Functions and linear transformations
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A more formal understanding of functions
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Vector Transformations
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Linear Transformations
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Matrix Vector Products as Linear Transformations
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Linear Transformations as Matrix Vector Products
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Image of a subset under a transformation
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im(T): Image of a Transformation
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Preimage of a set
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Preimage and Kernel Example
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Sums and Scalar Multiples of Linear Transformations
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More on Matrix Addition and Scalar Multiplication
Matrix Vector Products as Linear Transformations Matrix Vector Products as Linear Transformations
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- I think you're pretty familiar with the idea of matrix vector
- products and what I want to do in this video is show you that
- taking a product of a vector with a matrix is equivalent to
- a transformation.
- It's actually a linear transformation.
- Let's say we have some matrix A and let's say that its terms
- are, or its columns are v1-- column vector is v2,
- all the way to vn.
- So this guy has n columns.
- Let's say it has m rows.
- So it's an m by n matrix.
- And let's say I define some transformation.
- Let's say my transformation goes from Rn to Rm.
- This is the domain.
- I can take any vector in Rn and it will map it to some
- factor in Rm.
- And I define my transformation.
- So T of x where this is some vector in Rn, is equal to A--
- this is this A.
- Let me write it in this color right here.
- And it should be bolded.
- I kind of get careless sometimes with the bolding.
- But big bold A times the vector x.
- So the first thing you might, Sal, this transformation looks
- very odd relative to how we've been defining transformations
- or functions so far.
- So the first thing we have to just feel comfortable with is
- the idea that this is a transformation.
- So what are we doing?
- We're taking something from Rn and then
- what does A x produce?
- If we write A x like this, if this is x where it's x1, x2.
- It's going to have n terms because it's in Rn.
- This can be rewritten as x1 times v1 plus x2 times v2, all
- the way to xn times vn.
- So it's going to be a sum of a bunch of these column vectors.
- And each of these columns vectors, v1, v2, all the way
- to vn, what set are they members of?
- This is an m by n matrix, so they're going to have m-- the
- matrix has m rows, or each of these column
- vectors will have m entries.
- So all of these guys are members of Rm.
- So if I just take a linear combination of all of these
- guys, I'm going to get another member of Rm.
- So this guy right here is going to be a member of Rm,
- another vector.
- So clearly, by multiplying my vector x times a, I'm mapping,
- I'm creating a mapping from Rn-- and let me pick another
- color-- to Rm.
- And I'm saying it in very general terms. Maybe n is 3,
- maybe m is 5.
- Who knows?
- But I'm saying it in very general terms. And so if this
- is a particular instance, a particular member of set Rn,
- so it's that vector, our transformation or our function
- is going to map it to this guy right here.
- And this guy will be a member of Rm and we
- could call him a x.
- Or maybe if we said a x equaled b we could call him
- the vector b-- whatever.
- But this is our transformation mapping.
- So this does fit our kind of definition or our terminology
- for a function or a transformation as a mapping
- from one set to another.
- But it still might not be satisfying because everything
- we saw before looked kind of like this.
- If we had a transformation I would write it like the
- transformation of-- I would write, you know, x1 and x2 and
- xn is equal to.
- I'd write m terms here in commas.
- How does this relate to that?
- And to do that I'll do a specific example.
- So let's say that I had the matrix-- let me
- to a different letter.
- Let's say I have my matrix B and it is a
- fairly simple matrix.
- It's a 2, minus 1, 3 and 4.
- And I define some transformation.
- So I define some transformation T.
- And it goes from R2 to R2.
- And I define T.
- T of some vector x is equal to this matrix, B
- times that vector x.
- Now what would that equal?
- Well the matrix is right there.
- Let me write it in purple.
- 2, minus 1, 3, and 4 times x.
- x1, x2.
- And so what does this equal?
- Well this equals another vector.
- It equals a vector in the co-domain R2 where the first
- term is 2 times x1.
- I'm just doing the definition of matrix vector
- multiplication.
- 2 times x1 plus minus 1 times x2, or minus x2.
- That's that row times our vector.
- And then the second row times that factor.
- We get 3 times x1.
- Plus 4 times x2.
- So this is what we might be more familiar with.
- I could rewrite this transformation.
- I could rewrite this transformation as T of x1 x2
- is equal to 2x1 minus x2 comma-- let me scroll over a
- little bit, comma 3x1 plus 4x2.
- So hopefully you're satisfied that a matrix multiplication,
- it isn't some new, exotic form of transformation.
- That they really are just another way.
- This statement right here is just another way of writing
- this exact transformation right here.
- Now, the next question you might ask and I already told
- you the answer to this at the beginning of the video is, is
- multiplication by a matrix always going to be a linear
- transformation?
- Now what are the two constraints for being a linear
- transformation?
- We know that the transformation of two vectors,
- a plus b, the sum of two vectors should be equal to the
- sum of their transformations.
- The transformation of a plus the transformation of b.
- And then the other requirement is that the transformation of
- a scaled version of a vector should be equal to a scaled
- version of the transformation.
- These are our two requirements for being a linear
- transformation.
- So let's see if matrix multiplication applies there.
- And I've touched on this in the past and I've even told
- you that you should prove it.
- I've already assumed you know it, but I'll prove it to you
- here because I'm tired of telling you that you
- should prove it.
- I should do it at least once.
- So let's see, matrix multiplication.
- If I multiply a matrix A times some vector x, we know that--
- let me write it this way.
- We know that this is equivalent
- to-- I said our matrix.
- Let's say this is an m by n matrix.
- We can write any matrix as just a
- series of column vectors.
- So this guy could have n column vectors.
- So let's say it's v1, v2, all the way to vn column vectors.
- And each of these guys are going to have m components.
- Times x1, x2, all the way down to xn.
- And we've seen this multiple, multiple times before.
- This, by the definition of matrix vector multiplication
- is equal to x1 times v1.
- That times that.
- This scalar times that vector plus x2 times v2, all the way
- to plus xn times vn.
- This was by definition of a matrix vector multiplication.
- And of course, this is going to-- and I did this at the top
- of the video.
- This is going to have right here, this vector is going to
- be a member of Rm.
- It's going to have m components.
- So what happens if I take some matrix A, some m by n matrix
- A, and I multiply it times the sum of two vectors a plus b?
- So I could rewrite this as this thing right here.
- So my matrix A times.
- The sum of a plus b, the first term will just be a1 plus b1.
- Second term is a2 plus b2, all the way down to a n plus bn.
- This is the same thing as this.
- I'm not saying a of a plus b.
- I'm saying a times.
- Maybe I should put a dot right there.
- I'm multiplying the matrix.
- I want to be careful with my notation.
- This is the matrix vector multiplication.
- It's not some type of new matrix dot product.
- But this is the same thing as this
- multiplication right here.
- And based on what I just told you up here, which we've seen
- multiple, multiple times, this is the same thing as a1 plus
- b1 times the first column in a, which is that
- vector right there.
- This a is the same as this a.
- So times v1.
- Plus a2 plus b2 times v2, all the way to plus an
- plus bn times vn.
- Each xi term here is just being replaced by
- an ai plus bi term.
- So each x1 here is replaced by an a1 plus b1 here.
- This is equivalent to this.
- And then from the fact that we know that well vector products
- times scalars exhibit the distributive property, we can
- say that this is equal to a1 times v1.
- Let me actually write all of the a1 terms. Let me write
- this. a1 times v1 plus b1 times v1 plus a2 times v2 plus
- b2 times v2, all the way to plus a n times vn
- plus bn times vn.
- And then if we just re-associate this, if we just
- group all of the a's together, all of the a terms together,
- we get a1 plus a-- sorry.
- a1 plus-- let me write it this way. a1 times v1 plus a2 times
- v2 plus, all the way, a n times vn.
- I just grabbed all the a terms.
- We get that plus all the b terms. All the b terms I'll do
- in this color.
- All the b terms are like that.
- So plus b1 times v1 plus b2 times v2, all the way to plus
- bn times vn.
- That's that guy right there.
- Is equivalent to this statement up here; I just
- regrouped everything, which is of course, equivalent to that
- statement over there.
- But what's this equal to?
- This is equal to my vector-- these columns are remember,
- the column for the matrix capital A.
- So this is equal to the matrix capital A times a1, a2, all
- the way down to a n, which was our vector a.
- And what's this equal to?
- This is equal to plus these v1's.
- These are the columns for the a, so it's equal to the matrix
- A times my vector b.
- b1, b2, all the way down to bn.
- This is my vector b.
- We just showed you that if I add my two vectors, a and b,
- and then multiply it by the matrix, it's completely
- equivalent to multiplying each of the vectors times the
- matrix first and then adding them up.
- So we've satisfied-- and this is for an m by n matrix.
- So we've now satisfied this first condition right there.
- And then what about the second condition?
- And this one's even more straightforward to understand.
- c times a1, so let me write it this way.
- The vector a times-- sorry.
- The matrix capital A times the vector lowercase a-- let me do
- it this way because I want-- times the
- vector c lowercase a.
- So I'm multiplying my vector times the scalar first. Is
- equal to-- I can write my big matrix A.
- I've already labeled its columns.
- It's v1, v2, all the way to vn.
- That's my matrix a.
- And then, what does ca look like?
- ca, you just multiply its scalar times each of
- the terms of a.
- So it's ca1, ca2, all the way down to c a n.
- And what does this equal?
- We know this, we've seen this show multiple times before
- right there.
- So it just equals-- I'll write a little bit lower.
- That equals c a1 times this column vector, times v1.
- Plus c a2 times v2 times this guy, all the way to
- plus c a n times vn.
- And if you just factor this c out, once again, scalar
- multiplication times vectors exhibits the
- distributive property.
- I believe I've done a video on that, but it's
- very easy to prove.
- So this will be equal to c times-- I'll just stay in one
- color right now-- a1 v1 plus a2 v2 plus all
- the way to a n vn.
- And what is this thing equal to?
- Well that's just our matrix A times our vector-- or our
- matrix uppercase A.
- Maybe I'm overloading the letter A.
- My matrix uppercase A times my vector lowercase a.
- Where the lowercase a is just this thing right here, a1, a2
- and so forth.
- This thing up here was the same thing as that.
- So I just showed you that if I take my matrix and multiply it
- times some vector that was multiplied by a scalar first,
- that's equivalent to first multiplying the matrix times a
- vector and then multiplying by the scalar.
- So we've shown you that matrix times vector products or
- matrix vector products satisfied this condition of
- linear transformations and this condition.
- So the big takeaway right here is matrix multiplication.
- And this is a important takeaway.
- Matrix multiplication or matrix products with vectors
- is always a linear transformation.
- And this is a bit of a side note.
- In the next video I'm going to show you that any linear
- transformation-- this is incredibly powerful-- can be
- represented by a matrix product or by-- any
- transformation on any vector can be equivalently, I guess,
- written as a product of that vector with a matrix.
- Has huge repercussions and you know, just as a side note,
- kind of tying this back to your everyday life.
- You have your Xbox, your Sony Playstation and you know, you
- have these 3D graphic programs where you're running around
- and shooting at things.
- And the way that the software renders those programs where
- you can see things from every different angle, you have a
- cube then if you kind of move this way a little bit, the
- cube will look more like this and it gets rotated, and you
- move up and down, these are all
- transformations of matrices.
- And we'll do this in more detail.
- These are all transformations of vectors or the positions of
- vectors and I'll do that in a lot more detail.
- And all of that is really just matrix multiplication.
- So all of these things that you're doing in your fancy 3D
- games on your Xbox or your Playstation, they're all just
- matrix multiplications.
- And I'm going to prove that to you in the next video.
- And so when you have these graphics cards or these
- graphics engines, all they are-- you know, we're jumping
- away from the theoretical.
- But all these graphics processors are, are hard wired
- matrix multipliers.
- If I have just a generalized, some type of CPU, I have to in
- software write how to multiply matrices.
- But if I'm making an Xbox or something and 99% of what I'm
- doing is just rotating these abstract objects and
- displaying them in transformed ways, I should have a
- dedicated piece of hardware, a chip, that all it does-- it's
- hard wired into it-- is multiplying matrices.
- And that's what those graphics processors or graphics engines
- really are.
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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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