vector space (original) (raw)

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