 Welcome back to our lecture series, Linear Algebra Done Openly. As usual, I'm your professor today, Dr. Andrew Missildine. This is the first video for section 2.4 in our textbook entitled, Affine Geometry. Now before we get into the details here, I do wanna kind of specify that when you take your typical algebra class, like whether that's in high school, algebra one, algebra two, or sometimes they're called elementary algebra, intermediate algebra, and there's like a college algebra as well. Now we're in linear algebra. These classes, although they're called algebra classes, they're not purely algebra classes. There is so much emphasis placed on geometry. Algebra's one, two, and three often focus on functions. We talk about the graphs of functions. We graph circles and ellipses and hyperbolas and all these other graphs that come from equations and things like that. So this is often what's referred to as analytic geometry, which is essentially studying geometry from using algebraic equations and algebraic functions and the like, right? But when you take these high school algebra or college algebra classes, they really ought to be called high school algebra and geometry, college algebra and geometry. The reason that schools often avoid that terminology is they don't want you to confuse you with like a proof-based geometry class which is often taught in high school where you have like postulates about lines and planes and things like that. That's sort of like separate than the analytic geometry. But sure enough, they are, the algebra and the geometry are kind of married together and you have to study the two things together to kind of better understand one another, right? The algebra supports the geometry but the geometry supports the algebra. Linear algebra is no exception in that regard. A better name for this series would be linear algebra and geometry done openly, right? Because we can't divorce the geometry from the linear algebra either. And so in this section, I want to introduce sort of like the most primitive type of geometry which you can expect on a vector space and this is referred to as affine geometry. And so in this lecture, we're gonna talk about things which we call affine sets or just more simply, we're gonna call them flats. And this is gonna in many ways mimic the geometry we see in RN when we talk about things like points, lines, planes, et cetera. And so what is a flat? A flat, or again, these are sometimes called affine sets if we want to be a little bit more proper. These are first of all subsets of FN. So the flats live inside of FN but we are looking for things which are congruent to FM itself where M can fluctuate between zero and N right here. So congruent, right, this is a geometric term, right? Congruence is when you have a shape that is the same as some other shape. So like, you know, I mentioned that high school geometry class you often obsess about things like these two triangles congruent by the side angle, side condition and things like that, right? Even though the two triangles could be located in different places in the plane or in space, they have different vertices that determine the triangles. The two triangles are essentially the same thing because they have the same shape like the angles are the same, the side measurements the same. This is what one means by congruence that the two geometric objects are really the same thing. So an affine set is just supposed to be a set that is congruent to the vector space inside of the larger vector space. So like case in point, if you think of R2, right? R2 we think of as the real plane where we have like the x-axis and the y-axis and every point in the plane is just a point. You know, it can be determined by its x and y coordinates. What does one mean by a line? Like you see the blue one illustrated here on the screen. A line is just an object in the plane that looks like R1. What is R1? R1 would basically be the x-axis right here. The x-axis is a line, the y-axis is a line. This line we draw here is a line. And so a line is just things that look like the x-axis inside of sort of like this larger space. Backing up a little bit, let's think about like R3 first for a moment, right? If we were doing R3 geometrically, we might think of it as like we have the x-axis, we have the y-axis and we have the z-axis. Maybe we draw something like this, right? I guess I take that back. That doesn't follow the right hand rule, does it? We would want something. We wanna switch those things around, don't we? Or this is the x-axis and this is the y-axis. Cause now when you use the right hand rule that we've oriented to that correctly, sorry about that. But if we have like the x, y and z-axis, then your point, your points in space, they're just things like this. But what is a line? What does a line in this situation like before a line is gonna be something that looks like the x-axis? What is a plane? By a plane, we mean this object that kinda looks like R2. Although it might be floating somewhere else in space, a flat is just a plane. It's just something that looks like R2. That's what we mean by flats or these affine sets. And so what we're gonna do is try to describe flats in a general setting where our vector spaces might not be over the real numbers. We could do this over the complex numbers, over our finite fields or any other field that we could introduce but we're not going to in this series. Let's see what we mean by flats in this situation.