Zero vector
= Zero vector
{title2=$0$}
The <zero vector> of a <vector space> is its additive identity: $v+0=0+v=v$ for every <vector> $v$. It is unique, because if $0'$ is another such identity then $0=0+0'=0'$. Distributivity gives $\alpha0=\alpha(0+0)=\alpha0+\alpha0$, so $\alpha0=0$ after cancellation. In a coordinate <vector> space it has every coordinate zero.