This chapter provides the foundation for understanding the mathematical details. It is devoted to two fundamental topics of applied mathematics: metric spaces and matrices. The most commonly used techniques are iterative, which determine a sequence of real (or complex) numbers, vectors, or functions that converge to the desired solutions. The chapter outlines the theory of metric spaces. It presents the basic properties of linear structures, which will be needed in analyzing linear systems. The chapter discusses the norms, transformations, and function of matrices. Metric spaces and special mappings defined in metric spaces play very important roles in the solution methodologies of linear and nonlinear algebraic, difference, and differential equations. The solutions of these equations are real or complex vectors, scalars, or functions defined on discrete or continuous time scales. Therefore, the convergence analysis of iteration methods for solving such equations requires the concept of a certain kind of distance between vectors, scalars, and functions.