Spatio vectorial

Un spatio vectorial es un structura algebric.

Definition

Sia ${\displaystyle V}$ un insimul, ${\displaystyle (F;+;\cdot )}$ un corpore, ${\displaystyle \oplus :V\times V\to V}$ un operation binari (interior) e ${\displaystyle \odot :F\times V\to V}$ un operation binari (exterior). Alora ${\displaystyle (V;\oplus$ ;\odot )} es un spatio vectorial super le corpore ${\displaystyle F}$ si e solmente si, pro omne ${\displaystyle u,v,w\in V}$ e ${\displaystyle \alpha ,\beta \in F}$, pro le operation interior vale

V1: ${\displaystyle u\oplus (v\oplus w)=(u\oplus v)\oplus w}$,

V2: ${\displaystyle 0_{V}\in V}$ con ${\displaystyle v\oplus 0_{V}=0_{V}\oplus v=v}$,

V3: ${\displaystyle -v\in V}$ con ${\displaystyle v\oplus (-v)=(-v)\oplus v=0_{V}}$,

V4: ${\displaystyle v\oplus u=u\oplus v}$,

e pro le operation exterior vale

S1: ${\displaystyle \alpha \odot (u\oplus v)=(\alpha \odot u)\oplus (\alpha \odot v)}$,

S2: ${\displaystyle (\alpha +\beta )\odot v=(\alpha \odot v)\oplus (\beta \odot v)}$,

S3: ${\displaystyle (\alpha \cdot \beta )\odot v=\alpha \odot (\beta \odot v)}$,

S4: ${\displaystyle 1\odot v=v}$ con ${\displaystyle 1\in F}$ le elemento neutre multiplicative.

Le operation interior ${\displaystyle \oplus }$ es le addition vectorial, e le operation exterior ${\displaystyle \odot }$ es le multiplication scalar.

Vide etiam

• Tabella de leges algebric