Axioms for Addition Al Commutativity u+v=v+u. A2 Associativity (u+v)+0=u+(v+w) A3 Existence of the zero vector There exists a unique element 0 of V such that v+0=v,for all v∈V A4 Existence of an additive inverse For each v E V,there exists a vector -v such that v+(-v)=0. We will abbreviate u+(-v)for u-v,so we have defined subtraction.Axioms for Addition + A1 Commutativity u + v = v + u. A2 Associativity (u + v) + w = u + (v + w). A3 Existence of the zero vector There exists a unique element 0 of V such that v + 0 = v, for all v ∈ V. A4 Existence of an additive inverse For each v ∈ V , there exists a vector −v such that v + (−v) = 0. We will abbreviate u + (−v) for u − v, so we have defined subtraction