The following article presents a general outline of the genesis of the elementary concepts of vector space theory. It presents the main works that contributed to the development of these basic elements and analyzes how they developed and how they influenced each other. The study of systems of linear equations and the search for an intrinsic geometric analysis were the two main sources which gave rise to the theory of linearity. The fact of going beyond the third dimension in geometry in the middle of the 19th century, as well as the dialectical development between algebra and geometry from the creation of analytical geometry on, brought about the development of an initial unification of linear questions around the concept of determinant. This framework was generalized to the countably infinite dimension following work on functional analysis. Axiomatization, which was carried out at the end of the 19th century, although only really put to use after 1920, is a wider process which is part of the general development of mathematics in the beginning of the 20th century. I will analyze how this phenomenon came into existence and how it finally established its influence.