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Introduction to the notion of vector transformations. Created by Sal Khan.
Video transcript
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.