The teaching of linear algebra in France has undergone great modifications within the last thirty years. Today, linear algebra represents more or less a third of the mathematical contents taught in the first year of all French science universities. Traditionally, this teaching starts with the axiomatic definition of a vector space and finishes with the diagonalisation of linear operators. In a survey, Robert and Robinet (1989) showed that the main criticisms made by the students toward linear algebra concern the use of formalism, the overwhelming amount of new definitions and the lack of connection with what they already know in mathematics. It is quite clear that many students have the feeling of landing on a new planet and are not able to find their way in this new world. The general attitude of teachers consists more often of a compromise: there is less and less emphasis on the most formal part of the teaching (especially at the beginning) and most of the evaluation deals with the algorithmic tasks connected with the reduction of matrices of linear operators. However, this leads to a contradiction which cannot satisfy us. Indeed, the students may be able to find the Jordan reduced form of an operator, but, on the other hand, suffer from severe misunderstanding on elementary notions such as linear dependence, generators, or complementary subspaces
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