1. The definition

Definition. A vector space over a field $F$ is a set $V$ with

(i) a binary operation $+$, an element 0, and for each $v \in V$, an element $-v$,

(ii) an operation of ``multiplication by scalars'' so that $r \in F$ and $v \in V$ give an element $rv \in V$, such that

(a)	$v + w = w + v$	(commutative law for addition)
(b)	$v + (w + u) = (v + w) + u$	(associative law for addition)
(c)	$v +$   0$= v$	( 0 is a neutral element for addition)
(d)	$v + (-v) =$   0	(each element has a negative)
(e)	$1 v = v$	(1 is neutral for mult. by scalars)
(f)	$(rs)v = r(sv)$	(mult. in $F$ versus mult. by scalars)
(g)	$r(v+w) = rv + rw$	(mult. by scalars versus vector addition)
(h)	$(r+s)v = rv + sv$	(mult. by scalars versus addition in $F$)