 We'll begin our study of matrices by looking at the least important aspect of matrix algebra. So we want to look at what are called linear equations. A linear equation is, first of all, an equation, so we have to have an equal sign that lets this one out. We have to have one or more variables in the equation, so not this one. And finally, the terms of our equation have to either be constants or a constant multiplied by a variable raised to the first power. So these are equations, but they are not linear equations. If we have two or more linear equations, we also have what's called a system of linear equations, also known as a multilinear system of equations. It'll be most helpful if we put our equations in standard form. First, we agree on a specific ordering of the variables. So if I have a system of equations with three variables, x, y, and z, I might choose the ordering. How about x, y, z? Next, we'll want to make sure that all equations have the variable terms in order on one side of the equals and the constant on the other. So this equation has all the variable terms on the left-hand side, and they're in the order x, y, z. So it's already in standard form. However, this equation is not, and so we need to move that 5z to the other side, and the variable terms are in order. We have x coming before z. One more feature of standard form is that all equations should include all variables, so we'll need to use zero coefficients as necessary. It'll also be very convenient to write our equations as sums of terms, and so that means we'll want to use coefficients of 1 and negative numbers as necessary. So this equation we'll want to include as 0, y, so that it has all variables, and we'll change that minus 5z into a plus negative 5z. It's also convenient to write the coefficients of 1 that we ordinarily omit. And so we've rewritten our original system of equations as two equations in standard form. So what is the matrix? There's many different answers to this question, but we'll start with this one. A multilinear system of equations in standard form can be represented using an augmented coefficient matrix. Each row of the matrix is going to represent one equation, where the entries of the row are the variable coefficients in order, followed by the constant. So the first row of the matrix corresponds to the first equation, and the entries of the first row correspond to the coefficients in order, 3, 0, negative 2, followed by the constant, 1. And likewise, if I take a look at the second row of the matrix, these entries come from the second equation, the coefficients 2, 1, 0, and the constant 5. And finally the third row of the matrix corresponds to the third equation, with entries 1, negative 2, 0, and constant 3. In order to distinguish between the coefficients and the constant term, it is convenient to draw a vertical line separating them. And finally, we typically enclose the whole matrix either in brackets or in parentheses. In some cases, we might want to focus on just the coefficients. If we just include the coefficients, we have a coefficient matrix.