A computational definition of the notion of vectorial space

We usually define an algebraic structure by a set, some operations defined on this set and some propositions that the algebraic structure must validate. In some cases, we can replace these propositions by an algorithm on terms constructed upon these operations that the algebraic structure must validate. We show in this note that this is the case for the notions of vectorial space and bilinear operation. KEYWORDS: Rewrite system, vector space, bilinear operation, tensorial product, semantics, quantum programming languages, probabilistic programming languages.