 Okay, let's start things off with looking at systems of linear equations. Now you've got a textbook, you've got a lecturer, they've explained these things to you and so I don't want to get into the detail. I just want to get behind the concepts. Now some of the stuff that you are probably required to do is to solve these using gas, Jordan, Jordan elimination and you have to do all these row operations to bunch of them, do them for the test. I think it's kind of pointless. You're never going to do that in real life. It's more about understanding the concept than doing these. Look there are nice little puzzles to do but you make these small little arithmetical errors and it frustrates you and you're never going to do that in real life. So I feel it's a little bit pointless. Don't tell your lecturer, I said so. Anyway, let's just get to the concept of what's going on here. Now I just want to show you how these probably are created for many textbooks. I am going to do two equations and two unknowns. What you have to understand, those are going to be two lines in the plane where they intersect. That is going to be an X solution and a Y solution that solve both of those together. They are a system. Of course if the lines are parallel to each other, they're never going to cross so their slope is equal so they're never going to cross so there's not going to be no solution. Of course if they lie on top of each other there's an infinite number of solutions. You get that. That's really not that difficult. This is the way they're probably constructed. I'm going to choose the solution first. X is 2 and Y is 3 and I'm going to plug them in in this way. I'm going to say 2 times X minus 1 times Y and 1 times X plus 1 times Y and I put the X and Y values in there so I know what they are. So 2 minus 3 is 1, 2 plus 3 is 5. So I know that the two equations that I'm going to ask someone to solve is 2X minus Y, 2X minus Y is 1 and I'm going to say X plus Y equals 5. X plus Y is 5. Let's just break this chalk off. It's a terrible chalk. Anyway, what can you do? That's what I would give you and I would say would you solve this for X and Y? It's two systems. You're going to construct this. Well first of all that's not how you're probably going to get it. In terms of Y, so it's going to be Y equals 2X minus 1 and on this side you're going to get Y equals minus X plus 5, minus X plus 5. Something like that. I hope there's not a little mistake in there. So you have to bring it into that form. Now you're going to make this augmented matrix which is going to say a 2 and a minus 1 and a 1 and you're going to have a 1 and a 1 and a 5. You can even do those little things to remind you that this is an augmented matrix. The square brackets around parentheses doesn't matter. You're going to solve that. And as I say, I really think that that's kind of pointless exercise. There's so many more important things to get to, so many more interesting things, so many more real-world things. Why you'd have to waste time to do this Gauss-Jordan elimination? I don't know. It's easy to see what you have to do though if you just stick to the algebra. I mean it gets easier when this gets bigger to do it the matrix equation way It's easy to see what you need to do there. First row operation that you can do. You can just swap them around. No problems there. And number two is, I mean this is an equation. What is on the left-hand side is on the right-hand side. They are totally equal things. X plus Y and 5, they're totally equal things. So I can add this to that and I can add this to that. It's the exact same thing because these two things are equal. And if I want to eliminate an X, I'm going to multiply Y through out by negative 2, so I'm going to have something like 2X 2X minus Y equals 1. And then on this side I'm going to multiply throughout. That's my second row operation. I'm going to have minus 2X minus 2Y and that's going to equal minus 10. And as I said, what I do to the left-hand side, I can do it to the right-hand side. This left-hand side equals that. So if I add this to that, I can also add this to that. My X's disappear. I have a negative 3Y and I have a negative 9. Simply Y equals 3. If I plug Y equals 3 in there, it's simple to see that X is 2. Lo and behold, that's where I started off with. So you can create these for yourself. Three equations and three unknowns go nuts. You know, make your own and try and solve them. Do this as you're supposed to do. But as I said, that's not the issue. I'm going to show you, using the Wolfram Language in Mathematica, you know, sort of more modern way of going about it. Very easy to solve it of course with a computer and we're going to snowball from there using a computer to do all of these because that's real life, that's what you're going to get to. And as I say, doing this as a little recipe, that doesn't make you understand what's going on here, not at all. This sort of helps you understand what's going on. Okay, that's just a reflection of that. But the idea here is just to understand that these are lines and planes and hyperplanes and where they intersect, you get the solution, where they don't, you don't get a solution, where they lie on top of each other, you get an infinite many solutions. That's the important part, that's the important part. This, I can write many big matrices, sparse matrices. I can ask a computer to solve those for me to reduce this to row echelon form. Okay, here we are in Mathematica. I've got the desktop version here. All the links that I talked about, everything will be below. Use whichever you want. Do remember that I do have a course available. All that information also below if you want to learn how to use this. Just follow along for now. It is such an easy language to pick up. Just start playing. Really, there's nothing to it. Now, first of all, we had those two equations. We had y equals 2x plus 3 and y equals negative x plus 5. And I'm going to show you just simply how to plot that. So I'm just going to start typing plot, auto completes for me there. That plot is called a function. It's a keyword built inside the Wolfram language. There's over 6,000 of them. And you pass some information to this function. Those are called arguments. And the arguments all go inside of square brackets at the end of that function name. So I've got plot, I'm going to plot. I want to plot two little lines. And anything that I list, elements that I list, a list of things go inside curly braces. It's one of the things that you'll see. So in the one, we had 2 times x minus 3. The times, remember, most computer languages is this little star sign on my keyboard at shift 8. So 2 times x, what was it? Minus 1. And the comma, all the elements in the set are separated by a comma in a list, separated by a comma. So I have 2 times x minus 1. And the other one was minus x. And it was plus 5. And I'm going to close my curly braces. It's a list of two separate things. That's what I'm telling Mathematica. Because it's a list of things, I put them in curly braces and I separate them by commas. Then I'm going to put another comma, another curly brace. I'm just going to tell Mathematica what the range is on the x-axis. I should say the domain on the x-axis that I want plotted. I'm saying, please take x, say, let's take it from negative 1, say, to about 5. That's the domain on my x-axis. And then just to be very fancy, I'm going to use this argument called plot legends. See it pops up for me there. I can just hit Tab. It'll auto-complete for me. It puts that little arrow sign there. The way to get the arrow sign is just to do a minus and a greater than sign together. And I'm going to move my curses down to the expressions and hit return or enter just to have that. And I'm going to close my curly braces. The first time you see this, obviously you've got to look Greek. Do my course on Introduction to Mathematica. All of this makes sense. We'll just start playing like I'm playing here. And let's see what happens. And there we have it. We have our Cartesian coordinate system. I see the x-axis. I see the y-axis. And I see in blue my 2x minus 1 and my orange line there, y equals negative x plus 5. And we can see there where they intersect. We can see they intersect at x equals 2 and y equals about 3. And that's what these systems of equations are all about. I'm just solving these things. Now in Mathematica, I can create that augmented matrix. An augmented matrix is just a list of rows. And each element in that row also makes up a list. When I say things like a list, you know, in Wolfram language, think about curly braces. So the 1 was 2 comma negative 1 comma 1. Remember when I showed you the augmented matrix? The first row, the second row goes inside of its own curly braces. And that was 1 comma 1 comma 5. Remember that? Another curly brace. So the curly braces on the outside, it says this is one long list. And the curly braces on the inside says there's two nested lists in there. And each list is actually just a list in its own right. So when I hit shift and enter or shift and return, if you're weird and you're on a Mac, I'm going to get in trouble now. Anyway, that's what you see. But I can also do the following. I can say let's just do the exact same thing. 2 comma negative 1 comma 1. That was the 1. And the other one, what was it? 1 comma 1 comma 5. And I'm going to do this postfix notation, which is two forward slashes. If I do that, I can actually put the function, the function like plot, I can put it at the end. So it's called postfix instead of putting it in the front. But if I put it in the back, I've just got to tell Mathematica that I'm putting some function that should go in the front. I'm putting it in the back. So telling it by putting these two forward slashes. And the function, the keyword that I'm going to use is matrix form. There you see it. Matrix form. And Mathematica uses these nice little round brackets instead of the square ones. But there you can see it. It is the augmented matrix that we had before. And now let's do some simple Gus Jordan magic. Very easy to do. There is a function called row reduce. Row reduce. There we go. Remember, it's got to go inside of square brackets. All the information that I give this function inside of square brackets. I'm just going to quickly, I mean there's a way that you don't have to retype this, and I'll show you later. Anyway, so let's just put them in again. Negative 2, negative 1 comma 1 for the one row. The second row is 1 comma 1 comma 5. There we go. And I do all my curly braces nice and properly. And I close my function with a square bracket. And then I'm just going to do those two forward slashes at the end again and just say, well, all of that I just wanted in matrix form. And there's my row reduced. Look at that. Nice 1 0 2 0 1 3. And as I say, you've got a textbook. You've been to class, you know, X is 2 and Y is 3. It's as simple as that. And as I say, I don't tell anyone. Don't say I said it, but, you know, why you have to do that all on paper? It's beyond me. I'm sure. I mean, for me, it's a nice little puzzle. It keeps you busy. It keeps your mind going. You know, it sorts out the little erythmetical errors that you would make. Anyway, there you go. It's as simple as that. Of course, if you've got to pass an exam, you've got to do this, then use Mathematica just to check your results. It's as simple as that just to make sure that you've got the right answer before you submit. Now, that was two dimensions. Let's go into three dimensions. Now, I'm going to come to the screen on this side that I have on this side. I'm just going to... I've got two screens going here. I'm going to copy and paste here. Let's do that. And what I've entered here is just a special C in Mathematica. All these little things, they are cells. And these were code cells. So I wrote a piece of code and I got an output. What I've got here is I can tell Mathematica, hey, I want to write nice little textbook equations, please. And look how nice it formats that. And it tells me that's the first lot of equations that you've written. All sorts of formatting. Do my course. You'll see how that works. But now I've got three equations. So each equation is of a plane in three-dimensional space. And it's a system of equations. And we ask to solve that. And the same goes. We're just scaling up one extra dimension. Now, instead of a line, this is going to be a plane. There's three planes. And if they do intersect in a certain point, that is going to solve all three of them for value for x, y, and z, specifically some coordinate. If they are parallel to each other, you're never going to get. If they are on top of each other, they're infinitely many solutions. So these are planes. I could also draw lines in three-dimensional space. Doesn't matter. This is what you might be asked to do. And so what I'm going to show you here is let's just plot this. Show you how wonderful this is. Of course, now we're going up to three dimensions. Three dimensions. Same thing goes. I'm just going to do these three equations. Let's do them quickly. It's minus two times x. Minus two times x plus three times y. Where we go, minus three. That's my first one. So I put a comma for the second one. The second one is negative one over two. So how did I get that nice little fraction there? Well, I held down control or command and I hit the forward slash. Little key. And that got me that. So it's negative for half times x times x. And where do we go? Minus y plus seven comma. The last one is two times x plus y minus three. I'm going to close my curly braces because I've listed three equations that I've plotted. I hope I didn't make a little mistake. Now I've just got to tell Mathematica what I want. My domain for my x-axis and the y-axis. So let's make x go from negative five to five. It just tells you how much to plot. If you use the plotting calculator, I've got my favorite HP there. I still love those things. Anyway, I've just got to tell it the little window that it has to plot in. So I've got y also going from negative five to five. That looks fine. And let's also put plot legend. So we know which plane is which plane. Not labels, but legends is the one I want. And I want the expressions there. There we go. And I'm going to stall down. You can see my three equations there, the three colors that they are there. I'm just going to click on it and drag it out a bit so that we can see it in a very large way. And you can clearly see that point there where it intersects. Look at that. Look at that. Isn't that beautiful? We can see the three planes in three-dimensional space. And we can clearly see that they are really meeting up in that one little coordinate plane. I'm going to copy and paste again just to show you on this side. I'm going to say on this side copy and on this side paste. And there you go. You can see my second equation there. This is the augmented matrix from what we had before. And all we're going to do is just some row reduction now. So remember that that was row reduce. Row reduce. And I'm going to do my augmented matrix here. That was 2, negative 3, 1, negative 3. That was the one. The second one was work with me here. See that I don't make a mistake. 2, 7. And the last one was 2, 1, negative 1, 3. Was that it? Let's hope. Cross fingers. I'm not the world's best typist. And I want that in matrix form there. There we go. Row reduced form. And we see X is 1. Y is 2. And Z or Z, depending on where you live, is 1. Isn't that beautiful? And as I say, once again, I'm going to repeat this. It's not about doing Gastrodon elimination. It's about understanding what's going on here. That's the point that we are after. And it's simple to do and simple to understand and simple to visualize in Mathematica. Go on the internet. You can do it. You can sign up for free. Use it in your browser. You can do all of this in the browser. Sign up for yearly subscription. Go check out your university or your place of work. They might have a site license where you can get Mathematica for free or just buy it. As I say, for a powerful system that it is, powerful language that it is, it really is not that expensive. I love it. I use it for everything in my research unit here. Doing surgical research, of course. All the statistics, data analysis, machine learning, all done with Mathematica. Beautiful language. As I say, I don't get anything from them. I'm just plugging it because it is just a brilliant product and a brilliant computer language to use and learn, especially if you start out. Okay. There you go. You know what to do. You're going to go on to the page and just subscribe. Hit the like button. Well, if you didn't like it, you didn't like it. Nothing I can do about that. Try and do more of these. As I said, it's Christmas time here. Might get some more time in the evenings to make these. Certainly try my best and we'll do some other types of mathematics as well. It's just really a lot of fun to do. Come learn with me. Hit that subscribe button and also the little bell so that you know when the new ones come out and hopefully that bell rings a lot and hopefully this helps you. I really wanted just to help one person and that's what making these YouTube videos are all about. Just that one person who didn't get it when it was explained saw this video and it helped. A whole new world opens up, especially when you look at this using a computer language. A whole new world might open up for you. So hit that subscribe button.