Sia
V
{\displaystyle V}
insimul ,
(
F
;
+
;
⋅
)
{\displaystyle (F;+;\cdot )}
corpore ,
⊕
:
V
×
V
→
V
{\displaystyle \oplus :V\times V\to V}
operation binari (interior) e
⊙
:
F
×
V
→
V
{\displaystyle \odot :F\times V\to V}
(
V
;
⊕
;
⊙
)
{\displaystyle (V;\oplus ;\odot )}
spatio vectorial super le corpore
F
{\displaystyle F}
si e solmente si , pro omne
u
,
v
,
w
∈
V
{\displaystyle u,v,w\in V}
α
,
β
∈
F
{\displaystyle \alpha ,\beta \in F}
V1:
u
⊕
(
v
⊕
w
)
=
(
u
⊕
v
)
⊕
w
{\displaystyle u\oplus (v\oplus w)=(u\oplus v)\oplus w}
V2:
0
V
∈
V
{\displaystyle 0_{V}\in V}
v
⊕
0
V
=
0
V
⊕
v
=
v
{\displaystyle v\oplus 0_{V}=0_{V}\oplus v=v}
V3:
−
v
∈
V
{\displaystyle -v\in V}
v
⊕
(
−
v
)
=
(
−
v
)
⊕
v
=
0
V
{\displaystyle v\oplus (-v)=(-v)\oplus v=0_{V}}
V4:
v
⊕
u
=
u
⊕
v
{\displaystyle v\oplus u=u\oplus v}
e pro le operation exterior vale
S1:
α
⊙
(
u
⊕
v
)
=
(
α
⊙
u
)
⊕
(
α
⊙
v
)
{\displaystyle \alpha \odot (u\oplus v)=(\alpha \odot u)\oplus (\alpha \odot v)}
S2:
(
α
+
β
)
⊙
v
=
(
α
⊙
v
)
⊕
(
β
⊙
v
)
{\displaystyle (\alpha +\beta )\odot v=(\alpha \odot v)\oplus (\beta \odot v)}
S3:
(
α
⋅
β
)
⊙
v
=
α
⊙
(
β
⊙
v
)
{\displaystyle (\alpha \cdot \beta )\odot v=\alpha \odot (\beta \odot v)}
S4:
1
⊙
v
=
v
{\displaystyle 1\odot v=v}
1
∈
F
{\displaystyle 1\in F}
elemento neutre multiplicative. Le operation interior
⊕
{\displaystyle \oplus }
addition vectorial , e le operation exterior
⊙
{\displaystyle \odot }
multiplication scalar .
