= Solution
{c}
A <vector space> over a <field> $\mathbb F$ is a set $V$ with <vector addition> $V\times V\to V$ and <scalar multiplication> $\mathbb F\times V\to V$ satisfying the following <vector space axioms>. Under <vector addition>, $V$ is an <abelian group>: addition is associative and commutative, there is a <zero vector> $0_V$, and each $v$ has an additive inverse $-v$. For all $a,b\in\mathbb F$ and $u,v\in V$,
$$
a(u+v)=au+av,\qquad (a+b)v=av+bv,\qquad
(ab)v=a(bv),\qquad 1_{\mathbb F}v=v.
$$
The operations are defined for every indicated pair and take their values in $V$.
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