Vector space/Direct/Definition

Vector space

Let ${}K$ denote a field, and ${}V$ a set with a distinguished element ${}0\in V$, and with two mappings

$+\colon V\times V\longrightarrow V,(u,v)\longmapsto u+v,$

and

$\cdot \colon K\times V\longrightarrow V,(s,v)\longmapsto sv=s\cdot v.$

Then ${}V$ is called a ${}K$-vector space (or a vector space over ${}K$), if the following axioms hold (where ${}r,s\in K$ and ${}u,v,w\in V$ are arbitrary).

1. ${}u+v=v+u$,
2. ${}(u+v)+w=u+(v+w)$,
3. ${}v+0=v$,
4. For every ${}v$, there exists a ${}z$ such that ${}v+z=0$,
5. ${}1\cdot u=u$,
6. ${}r(su)=(rs)u$,
7. ${}r(u+v)=ru+rv$,
8. ${}(r+s)u=ru+su$.