Transformations, part 2 | Multivariable calculus | Khan Academy

So in the last video, I introduced Transformations and how you can think about functions as moving points in one space to points in another. Here, I want to show an example of what that looks like when the input space is two-dimensional.

This over here is the input space; it's just a copy of the XY plane. The output space is also two-dimensional, so the output space in this case is also the two-dimensional plane. What I'm going to do is just play an example of one of these Transformations and then go through the details of the underlying function and how you can understand the transformation as a result.

Here’s what it looks like; here’s what we’re going to be going towards: very complicated, a lot of points moving, lots of different things happening here. What’s common with this sort of thing when you’re thinking about moving from two Dimensions to two Dimensions, given that it’s really the same space, the XY plane, is that you often just think about the input and output space all at once.

Instead, just watch a copy of that plane move on to itself. And by the way, when I say watch, I don’t mean that you’ll always have an animation like this just sort of sitting in front of you. When I think about Transformations, it’s usually a very vague thought in the back of my mind somewhere. But it helps to understand what’s really going on with the function; I’ll talk about that more at the end.

First, let’s just go into what this function is. The one that I told the computer to animate here is f of XY as the input is equal to x² + y² as the X component of the output and x² - y² as the Y component of the output. To help start understanding this, let’s take a relatively simple point like the origin.

So here, the origin, which is (0, 0), and let’s think about what happens to that f of (0, 0). Well, X and Y are both zero, so that top is zero and the same with the bottom—the bottom also equals zero—which means it’s taking the (0, 0) to itself.

If you watch the transformation, what this means is that the point (0, 0) stays fixed; it’s like you can hold your thumb down on it and nothing really happens to it. In fact, we call this a fixed point of the function as a whole, and that kind of terminology doesn’t really make sense unless you’re thinking of the function as a transformation.

Let’s look at another example here. Let’s take a point like (1, 1), f of (1, 1). So in the input space, let’s just kind of start the thing over, so we’re only looking at the input. In the input space, (1, 1) is sitting right here, and we’re wondering where that’s going to move.

When we plug it in, x² + y², that’s going to be 1² + 1²; and then the bottom, x² - y², is 1² - 1². Plugging things in here, that’s (2, 0), which means we expect this point to move over to (2, 0) in some way. If we watch the transformation, we expect to watch that point move over to here.

Again, it can be hard to follow because there are a lot of moving parts, but if you’re careful as you watch it, the point will actually land right there. You can, in principle, do this for any given point and understand how it moves from one to another.

But you might ask, “Hey Grant, what is the point of all of this?” Right, we have other ways of visualizing functions that are more precise and kind of less confusing, to be honest. Vector fields are a great way for functions like this; graphs are a great way for functions with one input and one output. Why think in terms of Transformations?

The main reason is conceptual. It’s not like you’ll have an animation sitting in front of you, and it’s not like you’re going to by hand evaluate a bunch of points and think of how they move. But there are a lot of different concepts in math and with functions where, when you understand it in terms of a transformation, it gives you a more nuanced understanding.

Things like derivatives, or the variations of the derivative that you’re going to learn with multivariable calculus. There are different ways of understanding it in terms of stretching or squishing space, and things like this that don’t really have a good analog in terms of graphs or vector fields.

So it adds a new color to your understanding. Also, Transformations are a super important part of linear algebra. There will come a point when you start learning the connection between linear algebra and multivariable calculus, and if you have a strong conception of Transformations, both in the context of linear algebra and in the context of multivariable calculus, you’ll be in a much better position to understand the connection between those two fields.