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
Vector Transformations Introduction to the notion of vector transformations
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- In the last video we saw, a little bit more formally than
- you might have been exposed to the past, that a function is
- just a mapping of the members of one set to another set.
- So if this is my first set, x.
- We call that the domain.
- And the set that we're mapping to, y, in this case, that's
- called the codomain.
- And the function just maps each of the specific entries
- of x to an entry in y.
- When I say map, it really just creates an association.
- If we think of these in even less abstract terms -- which,
- on some levels is more abstract -- you could view x
- as a basket of bananas and y as a basket of apples.
- And for every banana, you're associating it
- with one of the apples.
- The mapping of going from each of those bananas to each of
- those apples would be a function.
- I don't know if that helps you or not, but I just want to
- broaden your preconceived notion of what a function is.
- I mean everything that you've probably seen before probably
- took a form that looks something like that.
- Where you said, oh, a function is, you just give me some
- number and I'll give you another number.
- Or I'll do something to that number.
- While it can be much more general than that.
- It's an association between any member of one set and some
- other members of another set.
- Now, we know that vectors are members of sets.
- Right?
- In particular, if we say that some vector x is a member of
- some set -- let me just say it's a member of rn, because
- that's what we deal with -- all that means is that this is
- just a particular representation of an n-tuple.
- Remember what rn was.
- rn we defined way back at the beginning of the linear
- algebra playlist. We defined it as the set of all n-tuples
- -- x1, x2, xn, where your x1's, x2's, all the way to
- xn's are a member of the real numbers.
- So your rn is most definitely a set.
- This could be rn.
- And obviously the use of the letter n is arbitrary.
- It could be rm, it could be rs.
- n is just kind of a placeholder for how many
- tuples we have. It could be r5.
- It could be 5 tuples.
- And when we say that a vector x is a member of rn, we're
- just saying that it's another way of writing
- one of these n-tuples.
- And all of our vectors so far are column vectors -- that's
- the only type that we've defined so far -- and we say
- it's this ordered list where each of the members are a
- member of r's.
- It's an ordered list of n -- it's an ordered list of
- n-components -- x1, x2, all the way to xn -- where each of
- those guys, or each of those x1's, x2's all the way to
- xn's, are a member of the real numbers.
- That's, by definition, what we mean when we say that x is a
- member of rn.
- So if x is a member of rn -- so let me draw two sets right
- here -- let's say that this set right here is rn and then
- let me just change, just to be general, let me create another
- set right there and call that set right there rm.
- Just a different number.
- It it could be the same as n, it could be different.
- This is m-tuples, that's n-tuples.
- We've defined that vectors can be members of rn.
- So you could have some vector here and then, if you
- associate with that vector in rn -- if you associate it with
- some vector in are rm -- if you associate it with, let's
- call that vector y, if you make this association, that
- too is a function.
- And that might have already been obvious to you and this
- would be a function that's mapping from rn to rm.
- And actually, I just want to make one little
- special note here.
- When I just drew the arrow like this, this shows that I'm
- mapping between two sets.
- I'm taking elements of this set and I'm associating with
- them with elements of that set.
- Now, in the last video you probably saw this.
- I wanted to do the side note note because I realized it
- might've been confusing.
- I introduced you to another way of writing a
- function like this.
- Where I said f could be defined as a mapping for any
- given x to x squared.
- And I just want to make a note on the notation.
- When I just have a regular arrow I'm going between sets.
- When I have this little vertical line at the base of
- the arrow, that's kind of the function definition.
- It tells me for any x you give me in the first set, in the
- second set I'm going to associate this x with, in this
- case, x squared.
- Anyway, I just wanted to make that side note.
- But the whole direction I was going in is that vectors are
- valid elements of sets, functions are just mappings
- between elements of sets, so you could have
- functions of vectors.
- And I even touched on that a little bit in the last video
- when I talked about vector-valued functions.
- If your codomain is a subset of rm, where m is greater than
- 1, then we say your function is vector-valued.
- It's not just mapping into the real numbers.
- It's mapping into some m-tuple of real numbers.
- So if you mapped two-dimensional space, you're
- dealing with a vector-valued function.
- Now I've been all abstract and whatnot, so let me actually
- deal with some vectors and it might make everything a little
- bit more concrete.
- So let's say I define the function f as f of x1, x2, and
- x3, is equal to x1 plus 2x2 and the second
- coordinate is just 3x3.
- And actually, I haven't formally defined coordinates
- for you yet, but I think you understand that just from your
- basic algebra training.
- So let's say that that's my function definition based on
- the notation that we've been introduced to.
- We could say that f is a mapping from -- its domain is
- r3 -- and it maps from r3, or it associates all values in r3
- with some value in r2.
- This is a 2-tuple.
- Right?
- So this is an r2 This is 3-tuple.
- Right?
- Or another way we could do this, if we just wanted to
- write it in vector notation, I could write that f -- if you
- pass f to some vector x1, x2, x3, I could say this will be
- equal to the vector -- and now it's going to have a
- two-component vector.
- It's going to be a vector in r2 where the first term is x1
- plus 2x2 and the second term is 3x3.
- So let's play around with this a little bit.
- See what it does for us, what it does to the vectors.
- So what is f of the vector 1, 1, 1?
- Well, I get 1 plus 2 times 1 is, I get the vector 3, and
- then my second term is just 3 times this one, so I get the
- vector 3, 3.
- Fair enough, let me do another one.
- Just to really experiment with this mapping.
- If I take the f of the vector in r3 2, 4, 1, what do I get?
- That equals 2 plus 2 times 4.
- That goes to the vector 10.
- 2 plus 2 times 4 and then 3 times the
- third term right there.
- So the vector 10, 3.
- So how can we visualize this?
- Well, three-dimensional vectors -- or vectors in r3 --
- are not always the easiest to visualize, but I think we can
- attempt to visualize these two guys.
- So let's say that this is the x1-axis, that's the x2-axis,
- that's the x3-axis.
- So this first vector right here, this yellow one, 1, 1,
- 1, it will look like this -- 1, 1, 1.
- And so if I were to go out here, then go out here, then
- go up 1, the point would be right there and if I were to
- draw it in standard position, I'd start at the origin, and
- the vector looks something like that.
- And then the second guy, 2, 4, 1, it would look like this.
- It would go 2 out here, we'd go 4 this
- direction, 1, 2, 3, 4.
- And then we'd go 1 up.
- So it looks something like this.
- 2, 4, 1.
- I think you get the idea.
- So I've drawn these two vectors that are essentially
- in my domain.
- Our domain is r3.
- This is our r3 right here.
- And let's see what our function maps
- these vectors to.
- So what's our codomain?
- Our codomain is r2, so this is much easier to
- visualize for us.
- So we just have to draw two axes like this.
- Let's call this x1 and let's call this x2.
- And so what does f of 1, 1, 1 -- of this yellow vector -- it
- becomes 3, 3.
- So if I do it in yellow, 1, 2, 3, 1, 2, 3.
- So it gets me this one.
- If I draw it in standard position the
- vector looks like this.
- So, literally, our function went from mapped from this
- vector in r3 to this vector in r2.
- That was what our function did.
- Likewise, if we take the other vector, we went from this 2,
- 4, 1 vector to sector 10, 3.
- So 1, 2, 3, 4, 5, 6, 7, 8, 9, 10.
- So it's going to look something like this.
- And it's going to be 3 up, so it's going to look
- something like this.
- So this vector right here, by our function, f, got mapped --
- let me do a different color -- to this vector.
- This vector right here in r3 got mapped to this vector in
- r2 by our function.
- Now this is just a switch of terminology.
- When we talk about functions of vectors the term that we
- tend to use is the word transformation.
- But it really is the exact same thing as a function.
- I don't want to confuse you, because if you watch the
- differential equations playlist, you saw the idea of
- a Laplace transformation, which is really an operation
- that takes a function as an argument.
- But in this case, and when we're dealing into linear
- algebra world, a transformation is really just
- a function operating on vectors -- or the way we're
- going to use it -- it's just a function operating on vectors.
- And so the general notation, instead of writing a lowercase
- f like that, for a function, people use an uppercase T to
- say it's a transformation.
- Although it doesn't have to be an uppercase T.
- But that's the one that people use the most. Just like this
- could be a g or an h, but people always
- use a lowercase f.
- So the same way we could have written this, we could have
- called this a transformation.
- And my sense of why, in the linear algebra world, they use
- this, is because you kind of imagine that this vector is
- being changed into that vector.
- Or that this is vector is being
- transformed into that vector.
- I think that's why they call it a transformation as opposed
- to a function.
- And it actually makes a lot more sense when you start
- going into things like video game programming.
- And a lot of what we're embarking on with our
- transformations is key to video game programming.
- But you're kind of transforming one image into
- another image if you're viewing at it from a different
- angle or whatever else you might want to do.
- We'll talk a lot more about that in the future.
- But I just wanted to introduce you to this notation.
- So these statements, I could have replaced all my f's with
- T's and I could have defined some transformation.
- And I just want to make you comfortable the notation.
- I could have defined it similarly, a transformation
- from r3 to r2 and I could have said that T of x1, x2, x3 is
- equal to the 2-tuple x1 plus 2 x1 comma 3x3 And I could have,
- just as similarly put a T up here because I have defined it
- the same way.
- I could have said T of my vector 1, 1, 1
- is equal to 3, 3.
- Now, you might say, hey Sal, why are you going through all
- this trouble of replacing T's with f's?
- I'm just doing this so you don't get confused.
- So that when you see in your linear algebra book, when you
- see linear algebra problems, when you see this big capital
- T and you're like, wow, I've never seen that before and
- they're using this fancy word called a transformation.
- This is completely identical to your notion of a function.
- It is a function.
- In the next video I'm going to talk about linear
- transformations.
- That's really just linear functions.
- And I'll define that a little bit more
- precisely in the next video.
- But hopefully by watching this video you at least have a
- sense that you can apply functions to vectors and, in
- the linear algebra world, we tend to call those
- transformations.
- And hopefully this example right here gives you, at
- least, a visual representation of why it's called a
- transformation.
- We're transforming from one vector to another.
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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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