The solution of linear and nonlinear programming problems involves the solution of a sequence of linear equation systems. In preparation for understanding the nature and the structure of algorithms for solving linear and nonlinear programming problems, we discuss in this chapter the fundamentals of a particular method of solving systems of linear equations that is known as the pivot method. We choose this algorithm because it articulates in great detail all the steps necessary to solve a system of equations with the opportunity of showing very clearly the structure of the linear system and of the steps leading to a solution, if it exists.

In the analysis and solution of a linear system of equations, the most important notion is that of a basis. A formal definition of a basis is given later. Here we discuss, in an informal way and borrowing from everyday life, the meaning of a basis. A basis is a system of measurement units. To measure temperature, we often use two systems of measurement units, the Fahrenheit and the Celsius systems. Equivalently, we could name them the Fahrenheit and the Celsius bases. Analogously, the decimal metric and the American systems are two different bases (systems of measurement units) for measuring length, surface, and volume. The essential role of a basis, therefore, is that of measuring objects. In an entirely similar way, the role of a basis in a given mathematical space is that of measuring objects (points, vectors) in that space.