In this investigation we have adopted as its main objective to understand and analyze in the framework of the theory of modes of thinking, reasoning from the theoretical or practical evidence from the college students to be exposed to concepts of linear dependence and independence vectors and solving a system of linear equations in R2 and R3 and the connections that establish these students in higher education between those concepts. Throughout the investigation we have shown that students tend to be in the same way of thinkingthat has to do with numerical and algebraic relationships that can be established with the concepts, so-called analytic-arithmetic thinking-even when the context otherwise favorable exercise of thought, such as synthetic-geometric. In addition, we found that the coordination and transition between different modes of thinking about linear dependence concepts and solution of a system of linear equations favor right connections between them.