 
Summary: Linear Algebra
Definition. A vector space (over R) is an ordered quadruple
(V, 0, , µ)
such that V is a set; 0 V ;
: V × V V and µ : R × V V ;
and the following eight axioms hold:
(i) ((u, v), w) = (u, (v, w)), u, v, w V ;
(ii) (v, 0) = v = (0, v), v V ;
(iii) for each v V there is w V such that (v, w) = 0 = (w, v);
(iv) (u, v) = (v, u), u, v V ;
(v) µ(c + d, v) = µ(c, v) + µ(d, v), c, d R, v V ;
(vi) µ(c, (u, v)) = (µ(c, u), µ(c, v)), c R, u, v V ;
(vii) µ(c, µ(d, v)) = µ(cd, v), c, d R, v V ;
(viii) µ(1, v) = v, v V .
Axioms (i),(ii),(iii) say that (V, 0, ) is an Abelian group. Axiom (iv) says that this group is Abelian.
One calls the elements of V vectors. From now on we write
u + v
for (u, v) and call this operation vector addition, and we write
cv
for µ(c, v), with the latter binding more tightly than the former, and call this operation scalar multiplica
