Vector space axioms

The vector space axioms define a vector space $V$ over a field $k$ as an abelian group $(V,+)$ equipped with scalar multiplication $k\times V\to V$, $(a,v)\mapsto av$, satisfying the same distributivity/associativity/unit conditions as the module axioms :

The field structure on $k$ (in particular, inverses for nonzero scalars) enables linear algebra constructions such as bases, dimension, and canonical forms.