University teachers often declare that using geometry, and " geometrical intuition ", would help students in their learning of linear algebra. Our work aims at investigating the questions raised by such a statement. In the first chapter, we present theoretical didactical elements we used in our study. A central tool is the work by Fischbein about intuition in mathematics, and especially the use of models; it allowed us to give a definition of "geometrical intuition". In the second chapter, we study the interventions of "geometrical intuition" in the birth and evolution of linear algebra. In the third chapter, we analyse the process of "didactical transposition that led to the introduction of linear algebra at university and even secondary school level during the reform of modern mathematics. We show that this process resulted, in France, in a stronger connection between geometry and linear algebra than observed in the historical evolution. Linear algebra is not taught in secondary school anymore. However some geometrical notions and tasks taught in secondary school are then studied within linear algebra. In the fourth chapter, we analyse the evolution in linear algebra courses of such notions and tasks. In the fifth chapter, we present and analyse questionnaires we submitted to teachers and post-graduate students, in order to draw out the different aspects of geometrical intuition and the various kinds of models they use in their courses and practices of linear algebra. Finally, on the basis of the different results we obtained, we make some proposals in order to "optimize" the use of geometry in the teaching of linear algebra.
