 All right. So we've learned how we can construct affine sets in a vector space using a particular vector on the flat and using spanners. That is, you have some span, which is parallel to the flat in hand. But what if we don't know who the spanners are necessarily? What if we just know vectors on the plane? How can we describe the flat internally without needing these external spanners to describe it? Well, in order to do that, I'm going to introduce the notion of an affine combination, which is similar to the idea of a linear combination we talked about earlier, but it's a slightly different thing, but not by much. So by an affine combination, what mean is the following? We have a set of vectors x0, x1 up to xm, and they all belong inside of fn right here. And we say that x is an affine combination of the other x's. If x is equal to a0, x0, or a0 is a scalar plus a1, x1, which of course a1 is also a scalar, all the way up to am, xm, like so. So if I were just to stop there and we see nothing else on this page, you'd be like, well, that sounds like a linear combination to me. Well, you're right. The affine combination is actually a special type of linear combination where the coefficients involved in the linear combination have the property that they sum up to one. So a0 plus a1 plus a2 plus a3 all the way up to am always adds up to one. That's not something we require for general affine combination or for linear combinations, but it is true for affine combinations. And so then we define the affine span, the affine span, which will denote this as aff of the set of vectors. So very similar to the spans we talked about before, which we could call those linear spans to distinguish between affine spans. So the affine span will be the set of all affine combinations of the vector, much like the span, aka the linear span is the set of linear combinations of all of those vectors there. And so the affine span by definition is just the set of all affine combinations. So you take linear combinations of the vectors so that the coefficients add up to be one. So for example, we could say that a vector is one half the first vector v1 plus one half the second vector v2. That would be an affine combination. We could take like one fourth v1 plus one fourth v2 plus one half v3. That would be an example of an affine combination. But also the coefficients don't have to be positive. They just have to add up to one. So we could take, for example, two v1 minus v2. That would be an example of an affine combination because two minus one is equals one. That's what we're requiring here. So the affine span, what this has to do with our, with the flats we've been talking about, is the affine span is actually the smallest affine set. It's so in itself is an affine set, but it's the smallest affine set that'll contain those specific vectors x0, x1, x2, all the way to xm. And to try to convince you about that, consider the following. If we have an affine combination, then in all reality, we could rewrite a zero as one minus a1 minus a2 minus a3 minus a4 all the way up to am. Basically, if I know what a1 is, a2, am, if I know all the coefficients except for a0, I can actually compute what a0 is because all the other numbers have to add up to one. So if I subtract them from one, that'll give me the remainder, which is our a0 right here. So that's true for every affine combination. This is the main reason why we care about them right now. Now, hey, if I were to distribute the x0 onto all of these things, and then combine terms not by the vector, but by the scalar, this equation would transform into this one right here. x is equal to x0 plus a1, x1 minus x0, plus a2, x2 minus x0, all the way up to am times xm minus x0 right here. So this second equation right here, if we ignore the assumption on the coefficients, this right here looks like the equation of an m-flat, right? We have us the general vector x is equal to a particular vector plus a linear combination of all these other vectors right here. And so no, for the general flat, we don't require that the coefficients have to add up to one. We could relax that condition, but notice that this equation is the equation of an m-flat and this m-flat will contain each and every one of those vectors. So for example, if you set all of the a's to be zero, if you set all the a's to be zero, you'll just get x0. So this m-flat contains x0. If you set the first coefficient to be one and all the other ones to be zero, then you're going to end up with x1 minus x0. The x0's cancel, so you get x1. Similarly, I could set the first one to be zero, the second to be one, all the others to be zero. Again, the x0's will cancel and you'll be left with just an x2. So this m-flat will contain each and every one of those vectors, x0, x1, x2, all the way up to xm. So it contains all of the vectors we want. And if we then expand from affine combinations to linear combinations, we can produce a bunch of other things as well. And so basically the moral of the story here is that if we have a set of points, we can pick our favorite point to be the representative of the flat. And then if we take the difference of all the other points from our representative, if we take the difference of all of those, these right here then form the spanners. We can form the spanners of the flat by subtracting particular points on the flat there. So let's look at some examples of this. So let us find the vector equation and the associated parametric equations for a line in R3 where we have two points, 1, 2, 3, and 2, negative 2, 0. So we've often heard the phrase that two points determine a line. We are now going to actually prove that that's possible, at least in R3 right here. So these points are points on the line. So we have some x0 and we have x1. It doesn't matter whose 1 and whose 0, just make a choice. And I'm going to pick x0 to be to be the first 1, 1, 2, 3. Now to find the line, I need a single spanner. I need some spanner, which we're going to call V. V is going to be formed by taking x1 minus x0. So we're going to take 2, negative 2, 0 and subtract from it 1, 2, 3. And when we do that, we get 1, we get negative 4, and we get negative 3. This then gives us the slope vector of the line. And so therefore our equation of the line will look like x is equal to x0 plus Tv. More specifically, x looks like x1, x2, x3, because we're in R3. Plus this is equal to x0. x0, we decided it was going to be 1, 2, 3. But we could have interchanged the roles. And this could have been x0 if we wanted to. It's just a selection there. And then plus T times our directional spanner here, we got 1, negative 4, negative 3. And so this then gives us the vector equation. If we want to switch this over to the parametric equation, so we get an equation for each coordinate. I'm writing them as x1, x2, x3, but you could call them x1 and z. That's more typical in a calculus class. x1 equals 1 plus T. x2 equals 2 minus 4T. And then finally, x3 equals 3 minus 3T. And so that's all there is to it. If you give me the points on the line, I can find the spanners and then I can use my vector equation to come up with the parametric equations if I want them. Let's look at another example of such a thing right here. So in this example, what we're going to do is we're going to find an equation of a plane inside of R4. And to determine a plane, we don't need two, but we need three points. We need one more point than the dimension, the flat. You might have noticed when we talked about those affine combinations, how the vectors were 0, x1, all the way up to xm, and these were useful to find an m-flat. In order to find an m-flat, you need one more point than the flat itself. So we need m plus 1. That's why you started counting at 0. So it doesn't matter who is who. So we'll just go in the order that they provided to us. This will be our x0, this will be our x1, and this will be our x2. The preference, the ordering doesn't really matter too much, but then we have to create our spanners by subtracting these things. So our first spanner we'll create by taking x1 minus x0. So that will look like negative 13, 3, 25, and negative 2. Take away negative 17, 6, 29, and 0. And so simplifying that actually gives us the spanner of 4, negative 3, negative 4, and negative 2. So that's going to be our first spanner. The second spanner we're going to form by taking x2, take away x0, like so. So x2 was listed above. We get negative 15, 6, 25, and negative 1. We're then going to subtract from it x0, which is still negative 17, negative 17, 6, 29, and 0. And then taking the difference of those things right there, negative 15, take away negative 17, so positive 2. You probably don't need me to narrate the entire arithmetic here. We're going to get 2, 0, negative 4, and negative 1. And so those are going to be our spanners. Therefore, using these spanners right here, our vector equation in general will look like x equals x0 plus su plus tv. And so if we plug in specifically what those things mean, we're talking about a plane in r4. So our x is going to look like x1, x2, x3, x4, like so. x0, as we've been using it here, we had a negative 17, 6, 29, and 0. For our first spanner here, we can't see it on the screen anymore, but we had 4, negative 3, negative 4, and negative 2. And then for the second spanner, we can still see it 2, 0, negative 4, and negative 1 right there. That's our spanner. And this gives us the vector equation. Well, this vector equation is a system of linear equations. We actually, be careful though, when you talk about the system of equation in this situation, I want you to be aware it's actually a system of 6 variables for equations because you get an equation for each of the coordinates x1, x2, x3, x4. Those are variables because they could be different numbers. Now, by construction, we're going to have four dependent variables, x1, x2, x3, x4, will then be dependent on the three variables, s and t, because x1 is equal to negative 17 plus 4s plus 2t. x2 is equal to, x2 is going to equal 6 minus 3s plus 0t, so that disappears. x3 is going to equal 29 minus 4s minus 4t. And then lastly, x4 is going to equal 0. We don't need to include that negative 2s minus t, like so. And so this gives us the parametric equations for this plane. If we don't have the spanners for a flat, like a line, a plane, a hyperplane, whatever, we can construct them by taking the difference of points on the flat, and those will then produce the spanners because of what we were talking about earlier with those affine combinations.