A vector space (also called a linear space) is a collection of objects called vectors, which may be added together and multiplied ("scaled") by numbers, called scalars.
Scalars are often taken to be real numbers, but there are also vector spaces with scalar multiplication by complex numbers, rational numbers, or generally any field.
The operations of vector addition and scalar multiplication must satisfy 8 axioms listed below:
- Associativity of addition: u + (v + w) = (u + v) + w
- Commutativity of addition: u + v = v + u
- Identity element of addition: there exists an element 0 ∈ V, called the zero vector, such that v + 0 = v for all v ∈ V
- Inverse elements of addition: for every v ∈ V, there exists an element −v ∈ V, called the additive inverse of v, such that v + (−v) = 0
- Compatibility of scalar multiplication with field multiplication: a(bv) = (ab)v
- Identity element of scalar multiplication: 1v = v, where 1 denotes the multiplicative identity in F
- Distributivity of scalar multiplication with respect to vector addition: a(u + v) = au + av
- Distributivity of scalar multiplication with respect to field addition: (a + b)v = av + bv