Um estudo acerca da inserção de aspectos históricos dos conceitos de dependência e independência linear em cursos de álgebra linear

Abstract

The present study aimed to investigate how the historical development of the concepts of Linear Dependence and Independence can be approached in Linear Algebra courses to enable a better understanding of these concepts by Mathematics undergraduates. To this end, we developed a Bibliographic Research with a qualitative approach for data analysis consisting of two moments. In the first moment, based on Dorier (1995b; 2000) and Moore (1995), we discuss the historical constitution of Linear Algebra, in which we identify four different preceding notions of the current concepts of Linear Dependence and Independence, whether they are inclusive dependence (Euler), unified dependence for equations and n-tuples (Frobenius), generalization of dependence to n-dimensional space (Grassmann) and axiomatization of dependence and linear independence (Peano). In the second moment, we present didactic suggestions, based on Mendes (2006; 2015; 2016) and Brandemberg (2018; 2021), on how to approach these different notions in Linear Algebra courses. Such suggestions aim to give students the opportunity to have contact with different aspects that allow them to broaden their understanding of linearity as a relationship between vectors, as well as to visualize the current definitions of Linear Dependence and Independence as a language that does not discard the notions given by Euler, Frobenius, Grassmann or Peano, but keep them in a unifying and generalizing character.