Systematic dimensional contracting: A methodological proposal for the calculation of equations and systems of M linear equations with N unknowns.

Abstract

Homogeneous linear systems integrate special properties that differentiate them from other linear systems and allow to simplify the search for solutions that, under certain conditions, promote general solutions of even heterogeneous systems and non-linear systems, hence their crucial importance in Mathematics, related sciences and in Engineering. From the Nine Chapters on the Mathematical Art of Ancient China to authors such as Seki Kowa, Leibniz, Cayley, Silvester, Bôcher, the resolutions of linear systems started to rely on matrix methods based on theoretical results such as the Gauss-Jordan elimination algorithm, Cramer's theorem, the Kronecker-Capelli theorem. Classic iterative methods such as Jacobi-Richardson, Gauss-Seidel, Cholesck factorization, the SOR method, conjugate gradient iterative methods, as well as graphical methods were also introduced. However, this article presents an innovative methodological alternative called systematic dimensional contraction, which is not based on matrices: it aims, among other dynamics, to systematically reduce the number of unknowns until the respective resolution is viable. In this view, the objective is to analyze the operability of this method of systematic dimensional contraction in the study of equations and linear systems, using homogeneous techniques. For this purpose, this article makes use of a theoretical-methodological research, of an explanatory typology, with bibliographic technical procedures and that uses the inductive-deductive method. Thus, the methodologic proposal for a systematic dimensional contraction method was constructed and applied to obtain original and exact solutions of homogeneous systems of linear equations, because solutions of this nature are a necessary condition for the construction of the homogeneous vector product and, in general, of the homogeneous theory of Vector Spaces.