2.3.1 Expressing Linear Systems in Matrix Form

When computational scientists discuss linear systems, they usually express the systems in matrix form. We will assume here that the reader is familiar with matrices and matrix operations. A review of these topics may be found in Chapter 7.

Consider the following system of equations.

(2.5)

The variables in the problem are x, y, and z. The left-hand side of each equation has one term associated with each variable. Each of these terms has the form coefficient · variable. If we allow coefficient to be positive, negative, or zero, then we can think of the left-hand side of an equation as being a sum of terms of the form coefficient · variable, where each term is associated with a different variable in the problem.

Since every equation in the problem refers to the same variables, the only information that we need the user to provide is the coefficients and the entries on the right-hand sides of the equations. Of course, this information will have to be provided in a way that tells us exactly where it belongs in the problem.

So let's take the coefficient information from (2.5) and arrange it in a matrix. Each row in the matrix will be associated with a particular equation. Each column in the matrix will be associated with a particular variable.

The matrix above is called the coefficient matrix associated with problem (2.5).

When convenient, we will use an augmented matrix, which appends an extra column containing right-hand side information to the coefficient matrix. The augmented matrix associated with problem (2.5) is