defining a vector space

Topics

• linear algebra
• vector space

A Vector space $\mathbb{V}$ over a field $\mathbb{F}$ (be it $\mathbb{R}$ or $\mathbb{C}$) is a set of vectors together with 2 operations vector addition, scalar multiplication that satisfy the following for all $u,v,w\in \mathbb{V}$ :

• Associativity : $u+(v+w)=(u+v)+w$
• Commutavity : $u+v=v+u$
• Zero vector : $\exists 0\in \mathbb{V}$ such that $\forall v\in \mathbb{V}:v+0=v$
• Inverse : $\forall v\in \mathbb{V}:\exists (-v)\in \mathbb{V}$ such that $v+(-v)=0$
• Compatibility : $\forall a,b\in \mathbb{F}:a(bv)=(ab)v$
• Identity : $1v=v$
• Distributivity : $\forall a,b\in \mathbb{F}:a(u+v)=au+av$ and $(a+b)v=av+bv$

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