A vector space is a set V that is an Abelian group under addition with the property that for each a ∈ F and v ∈ V, there is an element av ∈ V such that the following conditions hold for all a,b ∈ F and all u,v ∈ V:
1. a(u+v) = au + av .
2. (a+b)v = av +bv .
3. a(bv) = (ab)v .
4. 1v = v .