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