 In the previous video, we saw that a linear system can only have three types of solution sets. Either it's inconsistent, it has no solution, it could have unique solution, which would make it consistent, and we call this the independent case, or it could also have infinitely many solutions, which would make it consistent, and we refer to that as often as the dependent case. And so these three possibilities are the only possibilities when it comes to solving systems of linear equations of any dimension. In this video, I wanna talk a little bit about how three dimensional pictures might, how you might visualize them and think about these three possibilities here as well. Because the same three possibilities are, those are the only possibilities available for three dimensions as well. So if you recall a previous example we did in this lecture series, example 113, we had the following system of linear equations. It was a three equations, three, we were to graph the planes associated to each of these linear equations. You get something like the blue, orange, and green planes illustrated here on the screen. You would see that because of the incline of the three planes, there would be one point in an intersection common to all three of these points. You can see this point right here, which would be 234. And so in the three dimensional case, this is an instance of this independent consistent case because all of the planes in the system intersect at a unique point. Visualizing this can be seen as these three planes intersecting at that unique point. And one of course can move around, but that's the idea, they come to a single point. For the inconsistent case, this would occur because not all planes simultaneously have a single point. Imagine you're looking top down on three planes. You could have it that the three planes make some type of triangular region for which no point comes together for all of them, right? Or you could have like two of the planes are parallel to each other. And so another plane crosses. Again, we're looking from the top down on these planes right now. There might not be a common point of intersection for all of these. Now, these all kind of constitute the same issues that lines, if you had three lines of the plane would give you as well, but planes extending out of the screen coming towards your face. Look out. Okay, it's not gonna hit you. Let's think about the dependent case though, for a moment. So one of, like we saw with lines, the dependent case occurred when the two lines completely overlaid each other. And that is a possibility here as well. We could have a plane, right? We could have a plane and then we have a second plane that just lives entirely on the same plane, something like that. But like you can see in this image right here, it could be that the planes intersect each other. So they don't over, I mean, they intersect so that there's no complete overlay. There are points distinct for the different planes, right? So we could distinguish between the green, the red, or the green, the orange, and the blue one right here. But there still is a common intersection that's more than just a single point right here. We see that there's this common line that lives amongst all the planes. So in this situation, we still do have infinitely many solutions, but we don't necessarily have that the two or three planes are equal to each other in any capacity. So we could still get the dependent case even when the plane, and this graphic you see on the screen is actually a visualization of our example one-one-fourth a previous slide. And we have this three-by-three system, the two x minus y plus z equals 11, you can read the rest of it there. In which case we saw that this system had multiple solutions. Two of the solutions reduced earlier, we had five comma negative one zero. We had a six comma two comma one as a second one. But I claim that there's gonna be infinitely many solutions here. And in fact, one can find the so-called general solution in the following manner here. We could take as our general solution, pick any z-quant, z could be anything, anything, anything, pick your favorite number. Whatever that number is, the x-coordinate will be five plus that number you chose, and the y-coordinate will be three times that number, but take away one. And so we can construct the other two coordinates based upon the choice of z. So we could make up some more coordinates on the fly right now. Like if we wanted to, we could choose z to equal negative one. That would imply that x equals negative one plus five, in other words, four. And we would say that y would have to be three times negative one minus one, which would give us negative four. And so then we see that the point negative, or positive four, negative four comma negative one is yet a third point on the intersection of these three planes. And that's because the intersection is in fact this line. We can pick any point on this line and that would be a solution to the system of equations. This here in the yellow box is what we refer to as this general solution here. Some other vocabulary we wanna mention here is that in this system of equations, we're treating the z-coordinate as a so-called free variable. A free variable. The idea behind the free variable here is that we could choose anything we want for z. I picked negative one. In these previous examples, we took z to be zero, z to be one. You could choose z to be 17, the square root of seven if you wanted to. You could choose z to be whatever you want. But once you choose what z is, the x-coordinate and the y-coordinate of the solution will then be dependent on that choice. And so this is referred to as the dependent variables. And I should mention that the free variables are sometimes called independent variables, sometimes on the vocabulary you wanna use their same idea here. So this system has one free variable and two dependent variables. Once we choose the free variable, the two dependent variables will depend on that choice we made for z. And this is something we're gonna see often in the future that when we have this dependent case, that's because there is some non-zero set of free variables and then the dependent variables are then dependent upon. And so that's giving us some idea what this independence and dependence is all about, although there is more to it than just what I've said here. And so just a few things to kind of finish up section 1.1 here. I wanna mention a proposition, we'll call this proposition 118, that actually gives us some expectations of when a system of equation will have free variables or things of that capacity. So let's take a system of equations with M equations and N variables. So if we were to kind of list this thing out, our system of equations, we have one equation, two equations, three equations, no, dot, dot, four, M equations. So that's what we have here. We have M mini equations. And then the N value, N right here is gonna count your variables. You have like X1, X2, up to XN equals something B. And so the second number N is gonna be keeping track of the variables inside of our system. So this is what we said earlier. We call this an M by N linear system. M rows in columns, the way we kind of organize this thing. Now in the situation where we have N is greater than M, this means more variables. You have more variables than rows here. Then you're gonna have at least N minus M, many free variables. If you have too many variables, then some of them have to be free variables. That's what we wanna kind of get from this statement here. If N is greater than M, then I guarantee N minus M of them have to be free variables. Apply, there's multiple solutions here. All right, on the other hand, if our system of linear equations satisfies the condition that M is greater than N, this means we have more equations than variables. In that situation, if you take M minus N, we're always taking the bigger number minus the smallest one, then it turns out that M minus N many equations could be removed without modifying the solution set. So I wanna come back to this graphic we saw on the previous screen right here. So notice that we have these three planes that intersect on a common line. What if we kind of exit, just take it out. So I'm gonna use my white marker here, right? Because the background's white, so you can kind of, it kind of looks like I'm erasing it a little bit. Oh, you don't see the orange plane anymore. If you're like, well, what happened to the blue one? There's a big chunk out of it. Oh, no, no, it's there, right? This is really good graphics right here, I know. But it's like, oh, what if we took the orange plane out? You'll notice that the intersection of the green plane and the blue plane doesn't change when you take out the third one. And this is the dependence relation in action right here. The orange plane isn't really offering anything to the solution set. You can get the same solution set without it. And so condition two right here is giving us a situation which we can guarantee that there are needless equations in the system of equation. If you have too many equations, then it turns out you could remove without actually changing the solution set. Now, this first situation when N is greater than M, that is you have too many variables, this is what we often refer to as an under-determined system. Under-determined system. In contrast, if you have too many equations, this is what we refer to as an over-determined system. Because the idea is if you have too many equations, then there might be too many restrictions on the system of equations, so you have too many. And when you're over-determined, it starts to become very likely that your system is inconsistent. The more equations you add, the harder it is to solve the system because you have to find a solution for all of the equations. On the other hand, the more variables you add to the system, then it's easier to solve it because you have more freedom. You might be able to correct some of the defects that the equations have. And so having too many equations will over-determine, having too few equations, which means you have too many variables, would then under-determine the system. And there are some conditions we can guarantee from these under-determined systems and over-determined systems. One last bit of vocabulary that we'll give you in section 1.1 is the idea of a homogeneous or non-homogeneous system of equations. So we say the system of linear equations is homogeneous. When you take every single equation, it can be written into the form where the right-hand side looks like a zero, right? So you have all your variables on left-hand side. They will equal zero on the right-hand side. And this is what homogeneous means. The word homogeneous would translate as one gene, one family, everything's the same. So for a homogeneous system, the right-hand side, they're all zeros. And so we say that a system that's not homogeneous would be non-homogeneous. And we're just describing the right-hand side of the equations. We want those to be all zero. A quick example of that, consider this first system right here. You'll see that x plus 3y plus 5z equals zero. That looks good. 2x plus 4y plus 6z equals zero. That looks good. 4x plus 2y, there's no z's, okay, equals zero. Looking at the, on the right-hand side, you have only zeros. On the left-hand side, you have all the variables. And so this is an example of a homogeneous system of equations. This one's homogeneous. Now you have to be careful, right? Because when we say that a system of equations is homogeneous, all the right-hand sides have to be zero. One might be very tempted right here to be like, oh, this system of equation is homogeneous because all the variables are on the left and you have only zeros on the right-hand side. But gotcha here. You'll notice that there are some constants on the left-hand side. We can't declare whether a system is homogeneous or not until it's been put into that standard form. All of these friends need to move to the right-hand side. And when you do that, this system translates into the following. You get x plus 3y minus 2z equals four. That's what the first equation will look like. Then you get 4x plus 2y plus 3z equals five. And then you get negative x plus 5y minus 3z equals two. And so once you put this system of equations into standard form, we can then very quickly identify that, okay, this is a non-homogeneous system. It's non-homogeneous because the right-hand sides are not all zeros. They have to be all zeros to be considered a homogeneous system of equations. It's a very simple definition, but we do want to be aware of what this thing actually means. And what we can say about homogeneous systems, right, is that a homogeneous system of equations is always consistent. There's always a solution to a homogeneous system of linear equations. And I'll leave it up to you to kind of determine why is there always a solution, but you're gonna see that the solution to a homogeneous solution, it's kind of a trivial solution, but it's a pretty important solution that we'll be seeing over and over and over again in this course to you to kind of find what is that guaranteed solution to a homogeneous system. So that brings us to the end of section 1.1. Hopefully from the contents of these videos you'll be prepared to start working on homework 1.1 and the associated questions you'll find there. If there are of course any questions at all while you're studying this subject matter, feel free to post comments in the comments below here on YouTube, I'd be happy to answer any of your questions you see there. If you feel like you learned something, hit the like button. If you wanna get more updates about math videos in the future, feel free to subscribe, excuse me there. And I hope we'll have some fun together in this course. I'll see you next time everyone, bye.