PlanetPhysics/Vector Space 2

Let $$F$$ be a field (or, more generally, a division ring). A vector space $$V$$ over $$F$$ is a set with two operations, $$+: V \times V \longrightarrow V$$ and $$\cdot: F \times V \longrightarrow V$$, such that


 * 1) $$(\mathbf{u}+\mathbf{v})+\mathbf{w} = \mathbf{u}+(\mathbf{v}+\mathbf{w})$$ for all $$\mathbf{u},\mathbf{v},\mathbf{w} \in V$$
 * 2) $$\mathbf{u}+\mathbf{v}=\mathbf{v}+\mathbf{u}$$ for all $$\mathbf{u},\mathbf{v}\in V$$
 * 3) There exists an element $$\mathbf{0} \in V$$ such that $$\mathbf{u}+\mathbf{0}=\mathbf{u}$$ for all $$\mathbf{u} \in V$$
 * 4) For any $$\mathbf{u} \in V$$, there exists an element $$\mathbf{v} \in V$$ such that $$\mathbf{u}+\mathbf{v}=\mathbf{0}$$
 * 5) $$a \cdot (b \cdot \mathbf{u}) = (a \cdot b) \cdot \mathbf{u}$$ for all $$a,b \in F$$ and $$\mathbf{u} \in V$$
 * 6) $$1 \cdot \mathbf{u} = \mathbf{u}$$ for all $$\mathbf{u} \in V$$
 * 7) $$a \cdot (\mathbf{u}+\mathbf{v}) = (a \cdot \mathbf{u}) + (a \cdot \mathbf{v})$$ for all $$a \in F$$ and $$\mathbf{u},\mathbf{v} \in V$$
 * 8) $$(a+b) \cdot \mathbf{u} = (a \cdot \mathbf{u}) + (b \cdot \mathbf{u})$$ for all $$a,b \in F$$ and $$\mathbf{u} \in V$$

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 $$\mathbf{0} \in V$$ 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