 Welcome back to our lecture series, Math 1050, called Joudsburg for Students at Southern Utah University. As usual, I'll be a professor today, Dr. Angela Musseline. In lecture 13, we're going to continue learning about systems of linear equations that we learned about in lectures 11 and 12. Now, the primary method we've been utilizing for solving systems of linear equations are the methods of substitution and elimination. Frankly speaking, they do okay at solving systems of equations. Now, I say they do okay because we've only been focusing on two-by-two and three-by-three linear systems, two-by-two meaning that we have two equations with two unknowns, two variables. And for a three-by-three system, I mean we have three equations and three unknowns. All right? Our methods do okay in those settings, but there's a lot of other settings we have to consider. What happens if you have two unknowns but like a thousand equations, right? That's what we might call an over-determined system. What if you have five variables but only three equations? You have this under-determined system. We haven't really explored too much with that. And then, heck, let's talk about that. What if you have five variables? We have 10 variables. What if you have 25 variables? 100 variables. Our methods of substitution and elimination are actually kind of clunky when looking at larger systems of equations. And it turns out we can make the process much more efficient by taking off a lot of the baggage. And in order to do so, we're gonna introduce the notion of matrices. So what is a matrix? The simple answer is a matrix is a rectangular array of numbers. More precisely, if we have two positive integers, m and n, we say that an m by n matrix is a rectangular array of real or maybe complex numbers with m rows and n columns. An illustration of such a thing you can see right here. This is an example of a two by three matrix. When it comes to describing the dimensions of a matrix, the m by n, we always describe the rows first in the column second. It's like reverse alphabetical. The, if this is a two by two matrix, it means you have two rows like so and you have three columns. So the rows are telling you how far up and down your matrix is gonna go. The columns tell you how far to left and right your matrix is gonna go. And so a two by three matrix will have six numbers in it. Exactly two times three there. You have these six numbers. So this two by three matrix, you see that you have a one, negative two, negative one, negative one, three and five. We often refer to the numbers in the matrix by their position. So this number right here is in the first row, first column. So we call it the one, one position. This number is in the first row, second column. So it's in the one, two position. All right, this negative one is in the one, three position because it's the first row, third column. This one is in the two, one position. You can think of these as like coordinates that tell you where the number is located in the matrix. Second row, first column. You always talk about the rows first in a matrix. This is the two, two position. It's a little ballerina. And this one is the two, three position, second row, third column. And so this matrix is two by three matrix has these six positions there. All right, so the matrix is this rectangular array that's useful for capturing data. You can kind of think of it as just the mathematics of a spreadsheet. That's what a matrix is. It's just a spreadsheet here. Now, why is this relevant for systems linear equations is that whenever you have a system of linear equations put in standard form, we could encode this linear system as a matrix and we can solve the system using said matrix. Okay, so consider the following of what we would call a three by three linear system. It's three by three because it has three equations and three unknowns. This label of three by three is actually foreshadowing the dimensions of the coefficient matrix that we're gonna talk about right now. When you have a system of linear equations like this, we're gonna create a matrix, the so-called coefficient matrix, which is going to have as many rows as there are equations. So the rows of your matrix will coincide with your equations in the linear system. And you're gonna have a number of columns equal to the number of variables. So each column will coincide with a variable in the so-called coefficient matrix and each row is gonna coincide with an equation inside of the coefficient matrix. So looking at this linear system, the first equation is x minus two y plus z equals zero. For the moment, we're gonna ignore everything to the right of the equations here. That is to say that if you move all the variables to the left-hand side, combine like terms if necessary, put all the constants on the right-hand side, we're gonna ignore those constants on the right-hand side and just fixate on the variables for a moment. If I just look at the coefficients of those variables, the coefficient of x is a one, the coefficient of y is a negative two and the coefficient of z is a positive one. I'm gonna record just their coefficients in their same order and I'm gonna put them in alphabetical order x, y, and z. And so I record the coefficients one, negative two, and one. When you look at the second equation, if you record the coefficients of the variables and I'm gonna do this exact same order I did on the first row, which in this case is alphabetical, I don't see an x there, which actually suggests that the coefficient of x is zero. So I put a zero here in my matrix. The coefficient of y is a two and the coefficient of z is a negative eight here. Whenever you subtract a variable, that means its coefficient is negative and make sure that negative sign is put inside the coefficient matrix. The third row, the third row here, again, we record the variable coefficients, a negative four for x, a five for y, a nine for z. So each of these rows in the matrix coincides with an equation one, negative two, one. This one here, we get zero, two, negative eight, and then here we get negative four, five, and nine. Now each column of the matrix coincides with the variable. So the first column gives you the coefficients of x, one, zero, negative four. The second variable gives you the second column, negative two, two, and negative five there, excuse me, positive five. And then the third column is gonna give you the coefficients of z there, one, negative eight, nine. This is then the coefficient matrix of the linear system and it encodes all of the variables. It turns out if we organize the coefficients by rows and columns like we did, we don't actually have to write down the variables anymore because I know the first column is the first variable x and the first row is the first equation, okay? But also these constants on the right hand side are also necessary to know them as you solve the system of equations. And so what we're gonna do is we're gonna make a column just for the constant terms. So we write them over here, zero, eight, and negative nine. Then we're gonna draw a vertical line that separates the coefficient matrix from this extra column. And this vertical line is a substitute for the equations, the equal signs. So the vertical line goes exactly where the equal signs would be. And so the coefficient matrix plus this other column is what we call the augmented matrix. Augmented as suggested, we've added something to it. We've added something to the matrix, what it was the coefficient matrix plus the new column for the constants. That's where we get this augmented matrix. Now in a course like linear algebra math 2270, a lot of study goes into this coefficient matrix because the coefficient matrix says a lot about a lot of things. Now for our purposes, we're gonna be fixated on this augmented matrix because inside the augmented matrix, we have put all of the data that we need for the system of linear equations. We've encoded the linear system as a matrix. And now as we will see in the next videos or so, as we reduce and process this augmented matrix, we can create the solution to the linear system in a much more efficient manner than we've been doing with our methods of substitution and elimination.