Linear algebra is universally recognized as a very important subject not only within mathematics but also with regards to its applications to physics, chemistry, economics, etc. In the modern organization of mathematical subjects, a vector space is one of the simplest algebraic structures. Nevertheless this simplicity is not easily recognized by students who often have great difficulty at the start of a linear algebra course in the first year of university. This paper will focus on the French context of this teaching, but several research works have drawn out similar difficulties in various countries around the world