Vector Space

(VS 1) For all x, y in V, x + y = y + x (commutativity of addition).

(VS 2) For all x, y, z in V, (x + y) + z = x + (y + z) (associativity of addition).

(VS 3) There exists an element in V denoted by 0 such that x+0 = x for each x in V.

(VS 4) For each element x in V there exists an element y in V such that x + y = 0 .

(VS 5) For each element x in V, 1x = x.

(VS 6) For each pair of elements a, b in F and each element x in V, (ab)x = a(bx).

(VS 7) For each element a in F and each pair of elements x, y in V, a(x + y) = ax + ay.

(VS 8) For each pair of elements a, b in F and each element x in V, (a + b)x = ax + bx