This paper aims at studying the fundamental properties of fuzzy linear relations between vector spaces. The sum of two fuzzy relations and the scalar multiplication are defined, in a natural way, and some properties of this operations are established. Fuzzy linear relations are investigated and among the results obtained, there should be underlined a characterization of fuzzy linear relations and the fact that the inverse of a fuzzy linear relation is also a fuzzy linear relation. Moreover, the paper shows that the composition of two fuzzy linear relations is a fuzzy linear relation as well. Finally, the article highlights that the family of all fuzzy linear relations is closed under addition and it is closed under scalar multiplication.