Vector Spaces

A vector space is a set \( V \) along with an addition on \( V \) and a scalar multiplication on \( V \) such that the following properties hold. ¹

Commutativity

\[ u+v=v+u, \forall u,v\in V \]

Associativity

\( (u+v)+w=u+(v+w) \) and \( (ab)v = a(bv) \) for all \( u,v,w\in V \) and all \( a,b \in F \)

Additive Identity

There exists an element \( 0\in V \) such that \( v+0=v,\forall v\in V \)

Additive Inverse

\[ \forall v\in V: \exists w\in V, v+w=0 \]

Multiplicative Identity

\[ 1v=v, \forall v\in V \]

Distributive Properties

\( a(u+v)=au+av \) and \( (a+b)v=av+bv \) for all \( a,b\in F \) and all \( u,v\in F \)