A vector space over a field F is a set V (whose elements are called vectors) together with a field F (whose elements are called scalars) , equipped with two operations:

1. Vector Addition: An operation that takes two vectors `bb {u} , bb{v} in V` and returns a vector `bb {u + v} in V`.
2. Scalar Multiplication: An operation that takes a scalar `a in F` and a vector `bb {u } in V` and returns a vector `a bb{u} in V`.

Note: The set V and the field F are distinct entities.
The set V itself must satisfy the following ten axioms:

I. Axioms for Vector Addition (V is an Abelian Group)
The set V under the operation of vector addition must form an abelian group (or commutative group).
For all `bb {u , v , w} in V`:

1.Closure: `bb {u + v} in V`.
2. Commutativity: `bb {u + v = v + u}`.
3. Associativity: `bb {(u + v) + w = u + (v + w)}`.
4. Zero Vector (Additive Identity): There exists an element `bb{0} in V` such that `bb {u + 0 = 0
5. Additive Inverse: For every `bb{u in V}` , there exists an element `bb{-u} in V` such that `bb {u + (-u) = 0}`.