In this paper, we present the current teaching of linear algebra in first year of French science university and the main difficulties of the students. Then we give a brief account of our first steps towards the design of a teaching experiment: from a joint didactical and historical survey we draw out our first hypotheses : epistemological specificity, use of 'meta-lever', use of changes of settings and points of view, importance of the concept of rank. We present briefly the main aspects and objectives of the teaching design experimented over a whole teaching semester with around 200 students during five years. Finally, we explain the type of evaluations that have set up and the difficulties we encountered. Our conclusion deals with issues on the teaching and learning of linear algebra as well as issues on methodological and theoretical points in relation with the original didactical framework we have elaborated.