(VS8) Distributive property of scalar multiplication over scalar addition
(VS7) Distributive property of scalar multiplication over vector addition
(VS6)
Associativity of scalar multiplication
(VS5) Unit Property
(VS4) Existence of additive inverses in V
(VS3) Existence of a zero vector
(VS2) Associativity of addition
(VS1) Commutativity of addition
(C2) Closure under scalar multiplication
For each vector v∈V and each scalar a∈F, the scalar multiple av belongs to V
(C1) Closure under addition
For each pair of vectors u,v∈V, the sum u+v also belongs to V
Definition of a Vector Space
Let V be a nonempty set and let F be a field. Suppose that an addition operation and a scalar multiplication operation are defined on V, with scalars belonging to the field F. We call V a vector space over F provided that satisfies:
Closure under addition
Closure under scalar multiplication
Commutativity of addition
Associativity of addition
Existence of a zero vector
Existence of additive inverses in V
Unit Property
Associativity of scalar multiplication
Distributive property of scalar multiplication over vector addition
Distributive property of scalar multiplication over scalar addition
(F5) Distributivity of multiplication over addition
a·(b+c)=a·b+a·c
(F4) Inverses for addition and multiplication
For each element a in F and each nonzero element b in F, there exist elements c and d in F such that
a+c=0 and b·d=1
(F3) Additive and multiplicative identity elements
There exist distinct elements 0 and 1 in F such that