Now we are ready to give the formal definition of a vector space. A vector space is a set V along with an addition on V and a scalar multiplication on V such that the following properties hold:

commutativity
u + v = v + u for all u, v ∈ V;

associativity
(u+v)+w = u+(v +w) and (ab)v = a(bv) for all u,v,w ∈ V and all a,b ∈ F;

additive identity
there exists an element 0 ∈ V such that v +0 = v for all v ∈ V;

additive inverse
for every v ∈ V, there exists w ∈ V such that v + w = 0;

multiplicative identity
1v = v for all v ∈ V;

distributive properties
a(u +v) = au +av and (a +b)u = au +bu for all a,b ∈ F and all u,v ∈ V.