Let K be a field. A vector space V over the field K is an additive (abelian) group, together with a multiplication of elements of V by elements of K, i.e. an association $$ \left( {x,v} \right) \mapsto xv $$ of K × V into V, satisfying the following conditions: VS 1. If 1 is the unit element of K, then 1 υ = υ for all υ ∈ V. VS 2. If c ∈ K and υ, w ∈ V, then c(v + w) = cυ + cw. VS 3. If x, y ∈ K and υ ∈ V, then (x + y)υ = xυ + yv. VS 4. If x, y ∈ K and υ ∈ V, then (xy)υ = x(yυ).