WE commence our discussion of numerical algorithms by deriving ways to solve thelinear system of equations A~x = ~b. We will explore applications of these systems in Chapter 4, showing a variety of computational problems that can be approached by constructing appropriate A and ~b and solving for ~x. Furthermore, solving a linear system will serve as a basic step in larger methods for optimization, simulation, and other numerical tasks considered in almost all future chapters. For these reasons, a thorough treatment and understanding of linear systems is critical.