In the French tradition of Bourbaki, the theory of vector spaces is usually presented in a very formal setting, which causes severe difficulties to many students. The aim of this talk is to analyze the un-derlying reasons of these difficulties and to suggest some ways to make the first teaching of the the-ory of vector spaces less inefficient for many students. From a mathematical analysis with a historical perspective, we analyze the teaching and the apprehension of vector space theory in a new ap-proach. I particular, we w show that mistakes made by many students can be interpreted as a result of a lack of connection between the new formal concepts and their conceptions previously acquired in more restricted, but more intuitively based areas.