The primary motivation for the study of vectors and matrices is based on the study of solving systems of linear equations. The algorithms that enable us to find solutions are themselves based on certain kinds of matrix manipulations. In these algorithms, matrices serve as a shorthand for calculation, rather than as a basis for a theory. We will see later that these matrix manipulations do lead to a rich theory of how to solve systems of linear equations. But our first step is just to see how these equations are actually solved.
We begin with a discussion in Section ?? of how to write systems of linear equations in terms of matrices. We also show by example how complicated writing down the answer to such systems can be. In Section ??, we recall that solution sets to systems of linear equations in two and three variables are lines and planes.
The best known and probably the most efficient method for solving systems of linear equations (especially with a moderate to large number of unknowns) is Gaussian elimination. The idea behind this method, which is introduced in Section ??, is to manipulate matrices by elementary row operations to reduced echelon form. It is then possible just to look at the reduced echelon form matrix and to read off the solutions to the linear system, if any. The process of reading off the solutions is formalized in Section ??; see Theorem ??. Our discussion of solving linear equations is presented with equations whose coefficients are real numbers — though most of our examples have just integer coefficients. The methods work just as well with complex numbers, and this generalization is discussed in Section ??.
Throughout this chapter, we alternately discuss the theory and show how calculations that are tedious when done by hand can easily be performed by computer using MATLAB. The chapter ends with a proof of the uniqueness of row echelon form (a topic of theoretical importance) in Section ??. This section is included mainly for completeness and need not be covered on a first reading.