This book gives a complete and detailed account of the basic ideas and theorems of elementary linear algebra which represent the very least a competent French "bachelier" should know when embarking upon the "propedeutique" courses. The general approach and content of the book were determined by the desire to prepare the student to assimilate as easily as possible, the actual teaching he is receiving at this time, which he should, in turn, regard as a natural extension of the work he has already been doing. At the moment, it is unlikely that one in a thousand students could cope with this book, unaided and without considerable hard work. This state of affairs says a great deal about the lack of cohesion in our teaching programmes. For years, teachers everywhere have expressed their concern about the everidening gap between the methods and attitudes to the teaching of Mathematics in the secondary schools on the one hand and in the universities on the other. Having participated in many discussions on this problem, I have succeeded in convincing myself that even among those secondary school teachers who are most keenly aware of the need for reform, there exists a great uncertainty about the content of new courses, their structure and their connection with the various courses in the university. So it is, in fact, for these people that this book is primarily intended. I have devised it as a "teacher's book"; in other words, its aim is to provide a solid framework upon which to base lively oral teaching adapted to those pupils for whom it is intended.