We're still in section 7.2. It takes a little while to get through this section. Sometimes I go faster than other times. But we have to really take our time because there's a lot of unfamiliar stuff.

Maybe solving systems of equations is something you've done before. Usually people haven't done three or four variables unless you've done it in maybe an econ class. You might have done it a little bit there. So what we need to do is, now that we've practiced a few, we've kind of got a little bit of an intuition for it, now we want to develop a system, a methodology for attacking

these systems. Something that works every time and you can just rely on it as a formula in a way. So what we're going to look at here is a methodology for solving linear systems.

And we're going to do this, we'll codify it at the end, but sometimes in math, sometimes it's good to give the formula first and then do some examples. Sometimes it's good to do an example and just kind of work your way through it, see what you learn about it while you're doing the example, and then extract the important

information from the example. And that's what we're going to do this time. We're going to take a look at an example.

So there's a good old linear system. The word linear applies because all the variables only have powers of one, so that word there describes that phenomenon. And then the system means I have multiple equations.

And then I want to find one solution that satisfies every equation simultaneously. That's the goal, if it exists. Sometimes it doesn't exist. In this case it is going to exist, and in fact I'm going to tell you what the solution is. The solution, I'll try to give you solutions ahead of time if you want to try to work them out in lecture. You can try to solve them while I'm solving them.

You can get ahead or whatever you want to do. So anyway, the solution here is x equals two, y equals three, and z equals negative one. So that's the ahead of time I can give you the solution. Now we need to work towards that. Now what I want you to do is just think about this system, try to answer this question

in your mind. What's the most important part, or where is the information contained? That's probably a better question. Where is the information of this system contained? There's a few choices here for the answer. How about the variables? Are they important? Could I call them a, b, and c instead of x, y, z?

Be the same system, right? So I could call them Mickey Mouse, Donald Duck, and Goofy. It doesn't matter, right? I could just do anything there for the variables. Ultimately the goal is to find the variables in terms of a number, and that's the solution, but the variables at the outset do not contain any information. So what I want to do is when I solve this, what are some of the methods that we've

already looked at? The methods that we liked from high school were we could eliminate a variable, and then we could also substitute. Those were the two good methods. So really what's happening when I'm substituting and when I'm eliminating is I'm just playing

with the coefficients here. So the real information is contained in the coefficients. So to work with this, what I'm going to do is I'm just going to extract all the coefficients. It's like I take the system and I distill it, and I dunk it in something, and it eats away all the unimportant stuff. And what I'm left with are just the coefficients.

That's really where all the information is. Let's put a label to this. We're going to organize the coefficients in a matrix.

What's a matrix? A matrix is a big rectangular blob of numbers. So I'm just going to take all the important information. In front of that x up there, there's a 1 in the upper left-hand corner.

And then I'm just going to go across the first equation, and that's going to create what we're going to call the first row of the matrix. So I have a 1, a 1, a negative 1, and then I'm going to draw a line here to indicate the equal sign.

There's a lot of different things to talk about when it's the first time through. We'll go slowly here. Now I'm on to the second equation. I've extracted all the information from the first equation. Also, when I'm doing this, I'm speaking, I'm writing, and I'm reading, so sometimes

they don't all get coordinated. I end up with the wrong thing on the board. So try to pay attention, and since we're on film here, help me out if I write something incorrectly. Okay, so the second one there, I've got a 2 in front of the x, I've got a negative 1 in front of the y, I've got a 1 in front of the z, and then I've got a 0 on the other side of the equal sign.

The third equation, negative 1, 2, 3, 1. So the first thing is to just realize these two are interchangeable. I could give you a system like this, or I could just give it to you like this. Here's a matrix that represents a system of equations. Go solve it. So you could go back and forth.

If I gave you this, you should be able to write that equation down, or the system of equations. Okay, now let's also spend a little time and see what the solution looks like. If I had this solution, what would it look like in the matrix? So over here I have the solution, well, let's just follow the model here.

Whatever coefficients I have in front of the x, y, and z, I'm going to put them in the matrix. So what's that going to look like here? Well, for the x, I have a 1 there, but then there's no y, and there's no z, and x is equal to 2.

So that line right there represents x equals 2 because it's x plus 0y plus 0z equals 2. That's the same as x equals 2. So zeros, that's another lesson we're trying to learn early on, is that zeros represent

missing variables. If my goal is to eliminate variables, then my goal is to create zeros in the matrix. The next one, y, it's like having 0x plus y plus 0z, and then that equals 3, and then the last one will be 0, 0, 1, negative 1.

So that's what the solution looks like. The way you want to think of this is, this is a jumbled up Rubik's Cube, and this is the solved Rubik's Cube, and we need to learn how to go from here to here. If I can manipulate this in the proper way, I have to tell you what the rules are.

If I manipulate it in the proper way and I arrive at this, then I get my solution. So just like the Rubik's Cube, if I hand you the Rubik's Cube, you already know how to operate it, but there's certain rules, right? You can turn the edges, you can turn the top, but you can't turn it diagonally.

If you turn it diagonally, the whole thing breaks open. So there's certain rules I have to follow when I'm manipulating this. Those rules are going to come from what we already know how to do with equations. What do we do with equations? Let's just see for a second here. How would I eliminate that 2x?

Let's say my whole focus was to get rid of the x in the second equation. Well, see if you follow this, I could multiply the top equation by negative 2, right? That doesn't change it, I'm just going to multiply both sides by negative 2, and then add it, and that would get rid of that 2x. So what does that mean here?

That means I would multiply this row by 2, and add it to the second row, and then that would cause a zero here. Okay, so we're going to use the tools that we already know how to do up in the system of equations. We're going to apply that to the matrix situation.

Now I've rigged it here, where this 1 is going to be convenient for us to get started. Notice over here what my goal is. I'd like to have a 1 in the upper left-hand corner, and I'd like to have a bunch of zeros. So let's try to get this first.

And the way I'm going to do that is I'm going to eliminate these. This is the x in the second equation, I'm going to eliminate it, I'm going to eliminate that. And then I'm going to get, you might think of that as the first side of the Rubik's Cube. And just like with the Rubik's Cube, once you get one side, you don't want to mess it up. Even if you don't know how to do the Rubik's Cube, you can appreciate that if you get one side,

then you're trying to do the rest. If you mess up that side, you went backwards, right? So you have to keep the side and finish the rest of the cube. Same with this, I'm going to get one side, and then try to arrive at that. Okay, now this little guy here, this has a name.

Call that the pivot. And it's a little bit like your pivot foot in basketball, if you know what that is. In basketball, you have your pivot foot, and it's the foot that you're allowed to move around on. It stays fixed, and you use that foot to allow yourself to move around. That's your pivot foot. So we're going to use this number to move around in the matrix,

in the sense that I'm going to use this one to get rid of these two numbers. That's my goal. Okay, so I'm going to rewrite the whole matrix again, just so I have it down here. We'll start a little string of operations.

It's always dangerous recopying a matrix. You can leave out a minus or something. So, keep track of it for me. Okay, what are we going to do to change this matrix? In my mind, I'm just thinking it's the same as eliminating variables up there. So if I want, I can just say, how would I eliminate that 2x?

I multiply this by negative 2 and add it. So we have to have notation to depict what's going on here. What I'm going to do is I'm going to replace row 2. Okay, row 2 is going to get replaced, and I'll give you the language afterwards, but I'm going to write the symbolism first. Row 2, we're going to replace it with, we're going to go negative 2 row 1 plus row 2.

And that's the symbolism. Row 2 will be replaced by negative 2 row 1 plus row 2. I don't care what else happens. All I'm focused on is getting rid of that 2 here.

I want to make it a 0. I want to get one side of the Rubik's Cube. Okay, this one can be gotten rid of just by adding the 2. So I'm going to replace row 3 with just row 1 plus row 3. And if you like, let's write this out in English.

Get out your math to English dictionary. This here says replace row 3 with row 1 plus row 3.

You can write that out. Let's write it all out. Okay, so the R stands for rows. Okay, let's see what happens.

The new matrix that I get, usually when you're doing these steps, something stays the same. You go and get that done right away. I'm changing row 2 and I'm changing row 3, so row 1 isn't changed. Just rewrite that immediately. There's row 1 again. Then row 2 is going to become negative 2 row 1 plus row 2.

So I'm going to multiply by negative 2 and add it. That makes a 0 here. Then multiply by negative 2. Negative 2 plus negative 1 is negative 3. Multiply by negative 2, that makes positive 2. Add 1, that makes positive 3. And you do the same over here.

I'm multiplying by negative 2, that's negative 12. Add that, I get negative 12. Okay, that's the second row. And the whole goal was to start creating zeros and I've accomplished something there. Okay, row 3, I'm just going to add these two. So row 3 gets replaced by negative 1 and 1 added.

These two get added, I get a 3 here. These two get added, I have a 2. And then when these two get added, I get a 7. Let's see if you agree with that.

Okay, so what I've done essentially is I've gotten one side of the Rubik's Cube. I've made some progress here. I've essentially reduced these two. This is now two variables. If I were to write this back out in terms of x and y, it would be negative 3y plus 3z. And then 3y plus 2z equals 7.

So I've reduced it down to two equations, two variables, which you could go solve. But we want to practice this matrix method here. So let's do a few more things. Let's notice what's next. What else would I like in my Rubik's Cube? I'd like to have 0, 1, 0 for that column.

So what I'd like to do is I'd like to make this a 1. And let's do that by just multiplying by negative 1 third. This is essentially an equation, and I can multiply an equation by a constant. So in my next step, I'm going to replace row 2 by negative 1 third row 2.

Not a whole lot is going to happen in this step. Row 1 doesn't change. Row 2 does change by design. My whole goal was to get a 1 here. So I multiply by negative 1 third to cause that, and then I'll see what else happens. I get a negative 1 in this slot, then negative 1 third times this gives me 4.

And then the last row didn't change. See, look, I just did what I said not to do.

Now I'm in the situation that I was just in over here. You see, this 1 here, this pivot, I used it to eliminate these two guys. Now let's use this one here to eliminate those two guys. And notice what's going to happen is that this is not going to change, and neither are these.

Because as I add this, I'm just going to do 0 times 0 plus that, so that's not going to change. And then I'm going to be able to make these zeros, but none of the other things are going to change. So I'll keep that one side of my Rubik's Cube intact. So let's go over here. I'll reproduce it here.

1, negative 1, 6, 0, 1, negative 1, 4, 0, 3, 2, 7. So what am I going to do now? My whole goal is to make a 0 here and a 0 here. The way I do that is, for this one, I'm going to replace row 1.

What am I going to replace it with? Negative row 2 plus row 1. And the whole goal is to make that a 0, and I'll see what else happens. Yes?

Yeah, you have to be a little bit careful. What you want to do is you're incorporating this one into this one. So, yeah, I mean it really is row 1 minus row 2. It's the same thing. But the way I always draw this, I'll be consistent with this, is I'm always going to put the one that's replaced in the second term.

You have to be careful doing stuff like that in multiple steps because sometimes you change it and then you use it again later. So I just stick to this. Okay, so that's going to, let's just do that right away. Row 1's going to change, and all I'm doing is I'm multiplying this by a negative and adding it.

So that's 1, 0, I multiply by a negative here, that makes positive 1, so I add that, I get a 0. Looks like I got a free 0 there. That was nice. And then I multiply this by negative 1 and add it, I get a 2. Okay, row 2 isn't going to change, so let's just reproduce that. And then the last thing I want to do is I want to get rid of this 3 here.

So I'm going to replace row 3 with negative 3 row 2 plus row 3. Now some of you might be saying, well, why don't you use this one here? That's a legitimate question. Why wouldn't you use this one? Try to answer that in your mind.

Why am I using this one to eliminate that and that and not this one? So let's look ahead and see what happens if I do this. And this is something that you might have to work out when you're practicing these at home. You might do this and then you'll realize what happens. If I were to use this one, what's going to happen is I'm going to ruin these zeros. If I multiply this by something and add it, say negative 1, to get rid of this, I'm going to change that 0.

So once I've used this row to eliminate stuff, I want to move on to a different row. That's just my experience talking, but I can try to impart my experience to you. That's all I can do. Some of it you're going to have to go practice yourself, though.

Let's get our new row 3. We're going to multiply row 2 by negative 3 and add it. I get a 0 here, and then I get another 0. I'm multiplying by negative 3 here, so I get positive 3 and add 2, I get a 5. And then I'm going to multiply this by negative 3, that's negative 12, and add it, I get negative 5.

We're almost home free. We're almost at this situation. We've got this column, this column taken care of, let's do this one now.

Now we can be a little more nimble because we've done it a few times. All I'm going to do is I'm going to multiply by 1 5th here. So I'm going to replace row 3 by 1 5th row 3. Row 1 doesn't change. Row 2 doesn't change.

I get a 1 here and I get a negative 1. So far I've got this right here, this part. The only thing that's missing is this piece right there. Now I've got to get rid of this negative 1, and then I'll be done with my Rubik's Cube. So let's do that. What's going to happen? I'm going to replace row 2.

What should I replace row 2 with? How about if I just add row 3 to row 2? Row 3 doesn't change. This is a positive 1.

Row 2 is going to change. That's going to become 0 1 0. And then when I add these two, I get 3. Was that the answer? It was. And then this one's already solved.

So our Rubik's Cube is solved. That's the first one through, so it's always the hardest one. Are there any questions about that?

Now, let's just pause for a second. This was the complete solution, but you see, we don't always have to go all the way to the end. At first I want to take a few all the way to the end, but you'll learn as they get more complicated,

you want to learn how to stop part way and go back to the equations and get the job done without having to rewrite all these matrices. But we still need to practice this a little bit. Okay, so let's talk about what we just did here.

Try to codify some of the method that I just did. So what I've done is I've just reduced a matrix to what's called row echelon form of a matrix, sometimes called reduced row echelon form.

I don't want to make a distinction, so I'll just call it all row echelon form. All right, so as I've always done, we're going to investigate the vocabulary. Okay, we've got a matrix, so that's just talking about the rectangle of numbers. We've got a row here, so that's talking about these rows. We've been playing around with rows, so we kind of know what a row is.

And then what about this echelon word? How does that play a role here? Maybe it's somebody's name. Right, sometimes things are named after people. Not in this case, though. Echelon, so you have to ask yourself where in the English language have you heard that word?

Maybe like the upper echelons of society? What does that conjure up in your mind if I say the upper echelons or different echelons of society? What it should conjure up is that each time you move up the ladder in society, there's fewer people, so it creates kind of a pyramid.

Each time you go up in the echelons, there's fewer people, and so it's more elite, right? There's another place that this gets used in the military. Echelon formation.

Echelon formation is you can either have boats or planes. If you have ships, they travel like this when they're in a war setting. And the reason why is because if you're trying to fire on them, if you hone in on this guy, if they're all like this, then as soon as you hit this guy with the right distance,

you just turn your guns and you wipe out the whole fleet, right? And then if you do it in single file, then when you aim your guns, you'll hit this. Once you hone in on this guy, then all you have to do is just lengthen the gun and you wipe out the whole fleet. So to make them have to recalibrate every single time, they do in this staggered formation.

And that's called echelon formation. Planes have the advantage where they can actually do it vertically as well to make it even harder. Okay, but anyway, so that's where the word comes from. So where do we see echelons here? Here's your echelon. There's your echelon. So that's the word.

Now let's talk about what it is exactly. So let's have an example here. 1, 0, 0. I have my example from above, but I'm going to add to it a little bit. I'm going to add on a little row of zeros.

I have 2, negative 1, 3. So the first thing you're supposed to pay attention to is what I just drew there, that staggered effect. And that's what enabled us to solve the system having that effect.

And then the other thing that's new here is this row of zeros here. So I want to write down what all the key features are. And essentially what we're doing is we're establishing a method for how to solve the Rubik's Cube.

If you notice what I did was I picked that 1 and I made everything in its column zeros. Then I moved over to the next column and I got a 1 and I made everything 0. And then I moved over to the next column and I got a 1 and I made everything 0. So let's write that down a little bit. Now this didn't happen in the last one, but I have to put it in the features because you have to be able to do it.

When you're doing different problems. Okay, so rows with only zeros are at the bottom.

The first non-zero entry is a 1.

If I go to this row, the first non-zero entry is a 1. Go to this row, the first non-zero entry is a 1. Go to this row, the first non-zero entry is a 1.

Here, this one doesn't have a first non-zero entry, so it doesn't violate this. Okay, let's identify this with vocabulary. This is called the leading one.

Okay, so that's the second one. So those are all called leading ones. That's the vocabulary that we use to describe those ones. Okay, the third feature. I have to describe the fact that each one moves in. This echelon, I have to describe that feature there where you have a 1 and then it moves in.

So then, the leading one, one place to the right of the row above, you have a leading one.

Then as soon as you move down one, the leading one has to move over to the right.

That describes that echelon. And then here's the way we were solving the Rubik's Cube is each column with a leading one zeros elsewhere.

So notice that that's what happened. Here I had a leading one in that column and everything else was zero. Here's a leading one in this column, everything else was zero.

There's a leading one in the third column, everything else is zero. So that's that fourth feature. Let me ask you with a camera person, can you see this down here at the bottom of the board? Is that okay? Okay. So those are the four features. So let's practice it again.

You have to watch somebody do it. It's a little hard to watch somebody do it because it's arithmetic. So I recommend trying to do it with me just to kind of keep your attention going.

I guess I can write it right here. Anybody need this board here? This board I was writing when I finished it? Okay.

So each time I do a new system, I'm going to add some new features that, you know, if I put them all in the same one at the beginning, it'd just be too much. So we want to add different things. This last one, I made it very convenient by having this one in the upper left-hand corner. It allowed me to just get started right away. Here I'm going to hide it a little bit. We'll have to move things around.

I'll give you the solution. It's x equals 1, y equals 0, z equals negative 1.

Let's go over to a clean board and write our matrix.

Okay, let's just verify that we got everything copied correctly. All right, we're ready to go. So before you just dive right in, our goal is to get it in this form here so we have our solution. So it's pretty far from that. In fact, there are no zeros.

So one way to just have an intuition for this is you're just trying to create zeros. Especially at the beginning, just get some zeros going. In the first example, I solved this side over here, and I got a 1, 0, 0, but I don't have to do that. In fact, it's not so convenient to do that here. What was convenient last time was this 1.

So what I do is I just search for any 1. Any 1 is going to be good. Or even a negative 1 is good. So I can use this 1 to eliminate that negative 5. How? I multiply by 5 and I add it. I can use this 1 to eliminate that negative 2. How? Multiply by 2 and add it. So that's going to be my first move.

I'm going to replace row 2. My whole goal is to make zeros here. So I'm going to replace it specifically with 5 row 1 plus row 2. That's going to create that zero, and I'll see what else happens. I'm going to get rid of this negative 2, so I'm going to replace row 3 with 2 row 1 plus row 3.

And what do I get? First of all, the top row doesn't change, because I just used it. The second row is going to change, so let's go do that one. I'm multiplying everything by 5.

I'm going to multiply this by 5, that gives me 15, and add it, I get 17. Multiply this by 5, that's negative 5. And add that, that's negative 2. And then this is zero by design. Multiply this by 5, I get 10, and add 7, I get 17.

Let's do row 3. For row 3, I'm multiplying by 2 and adding it. That gives me a zero here. Then I multiply by 2 here, I get 6, and add it, I get a 3.

Multiply this by 2, I get negative 2, and add, I get a 4. And then multiply this by negative 2, that's negative 4, and add that, I get negative 5. Let's see if you believe me. All those steps there. I'll see if I believe myself.

I don't think I believe myself. I'm multiplying by positive 2 here, so that's positive 4, and I'm adding that, so that's a 3.

Do double check your work when you're doing this, because once you mess up one thing, you're lost. OK, but I made some progress. Yes? Why didn't you start by solving for the top left and then using that as a big point like you did in the first one? OK, good question. I tried to address it a little bit, but first of all, it's just my experience.

Now, why did I not want to do this? How would I make this a 1? I'd have to multiply by 1 third, and then that's going to cause a fraction here, a fraction here, and a fraction here, and I just don't like that. So I could, if I wanted to, but I don't want to involve fractions if I don't have to. So what you said is, why didn't I make a 1 here and then use that as a pivot?

You could, but you're going to have to multiply this row by 1 third, and that's going to create a bunch of fractions, and you want to avoid fractions, unless you're, you know, some kind of fraction genius or something. But most of us try to avoid fractions. OK, but I did make some progress here. I got, I got something. I got some zeros.

Now, let's think about this. Could I just take this column and move it over here? Could I just switch those two columns? Would that change the system of equations? What am I really doing? I'd just be moving the z's over into the front and the x's over to the right. That wouldn't change anything. The order of the variables doesn't matter.

So if I wanted to, I could actually switch these two. I'm not going to, because that's going to add some complexity. You have to remember, then, that the first column represents z and the last column represents x. So I don't want to do that, but you could. You know, once you get comfortable with this stuff, there's all kinds of things you can do. Let's keep going here.

So now what could I do? I don't really see any way to create a one without creating some fractions. So what I'm going to do here is I'm just going to notice this is a multiple of that two. So let's, let's just make this four a zero by multiplying by two here.

Now I don't want to use this one because what's going to happen is as I add, this one's going to change the zeros. I worked really hard to get those zeros, so I don't want to use this row anymore. I'm going to now move on to the second row. OK, what am I going to do? I'm going to replace row three with two row two plus row three.

Why did I choose to do that? I'm just trying to make some zeros. And I'm trying to avoid fractions also. OK, let's see what we get when we do that. The first row doesn't change, so that's reproduced there.

The second row doesn't change, so a lot of the work's done. The third row is going to change. I'm going to multiply this by two and add it. I get a zero here, I'm going to get a zero there. Multiply this by two, that's thirty-four, and then add the three, that's a thirty-seven.

And then I'm going to multiply this, it's the same thing, multiply this by two, that's thirty-four, and add three, that's thirty-seven. OK, so that's convenient now. I could multiply this row by one thirty-seventh. Let's do that. I'm going to replace row three by one thirty-seventh row three.

And the top row doesn't change, the second row doesn't change, and the third row becomes...

Well, if we look back at what we want, we'd like to have a one zero zero in the top there, right? So this one zero zero, let's move this to the top here. That's just like switching the equations. Would it have mattered if I had written the bottom equation in the top? You can switch the equations around. So here's my new move. I'm going to switch row one and row three.

OK, that's the notation, the double arrow there to indicate that I'm just switching the two rows. So now I have one zero zero one on top, on the bottom I have three, negative one, one, two, and the second row stays the same.

OK, now couldn't I use this one to now get rid of these two? Now I can do that, right? I can use this one to get rid of the seventeen and the three.

So let's do that. Now I'm going to replace row two with negative seventeen row one plus row two.

OK, let's do that. The top row doesn't change. OK, when I do that I get a zero here, I get a negative two here, and I get a zero here.

So this becomes zero, I'm doing negative seventeen plus this is zero, and then zero plus negative two is negative two, and then I get a zero here and zero there.

And then let's do the last one here. I'm going to replace row three by negative three row one plus row three. I get a zero in the first spot. Notice also, you can kind of do some short cuts, when I have zeros I'm not going to change anything,

because no matter what I multiply this by and add it, it doesn't change anything. So that's good for free. And then I multiply by negative three, so I'm going to multiply here by negative three and add it, I get a negative one. So we're getting pretty close to the final solution here. Pretty close, not completely there.

We've got a lot of zeros. This will be the last one that we do completely. But I just want to show you all the different row operations that we can do in one example.

So, what are we going to do next? Let's make a one here. Let's make a one there. So I'm going to do replace row two with negative one half row two.

Now I have one, zero, zero, one, zero, one, zero, zero, and then zero, negative one, one, negative one. And then we have one last step to get, let's get rid of this negative one, that'll be the solved Rubik's Cube now.

So I'll do, I'm going to replace row three with row two plus row three. Top row doesn't change, second row doesn't change, and then the last one becomes zero, zero, one.

And now we can read off, so you can read off the solution. That says x equals one, this one here says y equals zero, and this one says z equals negative one.

There's our solution. Now let's take a second and look where we could have saved ourselves some time. Now I claim all of this stuff was extra, just to get all the way down here. You didn't need any of this stuff right here. All of that could have been avoided. But you have to do it once to appreciate that it can be avoided, and then want to try to avoid it.

Right now you probably want to try to avoid it at this point. So, you see, when I was here I was done. So I have the experience to know. When I'm here I'm done, because this says 37x equals 37, so then I can find x.

Then once I have x, I can put it here, that's 17x, but I know x, and then I can solve for y. So we could have back substituted there. So let's just, since we don't have to do another example like this, let's just do that from here.

So from the point where we had this matrix here, let's just solve it. And this is really what you end up doing. I've got to make sure I copyright. Negative 1, 2, 17, negative 2, I've got a little glare up there.

37, 0, 0, 37. So here we were done, because we've done what we wanted to do. We wanted to eliminate two variables, right? That's eliminating two variables. So then I can solve for that variable.

So what you do is you just work your way up the matrix. So row 3, that tells me that 37x equals 37, which tells me that x equals 1. Then I go to row 2. What does row 2 say? It says 17x minus 2y equals 17, but then x equals 1, so that implies negative 2y.

Let's just write it out for your notes. So that implies 17 minus 2y equals 17, which means y is 0. And then the last one, row 3, that says 3x minus y plus z equals 2.

And x is 1, so that's 3 minus 0 plus z equals 2. That tells me that z equals 1. Negative 1.

So that's better, right? We don't really want to do all that stuff. I just wanted to show you different moves, like switching two rows, multiplying rows by constants.

But mainly, you can just get away with this move right here. Just doing a few of these moves reduces it down to the point where you can solve for one of them. Then you go back and you do back substitution. That's the real way to do this. Any questions about any of those steps? There's a lot of stuff there, so if you have some questions, go ahead and ask.

The first couple are hard because I have to explain everything. Now I've seen a couple, now I can turn up the heat a little bit. But we can also go a little more quickly. Oh, those are the constants in the equation.

So you have x plus y plus z equals something. The line represents the equal sign. This is what's called an augmented coefficient matrix. I'll write that down in a minute once I write down the generalization. Anyway, so what have we done here? Let's write down, we're going to stop our practice and we're going to go on and do some generalization.

No more arithmetic for a little bit. Generalization here. So what have we been doing? We've been solving systems of equations.

In this case, this was a four variable system with four equations. So if I'm going to generalize this situation, what I'm going to have is I don't know how many equations and I don't know how many variables. So let's say we have m equations.

Now this is a little bit of a problem in terms of notation. So let's start to write down a little bit. So here's, this is going to be equation one. I'm going to have, I have n variables, so I have to have x1, I have to have x2,

and then I have to go all the way down and have xn, and then that has to equal something. So let's say that's b1. Now these are all my variables, but let's just, before I put in coefficients, let's just get all the equations down. So equation two is going to be x1, x2, I've got to have all the same variables.

It's going to be equal to something slightly different, b2. And then, okay, well I don't want to sit here and go equation three, equation four, equation five. I'm going to go all the way down to the last equation, and it says here I have m equations.

So I go down to the last one, that's equation m. I still have all the same variables, x1, x2, xn. That's now equal to b sub m. And now I have to try to put in these coefficients.

So in front of each one of these guys, there's a letter, right? Or it's a number, but I'm going to represent it by a variable or a letter that represents the constant. So this is equation one, let's call this a. Now this is the coefficient in equation one in front of the first variable.

So I have to call it a double subscript, a11. It's the coefficient in the first equation and the first variable. Then the next guy is a, well it's still the first equation, but now it's the second variable. And then this guy is the first equation and now the nth variable.

So that double subscript, you can't avoid it. I ought to be able to identify where I am in the equation by which coefficient I have. And the only way to do that is to say which equation I'm in and which variable it's in front of. So you have to have two subscripts. This one here is equation two, variable one.

This is equation two, variable two. Then equation two, variable n. Then we have, okay now we go dot dot dot all the way down to the last equation. That's equation m, first variable.

Equation m, second variable. Equation m, nth variable. Okay, so that was a little bit of work. What's new is the coefficients and all the subscripts and everything.

Now let's put this in the matrix. So what you get out of this is the coefficient matrix.

And all you're trying to do here is just, like I said last time, is that part of this class is to get some mathematical maturity. You have to be able to deal with notation, deal with subscripts and things like that. Okay, so what's our coefficient matrix look like? Over here I'm going to have all the constants. This will be b1, b2, down to b sub m.

And then the numbers in front here will be a11. And then all along this row, this is row one and then nth variable. And then I go to a21, a2n.

All the way down to am1, amn. So there's your coefficient matrix. They call it the augmented coefficient matrix because we're augmenting it by putting the constants on the right side. So all of these are the coefficients and then when I add that, that's called the augmented coefficient matrix.

Then what do we do with this? We're going to manipulate this in order to solve the system. And what we saw in those previous examples is that we're doing row operations. So let's identify exactly what we can do is what are our legal moves to manipulate this matrix.

The row operations that we can do is I can replace some row. And I can't see, this is why doing it in general takes practice. Because I have to say it's just some row. I can't say it's row one or row two. It's some row. I'm going to replace it. And I'll take some other row and add it maybe to a multiple of it.

But this is what we've been doing, that row operation. That's the one we do over and over again where we take a multiple of one row and add it to the other row. I wish we could have finished that example here because I had a little bit more of a complicated one in there.

But I think I'll continue to abandon it. Okay, row operation number two. What else did we do in some of those other examples? I don't know if I have one up there right now. Yeah, I do. Multiplying a row by a constant just to make it better, right? Just to have a one or to get rid of a fraction.

So we can have, we can replace a row by a constant times the row. This one here is replace a row. I'm going to use some fancy language here with a linear combination of rows.

This is a linear combination. We'll talk more about what linear combinations are later. I just don't want to get too bogged down in that. It's just you multiply in a constant and multiply by a constant and add them.

That's what a linear combination is. And this one here is multiply a row by a constant. So that's a legal move. And then the last one was I could switch two rows.

Those are the three things that I did in doing those. And that got us down to the row echelon form. In practice, you're really just going to use this one, okay? Because what happens is if you use this one, then you reduce it down a certain point, then you back substitute.

You don't have to always do all these other steps. But these are the legal moves that you're allowed to do for a matrix. Or you're trying to solve it. Let's look at an underdetermined.

So what's an underdetermined? That means I have more equations than, or fewer equations than variables.

So that would be something like x plus y minus z equals 3. And 2x minus y plus 4z equals negative 3. So I have more variables than equations. So let's do a matrix for this. 1, 1, negative 1, 3.

2, negative 1, 4, negative 3. Here we have fewer steps to get going here. I just want to, um, let's get this reduced in this row echelon form. So the first step here is I'm going to use this one here to eliminate that 2.

So I'll go row 2 is replaced by negative 2 row 1 plus row 2. Okay, I'm multiplying by negative 2. So that makes a 0 here. Multiply by negative 2, I get negative 3.

Multiply by negative 2, that's positive 2. And add it, I get a 6. Multiply by negative 2, it's negative 6. And add it, I get a negative 9. So now I'm going to make a 1 here.

Let's go here. We'll go replace row 2 with negative 1 third row 2. I just want to get a 1 here. Let me add here for the undetermined system. Here, reduce to row echelon form.

This is the one where you have to reduce it all the way. You don't have to, but it's helpful to reduce it all the way because we're going to end up with infinitely many answers. If you look back at your, this is the one I asked you to try to do yourself before you came to class next time. So we're practicing it again, but it's helpful to reduce this one all the way.

Don't count on doing back substitution. Let's go here. Okay, so I'm going to do replace row 2 with negative 1 third row 2. And I get this guy.

And then one last step, I'm going to use this one to eliminate that, and that will be row echelon form. Let's do replace row 1 with negative row 2 plus row 1.

And we're home free. So, 1, 0. Okay, I'm multiplying by negative here, so that's positive 2. Add it, I get a 1. Multiply by negative 2, that's negative 6. And add it, I get negative 3.

See if you believe me. I'm just multiplying by negative 1. Multiply by negative 1, so this is 0. And then, that's it. Okay, so this is your row echelon form. Notice it solves all the conditions.

You've got your leading ones, and then also everything in its column is 0. So this doesn't matter. If you try to eliminate those, you're going to ruin those zeros. So that's the reduced form here. Okay, and then let's just see how we get our final solution.

Okay, so from row 2, I get y minus 2z equals 3, or y equals 3 plus 2z.

And then row 1, I have x plus z equals 0, or x equals negative z. Okay, so that's our final solution.

you see what happens is i've got everything in terms of z so now let's do that parameter thing we did last time we'll set z equal to t where t is any real number and then i can say what x and y are in terms of that x equals negative t y equals three plus two t

and then you can say z equals t there's your solution that takes practice, writing things down in terms of parameters takes practice because it's kind of abstract at first, it's kind of a new thing

that's why i invited you last time to practice that yourself you need to you can't just watch somebody do that and expect to be able to master it, you have to practice it any questions right now?

one last little concept as far as solving these systems there's another special kind of system that we're going to need to look at

we've got a new word here, we know what systems of equations are we're talking about linear systems here homogeneous is the word, it's not homogeneous like your milk it's homogeneous what that really means is it's like the same family, same family of

equations, so this is a homogeneous system an example, let's have an example the key to the homogeneous system is that all the constants are zero on the right side, that's a special case and it's going to come up later in our studies so

we have to study it a little bit now while we're in this context so there's the second equation, there's the first one, and the third one is constants zero, that's the key, that word here

this is this can you do this in your head? I claim you can

find a solution to that, find a solution in your head well if you're going to do it in your head it's got to be pretty simple, right?

how about zero, zero, and zero? don't those all work? if I plug in x, y, and z as zero, all three of those equations work and that's because you have zero on the right side so x equals zero, y equals zero, z equals zero is a solution

all right, well so are we done? well no, there might be other solutions, this might actually reduce down to the underdetermined system so we might get infinitely many so we have to worry about that a little bit here's the question

are there other solutions? and the answer is if so, there are infinitely

so if there is another solution, there's actually infinitely many

so you still go and solve these like you do the other ones we'll do that, probably try to avoid that board if I can

so let's see that we get infinitely many and then let's also, that gives us another chance to practice the organization make sure I copied it right

otherwise I'll get a bogus one okay, so let's put it in a matrix here one, one, two, zero, zero, zero and then I have three, negative one, two, negative two, five, three and we've practiced this a bunch now so let's hone in on the pivot there

I conveniently put it there so I've got two steps to start with I'm going to do replace row two I'm just going to get rid of this three so I'm going to multiply this by negative three and add it, so I'm going to replace row two with negative three, row one plus row two

top row doesn't change let's do this right now I get a zero here multiply by negative three and add it I get negative four multiply by negative three here is negative six and add it

so far so good okay, now I want to get rid of this negative two so I'm going to replace row three with two, row one plus row three that creates a zero here let's do the rest now

I'm doing two times this one plus five is seven and then I'm doing two times two is four plus three is seven and notice the zeros never change I can multiply those and add them all day they're always zero so we can stop right here

let's do one little extra step let's make this a one here let's do replace row three with one-seventh row three one, two, zero zero, negative

we can also we can multiply this by negative one-fourth two let's do that just to make it nicer so we'll replace row two with negative one-fourth row two so we'll do two operations there in one step and now you can see what's going to happen is

this row can be eliminated so now I'll do replace row three with negative row two plus row three

top row doesn't change second row doesn't change and then the last row just becomes negative and then it becomes a row of zeros and now we'll do one more step to get it into row echelon form

so we'll do replace row one with negative row two plus row one so I'm multiplying this by negative one and adding it

multiply by negative one and add it so there's your reduced row echelon form and now we can extract the necessary information from that we're going to do the same thing that we just did in the previous one

we're going to get a parameter for this so row two what does that give us? that gives us that y plus z equals zero which implies y equals negative z

and then row one says x plus z equals zero so that implies that x equals negative z alright, now that last step to organize all this

was to set up a parameter so let's let z equal t and then we have x that's negative t y is equal to negative t and z equals t there's your final solution

so you give me a value of t, like five then negative five, negative five, five is the solution to this original equation now that's a lot of stuff a lot of watching somebody do arithmetic

it's hard to watch somebody do that you really do have to go home and practice that but I kind of went through all the different possibilities I'm going to kind of switch gears it's a little easier that was a lot of stuff

so the switching of gears let's get a clean board here

we need to get into chapter eight

so chapter eight is all about matrices

now we've already seen what a matrix is but we didn't formally introduce it we just used it as a tool now we're going to start talking about the mathematics just of matrices alone as I said before what a matrix is, is it's a rectangle of numbers

formal language for that is it's a rectangular of numbers and we use capital letters

traditionally a couple of things here too

I've got the plural and the singular version matrix is singular and matrices is plural let's have a few examples here

that's a matrix now notice I'm just giving you the blob of numbers we're not attaching it to an equation but behind the scenes you could think of this as having come from the coefficients of an equation often that's how you get matrices to start with but in this case we're just going to study matrices by themselves

notice I'm using capital letters so that's what I just said here we use capital letters to denote matrices so now I just call it capital A rather than a little a or capital B I said rectangular and then this one was square so I'll at least show you a rectangular one

we also had a rectangular one in the last example it's not up there anymore so those are two matrices we're going to have to learn what to do with them and let's also write down a general matrix so if I have I have to specify here

this one here has two rows and two columns this one has two rows and three columns so in general I might have n columns and I might have m rows

so now let's write in what all these little entries will be the very first one is in row one column one this is the same notation I was using before that's why I introduced it before so this is row one column one

I'm now in row one column n because there's n columns and then I go down here to the last one this is row m column one and then I have row m column n

we organize this by saying that A is an m by n matrix

so that tells you the size of the rectangle if it's a two by three this one will be a two by three and this is a two by two matrix in English we say

so if A is m by n if we say that in English you're supposed to imagine this situation where you have m rows n columns so that's the order that we

we decide that we do rows first because we could switch it, right? so we have to have a convention we decide that rows come first then you can also say this a couple of different ways you could say A has dimensions

that's an equivalent statement has dimensions n by n and you could also say A has order those are the words we use to describe

the size of a matrix so let's hone in on some special ones if I have a one by n matrix so what's a one by n so right away we have to know which is which this is one row

n columns so what are some examples of this maybe one, two, three it's got one row, three columns this is a one by three or we could have one, two, three, four, five that's a one by five so a one by n is just n of them

so those are some examples but we have this particular vocabulary that we use for this this is called a row vector so it's still a matrix you could call it a matrix it's not wrong it's just that the convention is that we call things that have just a single row and then we'll also have one for a single column

we call those vectors we'll get used to this term by the end of the course because we're going to see it all the time then what's the opposite of that it's going to be an m by one matrix

so that should be it's got m rows and only one column so these guys are written like this that's a three by one and then I could have a five by one one, two, three, four, five and guess what

those are called column vectors let me just finish up with

just a couple more things and then we'll call it a day

this would probably be a good place to stop we'll stop here we'll just stop I'm leaving off in section 8.1 we'll finish up 8.1 that'll be quick next time

and then 8.2 we'll have some stuff to do there yeah I'll be up to the TA he should have told you he'll tell you in section after this class yeah I mean it's going to be on section 7.1 7.2

but specifically what he's going to put on there ask him because he makes up the quizzes any other questions? alright I guess you guys don't really leave because you have discussion right after this I get to leave