Vector space

Mathematical definition

Set $\mathbf{V}$ is a vector space over $\mathbb{F}$ if:

1. Are defined operations

• • of addition: $\mathbf{V} \times \mathbf{V} \to \mathbf{V}$
  • of scalar multiplication: $\mathbb{F} \times \mathbf{V} \to \mathbf{V}$

2. Are satisfied the properties

• • Commutativity: $\forall u,v\in\mathbf{V},\; u+v=v+u$;
  • Associativity: $\forall u,v,w\in\mathbf{V},\;\forall a,b\in\mathbb{F},\; (u+v)+w=u+(v+w)\;\land\;(ab)v=a(bv)$;
  • Additive identity: $\exists \vec 0\in\mathbf{V},\;\forall v\in\mathbf{V},\; \vec 0+v=v$;
  • Additive inverse: $\forall v\in\mathbf{V},\;\exists w\in\mathbf{V},\; v+w=0$;
  • Multiplicative identity: $\forall v\in\mathbf{V},\; 1v=v$;
  • Distributivity: $\forall u,v\in\mathbf{V},\;\forall a,b\in\mathbb{F},\; a(u+v)=au+av\;\land\;(a+b)u=au+bu$.