u + v = v + uBukti :u + v = (u1, u2, u3) + (v1, v2, v3)= (u1 + v1, u2 + v2, u3 + v3)= (v1 + u1, v2 + u2, v3 + u3) [sifat komutatif bil.riil]= (v1, v2, v3) + (u1, u2, u3)= v + u (u + v) + w = u + (v + w)Bukti :(u + v) + w = [(u1, u2, u3) + (v1, v2, v3)] + (w1, w2, w3)= (u1 + v1, u2 + v2, u3 + v3) + (w1, w2, w3)= ([u1 + v1] + w1, [u2 + v2] + w2, [u3 + v3] + w3)= (u1 + [v1 + w1], u2 + [v2 + w2], u3 + [v3 + w3]) [sifat asosiatif bil.riil]= (u1, u2, u3) + (v1 + w1, v2 + w2, v3 + w3)= (u1, u2, u3) + [(v1, v2, v3) + (w1, w2, w3)]= u + (v + w) u + 0 = 0 + u = 0Bukti :u + 0 = (u1, u2, u3) + (0, 0, 0)= (u1 + 0, u2 + 0, u3 + 0)= (0 + u1, 0 + u2, 0 + u3) [sifat komutatif bil.riil]= (0, 0, 0) + (u1, u2, u3)= 0 + u0 + u = (0, 0, 0) + (u1, u2, u3)= (0 + u1, 0 + u2, 0 + u3)= (u1, u2, u3) [sifat penjumlahan bilangan nol]= uu + 0 = 0 + u = u u + (-u) = 0Bukti :u + (-u) = (u1, u2, u3) + (-u1, -u2, -u3)= (u1 u1, u2 u2, u3 u3)= (0, 0, 0) [sifat pengurangan bil.riil]= 0 k(lu) = (kl)uBukti :k(lu) = k[l(u1, u2, u3)]= k(lu1, lu2, lu3)= (k[lu1], k[lu2], k[lu3])= ([kl]u1, [kl]u2, [kl]u3) [sifat asosiatif bil.riil]= (kl)(u1, u2, u3)= (kl)u k(u + v) = ku + kvBukti :k(u + v) = k[(u1, u2, u3) + (v1, v2, v3)]= k(u1 + v1, u2 + v2, u3 + v3)= (k[v1 + u1], k[v2 + u2], k[v3 + u3])= (ku1 + kv1, ku2 + kv2, ku3 + kv3) [sifat distributif bil.riil]= (ku1, ku2, ku3) + (kv1, kv2, kv3)= k(u1, u2, u3) + k(v1, v2, v3)= ku + kv (k + l)u = ku + luBukti :(k + l)u = (k + l)(u1, u2, u3)= ([k + l]u1, [k + l]u2, [k + l]u3)= (ku1 + lu1, ku2 + lu2, ku3 + lu3) [sifat distributif bil.riil]= (ku1, ku2, ku3) + (lu1, lu2, lu3)= ku + lu 1u = uBukti :1u = 1(u1, u2, u3)= (1u1, 1u2, 1u3)= (u1, u2, u3) [sifat identitas perkalian bil.rill]= u