 In the previous video, we talked about the top-down approach to describing affine sets. What we're going to do in this video is describe what we call the bottom-up approach to describing flats. So instead of starting at the whole universe and going downward, we're going to start with basically nothing and build up a flat from there. And so remember when we talk about a flat, a flat is something that's going to be congruent to the vector space fp, where p is like the dimension of the vector space. It's some parameter for our affine set. So we'll have like one flats, two flats, three flats, four flats, things like that. And so using that parameter, we can kind of build a flat. So let's start at the very bottom. What happens when our parameter is equal to zero? That basically means we need something, we are creating a set that looks like f0, which really is nothing to do with Captain Falcon or anything like that. Instead, this is just a point in the space. So a point in fn is what we mean by a zero flat. A zero flat is just a point. And so we can describe that using a vector equation x equals x0. That's a very easy equation to solve, right? Like in a high school algebra class, if I give you the equation x equals 2 and I ask you to solve it, you're like, uh, is it 2? Yes, you did it. The same thing can also be done for vector equations. So I'm like, well, x equals the vector 1, 2, 3, then you're like, uh, is the vector 1, 2, 3, and I'm like, yes, you did it. So describing a zero flat comes down to solving this very trivial vector equation. The variable x is equal to a fixed vector x0. And this will then give us a zero flat, which is congruent to the zero space, the zero dimensional vector space, which is the span of the empty set. If you have no vectors and you take a span, you always get the zero vector. But other than that, you might not know. So that is what that's what we mean by a zero flat is exactly just a point. Well, what's a one flat? Then when we set the parameter to equal one, we're looking for something that's congruent to F1. So what would be congruent to F1? Well, that's what we mean by an F flat, which if an F of a zero flat is just a point. A one flat is just going to be a line. And so how does one describe a one flat a line and a general vector space? So a line L turns out we have to know a point on the line and we have to know the slope of the line. If you think about it like college algebra or high school algebra, right? We describe lines as the equation y equals mx plus b, where b is some point of reference. The y intercept, this is a point on the line. And we have to know the direction of the line, which is called a slope. We can mimic this when it comes to vector equations where we take the following vector equation, x. This is our general point on the line. It'll be given as x zero, which is some point, a specific point, a particular point on the line, plus t times v, where v is some directional vector. It's kind of acting like the slope of the line here. v is the slope vector of the line here. When you talk about lines in a high school algebra class or college algebra class, we don't usually think of a slope as a vector. We think of it as a number, but that's because in that case, when we do single variable algebra, single variable calculus, your slope is just a one dimensional vector, which is no different from a scalar. You can't tell that it's a vector quantity. But in multivariable settings, we start to see that the slope of a line is actually a vector, the slope vector v. This is some fixed vector, and then x zero is also a fixed vector on the line. And every point on the line is then the combination of a specific point, a particular point, plus some scalar multiple of your slope vector. That gives you a one flat, a.k.a. a line. Well, what about a two flat? A two flat p2, p equals two, we're looking for an object that looks like f2 inside of our vector space. This is what we mean by two flat, but this is what we call a plane. So a plane in fn is supposed to be something that looks like the plane where we think of, right? What we're going to do is we're going to take the combination of two independent vectors. So consider the span of two linearly independent vectors. But then if you translate that span using some specific vector that's on there, x zero, right? A plane is described as the solutions to the vector equation x, which is unknown, is equal to x zero, which is a specific vector on the plane, plus some scalar, unspecified scalar s, u, plus some other scalar t times the vector v. So much like the line here, the vectors u and v kind of act like the slope vectors. They're determining which direction is the slope inclined towards. So we have to kind of measure like what's the incline along the x-axis, what's the incline along the y-axis. But we can use other vectors besides the x and y-axis right here. And so these vectors u and v, more generally we're going to call these, you could call these the slope vectors some more. But we're going to call these the spanners, the spanners of our plane right here. So to describe a plane, what we have to do is we have to know the spanners. You need two independent spanners to form a plane because that's what makes it a two-flat. And you have to know a specific vector on the plane. If you do that, then you have this vector equation that describes your plane. And then if we want to go to a three-flat, we would do the same thing. We have to know a specific vector on the three-flat. And we have to have three independent spanners. And so if we take arbitrary linear combinations of the three spanners plus a specific vector on there, we then get a three-flat. A three-flat is determined by the vector equation x equals x-naught plus su plus tv plus, I need some more letters here. We're going to do rw. And this would give us a three-flat in our ambient space. To do a four-flat, we would take four independent spanners plus a specific vector on the flat. And that would give us an equation here. A metaphor I often like to use here would be, what if we'd have like an RC car, right? Let's think about like an RC car, like a remote control vehicle of some kind. Now, my toddler has a remote control car that's pretty easy to drive because he's a toddler. It can't be too complicated. And basically on the remote, it only has two buttons. There's a button that makes the car go forward and there's a button that makes it go backward. So basically, his remote control is one-dimensional. They don't want to make it too hard for the three-year-old here. It's his one-dimensional remote control. He can only drive the car forward and back. And basically the idea is if he bumps the wall hard enough or he bumps chairs on the floor, that'll cause it to move around. The car has some bumpers on it. And that's how he can kind of mimic two-dimensional travel. But essentially, his remote control only has one travel. You can either go forward or backward. There's only one joystick. And that's because it has only the one spanner on the remote control. Now, my older son, you know, he's 10 years old. And so he can drive much more sophisticated RC cars, right? And so his remote control, unlike the toddler, which just had the one joystick that just goes forward and back, my older son, on his remote control, he's got two joysticks, right? Two joysticks. Pushing one makes you go forward and back. Pushing the other one makes you go left and right. And so this actually makes it possible that the car can truly drive in two dimensions because it can go forward and back as much as it wants. It can go left and right. And so this remote control essentially has two spanners. Think of the joysticks on the remote control as your spanning vectors. The RC car can drive on a plane because it can go left, right, up and down. Sorry, excuse me, forward, back, left and right. It has two-dimensional travel because we have these two joysticks. Now, my 10-year-old, he wants to upgrade his RC car. He wants like a little helicopter drone, right? The remote control for that drone is going to basically look like the following. It's going to have a remote that controls up, forward and back. It's going to have a remote that controls left and right. And it also is going to have a control that goes up and down. This remote, this helicopter, it actually can travel in three dimensions because we have these three independent spanners. One that does forward and back, one that does left and right, and one that goes up and down. And so therefore, a three-flat is going to look like something that is three-dimensional even if the ambient space is larger than three dimensions. And I understand that this can get kind of confusing because we as three-dimensional creatures struggle to understand things that are fourth-dimensional, five-dimensional, three-dimensional, sorry, seven-dimensional, 12-dimensional, right? But if we think about it in terms of algebra, we can describe exactly the algebra without needing any of the geometric intuition, right? Because a three-flat is just a vector equation with three independent spanners, even if the ambient space is larger. And so what is an M-flat going to be? An M-flat just means we're going to have a sum, a linear combination, of M-independent spanning vectors. And if we specify a specific vector on the flat, that will uniquely determine the flat right there. And therefore, this then describes, this process summarizes this bottom-up approach. We can build flats by just adding more joysticks to the remote. We just add more and more spanning vectors to increase the span of the flat in consideration. And so this equation we see right here is often referred to as the vector form of the flat. But this equation right here is a vector equation. We can turn it into a system of linear equations. And when you translate your vector form into a system of linear equations, these are often referred to as the parametric equations of the affine set. And these parametric equations, this flat is the general solution to the top-down approach, that linear system. So there is this connection between it. As we've been solving linear systems that have multiple solutions, we often start off with this linear system, which is the top-down approach. And then we describe the solution set as, you know, using free variables and such. That's the bottom-up approach to describing the same flat. So solving a linear system, you go from a linear system to a general solution set, is basically converting from the top-down approach to the bottom-up approach of these affine sets. And so in this section, we want to kind of go the other way around. We want to focus on the bottom-up approach as opposed to the top-down, which we have traditionally been starting with, with these geometric problems.