Let F be a field (or, more generally, a division ring). A vector space V over F is a set with two operations, + : V × V →V and ⋅ : F × V →V , such that

1. (u + v) + w = u + (v + w) for all u,v,w ∈ V
2. u + v = v + u for all u,v ∈ V
3. There exists an element 0 ∈ V such that u + 0 = u for all u ∈ V
4. For any u ∈ V , there exists an element v ∈ V such that u + v = 0
5. a ⋅ (b ⋅ u) = (a ⋅ b) ⋅ u for all a,b ∈ F and u ∈ V
6. 1 ⋅ u = u for all u ∈ V
7. a ⋅ (u + v) = (a ⋅ u) + (a ⋅ v) for all a ∈ F and u,v ∈ V
8. (a + b) ⋅ u = (a ⋅ u) + (b ⋅ u) for all a,b ∈ F and u ∈ V

Equivalently, a vector space is a module V over a ring F which is a field (or, more generally, a division ring).

The elements of V are called vectors, and the element 0 ∈ V is called the zero vector of V .

This entry is a copy of the GNU FDL vector space article from PlanetMath. Author of the original article: djao. History page of the original is here