Contribution à l'étude de l'enseignement à l'université des premiers concepts d'algèbre linéaire. Approches historique et didactique.
DORIER Jean-Luc, 29 juin 1990

Linear algebra is a field which holds a central position in the organization of modern mathematics. Meanwhile experience shows that  difficulties, often persistent, appear in the teaching at university of the first notions of vector space theory. Our research presents the first steps toward a full didactical analysis of this problem.
We first present a historical analysis of the genesis of elementary concepts of linear algebra, which attempts to draw out their epistemological nature. We also show what make this work useful for a research in Didactics.
We then try, through an analysis of students' work, to clarify the nature of the teaching of the beginning of linear algebra as it exists, and to show which procedures students use according to the tasks and to some of their previous abilities. Along with this sudy we give some elements for a reflexion about the method we used and the benefits it received from tools and methods of didactical theories.
Finally we give a few general propositions for the teaching of linear algebra to beginners. Through this work,we raise the question of the adaptation of some didactical theories to the specificities of university teaching and of the epistemological nature of linear algebra concepts ;  as an answer, we give concrete proposals for the use of metamathematical point of view, in  teaching sequences.