Vector space axioms
= Vector space axioms
Addition makes $V$ an <Abelian group>: for all $u,v,w\in V$, $(u+v)+w=u+(v+w)$ and $u+v=v+u$, there is a <zero vector> $0$, and every $v$ has an additive inverse $-v$. <Scalar multiplication> satisfies, for $a,b\in\mathbb F$,
$$
1v=v,\qquad (ab)v=a(bv),\qquad a(u+v)=au+av,\qquad(a+b)v=av+bv.
$$
Both operations take values in $V$. These requirements make manipulating <linear combinations> independent of how sums are grouped or expanded.