was
- the transformation f applied to x.
- So g of f of x.
- That's what this is equal to.
- The composition of h with g, and the composition
- competition of f with h, the composition of h and g, all of
- that applied to x is equal to h of g of f of x.
- Now what is this equal to?
- Well this is equal to -- I'll do it right here -- this is
- equal to h, the transformation h, applied to -- what is this
- term right here?
- I'll do it in pink.
- What is this?
- That is the composition of g and f applied to x.
- You can just replace s with g, and f with T, and you'll get
- that right there.
- So this is just equal to the composition of g with f
- applied to x.
- That's all that is.
- Now, what is this equal to right there?
- And it's probably confusing to see two parentheses in
- different colors, but you get the idea.
- What is this equal to?
- Well, just go back to your definition of the composition
- -- I just want to make it very clear what we're doing.
- This is, if you imagine this being your T and then this
- being your s, this is just the composition of s with T,
- applied to x.
- So this is just equal to -- I'll write it this way.
- This is equal to -- I shouldn't write s's -- this is
- a composition of h with the composition of g and f.
- And then all of that applied to x.
- Now why did I do all of this?
- Well one, to show you that the composition is associative.
- I went all the way here and then I went all the way back.
- And essentially it doesn't matter where you put the
- parentheses.
- The composition of h with g with f, is equivalent to the
- composition of h with the composition of g and f.
- That these two things are equivalent, and essentially
- these two things, you can just re-write them.
- The parentheses are essentially unnecessary.
- You can write this as a composition of h with g with
- f, all of that applied to x.
- Now, I took the time to say that each of these linear
- transformations I can represent as matrix
- multiplications.
- Why did I do that?
- Well we saw before, that any composition, when you take the
- composition of s with T, the matrix version of this
- transformation of this composition is going to be
- equal to the product -- by our definition of matrix matrix
- products -- the product of the s's transformation matrix and
- T's transformation matrix.
- So what are these going to be equal to?
- So this one right here -- if you think of this
- transformation right here, this statement right here, its
- matrix version of it-- so let me write that.
- A matrix version of the composition of h with g, and
- then the composition of that with f, applied to x, is going
- to be equal to -- and we've seen this before -- the
- product of these matrices.
- So this composition, its matrix is going to be a b.
- h and g, their matrices are a and b.
- So it's going to be a b -- and I'll do it in parentheses.
- And then you take that matrix, and you take the product -- so
- this guy's matrix representation is a b, right?
- And this guy's matrix representation is c.
- So the matrix representation of this whole thing is this
- guy, taking the product of a b, and then taking the product
- of that with c.
- So a b.
- and then c.
- And then if you look at this guy right here -- and of
- course all of that times a vector x, all of that time
- some vector x, right there.
- That's the vector x.
- Now let's look at this one right here.
- If we take the composition of h with the composition of g
- and f, and apply all of that to some vector x, what is that
- equivalent to?
- Well this composition right here, the matrix version of
- it, I guess we can say, is going to be the product b c.
- And we're going to apply that to x.
- So we're going to have the product b c.
- And then we're going to take the product of that with this
- guy's matrix representation, which is a.
- And we've shown this before.
- We never showed it with three, but it extends.
- I kind of showed it extends, so you can just keep applying
- the definition.
- You can keep applying this property right here, and so
- it'll just naturally extend.
- Because every time, we're just taking the
- composition of two things.
- Even though it looks like we're taking the composition
- of three, we're taking the composition of two things
- first here.
- And then we get its matrix representation.
- And then we take the composition of that with this
- other thing.
- So the matrix representation of the entire composition is
- going to be this matrix times this matrix.
- Which I did here.
- Similarly, here we take first the composition of these two
- linear transformations, and their matrix representation
- will be that right there.
- And then we take the composition of that with that.
- So its entire matrix representation is going to be
- guy's matrix times this guy's matrix.
- So a times b c.
- And of course, all of that applied to the vector x.
- Now, in this video I've showed you that these two things are
- equivalent.
- If anything, the parentheses are completely unnecessary.
- And I showed you that there.
- They both essentially boil down to h of g of f of x.
- So these two things are equivalent.
- So we could say, essentially, that these two things over
- here are equivalent.
- Or that AB, the product AB, and then taking the product of
- that matrix with the matrix C, is equivalent to taking the
- product A with the matrix BC.
- Which is just another product matrix.
- Or another way of saying it is that these parentheses don't
- matter, that all of these is just equal to ABC.
- Or -- I mean, this is just a statement that matrix products
- exhibit the associative property.
- It doesn't matter where you put the parentheses.
- And you know, sometimes it's confusing me, the word
- associative.
- It just means it doesn't matter where you put the
- parentheses.
- Matrix products do not exhibit the commutative property.
- We saw that in the last video.
- In general, we cannot make the statement that AB
- is equal to BA.
- We cannot do that.
- And in fact in the last video -- I think it was the last
- video -- I showed you that if AB is defined, sometimes BA
- is not even defined.
- Or if b a is defined, sometimes a b isn't defined.
- So it's not commutative.
- It is associative, though.
- In the next video, I'll see if matrix products are actually
- distributive.
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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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