Vector Spaces

For each vector v∈V and each scalar a∈F, the scalar multiple av belongs to V

For each pair of vectors u,v∈V, the sum u+v also belongs to V

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

a·(b+c)=a·b+a·c

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

There exist distinct elements 0 and 1 in F such that

0+a=a and 1·a=a

(a+b)+c=a+(b+c) and (a·b)·c=a·(b·c)

a+b=b+a and a·b=b·a

• Commutativity of addition and multiplication
• Associativity of addition and multiplication
• Additive and multiplicative identity elements
• Inverses for addition and multiplication
• Distributivity of multiplication over addition