IPA : [ ˈɒpstrɒktvɛktorteːr]
absztrakt vektortér
( matematika , lineáris algebra ) Legyen V egy halmaz,
Γ
{\displaystyle \Gamma }
V
×
V
×
V
{\displaystyle V\times V\times V}
⋅
:
Γ
→
V
×
V
{\displaystyle \cdot :\Gamma \to V\times V}
a
,
b
,
c
∈
V
,
λ
,
μ
∈
Γ
{\displaystyle a,b,c\in V,\lambda ,\mu \in \Gamma }
V1:
(
a
+
b
)
+
c
=
a
+
(
b
+
c
)
{\displaystyle (a+b)+c=a+(b+c)}
V2:
a
+
b
=
b
+
a
{\displaystyle a+b=b+a}
V3: Létezik olyan
o
∈
V
{\displaystyle o\in V}
a
∈
V
{\displaystyle a\in V}
a
+
o
=
a
{\displaystyle a+o=a}
V4: Bármely
a
∈
V
{\displaystyle a\in V}
a
′
∈
V
{\displaystyle a'\in V}
a
+
a
′
=
o
,
{\displaystyle a+a'=o,}
a
′
=
(
−
1
)
⋅
a
{\displaystyle a'=(-1)\cdot a}
a
{\displaystyle a}
V5:
(
λ
+
μ
)
⋅
a
=
λ
⋅
a
+
μ
⋅
a
{\displaystyle (\lambda +\mu )\cdot a=\lambda \cdot a+\mu \cdot a}
V6:
λ
⋅
(
a
+
b
)
=
λ
⋅
a
+
λ
⋅
b
{\displaystyle \lambda \cdot (a+b)=\lambda \cdot a+\lambda \cdot b}
V7:
λ
⋅
(
μ
⋅
a
)
=
(
λ
⋅
μ
)
⋅
a
{\displaystyle \lambda \cdot (\mu \cdot a)=(\lambda \cdot \mu )\cdot a}
V8:
1
⋅
a
=
a
{\displaystyle 1\cdot a=a}
Ekkor V-t a
Γ
{\displaystyle \Gamma }
test feletti vektortérnek ,
V
{\displaystyle V}
Γ
{\displaystyle \Gamma }
Γ
=
R
{\displaystyle \Gamma =\mathbb {R} }
valós vektortérről ,
Γ
=
C
{\displaystyle \Gamma =\mathbb {C} }
komplex vektortérről beszélünk.
A V1-V8 tulajdonságokat vektortér-axiómáknak nevezzük.
