Exploring Linear and Nonlinear Maps: Properties, Classifications and Numerical Solution Techniques

Abstract

Linear maps, or linear functions, are foundational constructs in mathematics characterized by the principles of additivity and homogeneity. This paper delves into the defining properties of linear maps, distinguishing them from their nonlinear counterparts. We explore the necessity of homogeneity under both normal and genuine scalars, illustrating scenarios where additivity does not imply homogeneity, particularly in antilinear maps. The discussion extends to various types of nonlinear equations, including polynomial, trigonometric, exponential, logarithmic and transcendental forms, highlighting their applications in fields such as regression analysis, logarithmic scaling and project management through S-curve models. Furthermore, the paper examines numerical methods for solving nonlinear equations, with a focus on the bisection and Regula-Falsi methods. Detailed algorithms and solved examples demonstrate the practical implementation of these techniques in finding real roots of complex equations. By bridging theoretical concepts with computational strategies, this study provides a comprehensive overview of linear and nonlinear mappings, offering valuable insights for both mathematical theory and applied problem-solving.