Définition d'un espace vectoriel

${\displaystyle K}$ représente le corps des réels ou des complexes. (${\displaystyle \mathrm {K} =\mathbb {R} \quad ou\quad \mathrm {K} =\mathbb {C} }$)

Soit ${\displaystyle E}$ un ensemble muni d'une loi interne notée ${\displaystyle +}$ et d'une loi externe ${\displaystyle K\times E\rightarrow E}$ notée ${\displaystyle \cdot }$

Il existe 10 Axiomes pour définir un ${\displaystyle \mathrm {K} }$-e.v (${\displaystyle K}$-espace vectoriel) :

${\displaystyle {\begin{array}{lcl}+:\quad E\times E\qquad \rightarrow \qquad E\\\qquad (u_{1},u_{2})\qquad \rightarrow \qquad u_{1}+u_{2}\end{array}}}$${\displaystyle \qquad }$et${\displaystyle \qquad }$${\displaystyle {\begin{array}{lcl}\centerdot \quad :\quad \mathrm {K} \times E\qquad \rightarrow \qquad E\\\quad \qquad (\lambda ,u_{2})\qquad \rightarrow \qquad \lambda \cdot u_{2}\end{array}}}$

${\displaystyle \qquad }$${\displaystyle \qquad }$${\displaystyle \qquad }$( Loi Interne)${\displaystyle \qquad }$${\displaystyle \qquad }$${\displaystyle \qquad }$${\displaystyle \qquad }$${\displaystyle \qquad }$${\displaystyle \qquad }$${\displaystyle \qquad }$${\displaystyle \qquad }$(Loi Externe)

1. ${\displaystyle \forall (u_{1},u_{2})\in E^{2}:u_{1}+u_{2}=u_{2}+u_{1}}$.
2. ${\displaystyle \forall (u_{1},u_{2},u_{3})\in E^{3}:u_{1}+(u_{2}+u_{3})=(u_{1}+u_{2})+u_{3}}$.
3. ${\displaystyle E}$ admet un élément neutre ${\displaystyle 0_{E}\in E}$ , c'est-à-dire ${\displaystyle \forall u\in E,u+0_{E}=u}$.
4. Tout élément ${\displaystyle u\in E}$ admet un symétrique ${\displaystyle u'}$ tel que ${\displaystyle u+u'=0_{E}}$. On note ${\displaystyle u'=-u}$.
5. ${\displaystyle \forall u\in E:1\cdot u=u}$
6. ${\displaystyle \forall (\lambda ,\mu )\in \mathrm {K} ^{2},\forall u\in E:\lambda \cdot (\mu \cdot u)=(\lambda \cdot \mu )\cdot u}$
7. ${\displaystyle \forall \lambda ,\in \mathrm {K} ,\forall (u_{1},u_{2})\in E^{2}:\lambda \cdot (u_{1}+u_{2})=\lambda \cdot u_{1}+\lambda \cdot u_{2}}$
8. ${\displaystyle {\displaystyle \forall (\lambda ,\mu ),\in \mathrm {K} ^{2},\forall u\in E:(\lambda +\mu )\cdot u=\lambda \cdot u+\mu \cdot u}}$