Axioms

Axioms of vector spaces - given that the vector space is called V

Axiom 1

Closure under addition - u + v is an element of V

Axiom 2

Addition is commutative - u +v = v + u

Axiom 3

Addition is associative - (u+v) + w = u + (v+w)

Axiom 4

Zero - the vector space contains the zero vector, such that u + 0 = u

Axiom 5

Negatives - for each element in the vector space, there exists a negative of that element. u + (-u) = 0

Axiom 6

closure under scalar multiplication - all scalar multiples of a vector exist within the vector space

Axiom 7

distributivity of vector addition - k(u + v) = ku + kv where k is a scalar and u and v are vectors

Axiom 8

distributivity of scalar addition - (k + m)u = ku +mu where k and m are scalars and u is a vector

Axiom 9

scalar multiplication is associative - k(mu) = (km)u

Axiom 10

scalar identity - 1u = u