 So I'm going to introduce two methods of trying to describe flats in a general vector space. And the names aren't super clever. We're going to have the top down approach versus the bottom up approach. In this video we're going to discuss what we mean by the top down approach. Now the top down approach for describing flats is to use linear equations and we're going to start adding more linear equations. So let's say we have a single linear equation and use the example of R3 to help motivate what happens here. If you have a single linear equation like AX plus BY plus CZ equals some number D, this is going to afford you some type of plane that lives inside of R3. The angle will depend on the coefficients ABCD and things like that, but we'll have some plane that lives inside of 3 space. Because the solution to a single linear equation and R3 is a plane, what we do is in general, we say that the solution to a single linear equation in Rn, or in fact in any vector space fn, we call this a hyperplane. The prefix hyper here is kind of describing that it might be transcending our usual notion of three-dimensional geometry. But really what a hyperplane means is that a hyperplane is something, so like our hyperplane we'll call it H. This is something that's congruent to fn-1 when we look inside of the vector space fn. It's one, so to speak, dimension smaller than the universe that's living inside of it. So for example in R3, a hyperplane is just a genuine plane because a plane is essentially congruent to f2. This two-dimensional space. And when you look at the linear equation by itself, everything except for the first variable could be treated as a free variable after all. A single linear equation in R3 has two free variables and that two-ness is kind of with predicting that we actually had a plane right here. On the other hand, when you have a linear equation in fn, you're gonna have n-1 free variables and that's essentially why the hyperplane's gonna look like f to the n-1. And so this is an affine set that looks like fn-1 inside of fn. That's what a hyperplane is. Well what happens if we take a second plane inside of R3? If we take a second linear equation, it produces a second plane and we then could draw maybe something like the following. Looks like this maybe. And these planes of course we can see through them. And so when you look at the two planes, they're gonna intersect each other to form some type of line, right? And so maybe if we draw this a little bit better, like this right here is on the other side of the plane. So these things, whoops, no that's not what I wanted to do, I want to do something like this, yes. So when we look at these, whoops and I did that wrong as well, that part, there you go. So when you look at these planes, they intersect at each other at a line and I apologize for the crudeness of this drawing here. Clearly I never got a PhD in art, I already think like that. But when we intersect two planes, their common intersection would probably be a line, which a line is another type of flat, isn't it? So when we put the two equations together, so let's imagine we have now this system of equations where we had the first equation a11x1 plus a12x2 all the way up to a1nxn equals something we'll call b1. We didn't throw in the second equation a21x1 plus a22x2 all the way down to a2nxn equals b2. So when we look at this system of equations, two equations in unknowns, we've now intersected two hyperplanes. So we have the intersection, the intersection of two hyperplanes. And when we take the two hyperplanes together, their intersection is not a hyperplane. In some respect, the intersection is one dimension smaller. If we take two planes, which we think of as two dimensional, when we intersect them, we get this one dimensional object, which in R3 we'd call that a line. But in general, this is going to give you something that's congruent to f in minus two. We sort of lost two dimensions when we intersect these two hyperplanes. So one has to be careful that if these two equations, if they're not linearly independent, then adding the second equation actually doesn't change the first one because they just overlay each other. So if we have two independent equations, their intersection will give you a flat that is two dimensions lower than the ambient space. And what if we keep on doing this? What if we keep on adding more and more equations to the system of equations? What if we add another equation A31x1 plus A32x2 all the way up to A3nxn equals B3? And what if we add another one, A41x1 plus A42x2 all the way up to A4nxn is equal to B4, right? We could keep on doing this. We could keep on adding it. And every time we add a new linear equation, assuming that it was linearly independent to the equations we pretty easily have, right? The first one gave us a hyperplane f in minus one. But then the second one, when we put them together, gives us a flat that looks like f in minus two. These are just congruence, right? Then we do the first three, you get something that looks like f in minus three. When you do the next one, you're going to get f in minus four. And so as you keep on adding more and more and more, eventually this kind of like terminates. There's a limit on how many times you can add a linearly independent equation, right? You end up with nnx1 plus An2... Sorry, not. That should be An1x1 plus An2x2 all the way down to Annxn equals Bn. And so if you look at this all together, you're going to end up with, again, assuming all the equations were linearly independent the entire time. We didn't add an equation that was already covered by combinations of the previous ones. When we combine this together, this should give us something congruent. The solution set should give us something congruent to f0, which f0 is just the vector space containing only the zero vector space. It's just a single point, a zero dimensional vector space. And so when we add this final equation, you get something that's just a point, which would be a unique solution. This is kind of why we love this square linear system when you're in by n. If your equations were chosen to be linearly independent, having the same number of equations as you have variables is the minimal number of equations you have to have to have a unique solution. If you have too few equations, if it's an underdetermined system, then you will not have unique solutions that will have to be multiple solutions. Because if you have too few equations, you have too many variables, which would represent this linear, you necessarily have to have this linear independence on the variables. That is, there have to be a free variable in the system. So we can describe flats as intersections of hyperplanes. You have one hyperplane, you're going to get fn-1. Two hyperplanes, if they're independent, will give you fn-2. If you have three hyperplanes, if the equations are independent, you'll get fn-3. You can keep on adding more and more equations until at the end you have a single point. Well, once you start adding another equation after the nth spot, you can't guarantee independence anymore. You're going to add in another equation. Well, you're going to get an overdetermined system in that situation. So you potentially, you're going to get inconsistency, right? So the only way you proceed is you add equations which are dependent to the previous equations. But you could also add them in such a way that you get inconsistent, right? So along this whole journey, we're assuming we were consistent and the equations were independent along the way. And so there is sort of like a limit in describing these affine sets. And this summarizes what we mean by the top-down approach describing a flat.