ABSTRACT
Let V be a nonempty set of objects on which two operations are defined: addition and scalar multiplication. If the following properties hold for all u, v, and w in V and all scalars k and l, then V is a vector space.
(Closure under addition) If u and v are in V then u + v is in V.
(Closure under scalar multiplication) If u is in V then ku is in V.
(Commutativity) u + v = v + u.
(Associativity) u + (v + w) = (u + v) + w.
(Additive identity) An additive identity, usually represented by 0, exists and is in V.
(Additive inverse) If u is in V then –u is in V.
k(u + v) = ku + kv.
(k + l)u = ku + lu.
k(lu) = (kl)u.
1u = u.
