Definición: Diremos que
F
≠
∅
{\displaystyle F\neq \emptyset }
F
{\displaystyle F}
+ cumple con:
a)
(
∃
0
F
∈
F
)
:
(
∀
x
∈
F
)
,
0
F
+
x
=
x
+
0
F
=
x
{\displaystyle (\exists 0_{F}\in F):(\forall x\in F),0_{F}+x=x+0_{F}=x}
b)
(
∀
x
∈
F
)
,
(
∃
(
−
x
)
∈
F
)
:
x
+
(
−
x
)
=
0
F
{\displaystyle (\forall x\in F),(\exists (-x)\in F):x+(-x)=0_{F}}
· cumple con:
a)
(
∃
1
F
∈
F
)
:
(
∀
x
∈
F
)
1
F
⋅
x
=
x
⋅
1
F
=
x
{\displaystyle (\exists 1_{F}\in F):(\forall x\in F)1_{F}\cdot x=x\cdot 1_{F}=x}
b)
(
∀
x
∈
F
,
x
≠
0
)
∃
(
x
)
−
1
∈
F
:
x
⋅
x
−
1
=
1
F
{\displaystyle (\forall x\in F,x\neq 0)\exists (x)^{-1}\in F:x\cdot x^{-1}=1_{F}}
Además,
(
∀
x
1
,
x
2
,
x
3
∈
F
)
x
1
⋅
(
x
2
+
x
3
)
=
x
1
⋅
x
2
+
x
1
⋅
x
3
{\displaystyle (\forall x_{1},x_{2},x_{3}\in F)x_{1}\cdot (x_{2}+x_{3})=x_{1}\cdot x_{2}+x_{1}\cdot x_{3}}
Sea V un conjunto de vectores y F un cuerpo. Sean
x
=
(
x
1
,
x
2
,
.
.
.
,
x
n
)
y
=
(
y
1
,
y
2
,
.
.
.
,
y
n
)
{\displaystyle x=(x_{1},x_{2},...,x_{n})\,\,y=(y_{1},y_{2},...,y_{n})}
x
+
y
=
(
x
1
+
y
1
,
x
2
+
y
2
,
.
.
.
,
x
n
+
y
n
)
∈
V
{\displaystyle x+y=(x_{1}+y_{1},x_{2}+y_{2},...,x_{n}+y_{n})\in V}
Sea
λ
∈
F
{\displaystyle \lambda \in F}
λ
x
=
λ
(
x
1
,
x
2
,
.
.
.
,
x
n
)
=
(
λ
x
1
,
λ
x
2
,
.
.
.
,
λ
x
n
)
∈
V
{\displaystyle \lambda x=\lambda (x_{1},x_{2},...,x_{n})=(\lambda x_{1},\lambda x_{2},...,\lambda x_{n})\in V}
Definición:
Sea
V
≠
∅
{\displaystyle V\neq \emptyset }
Espacio Vectorial sobre el cuerpo F, si se pueden definir en V operaciones {+, ·} tales que:
+
:
V
×
V
→
V
{\displaystyle +:V\times V\to V}
(
x
,
y
)
→
x
+
y
{\displaystyle (x,y)\to x+y}
Con las siguientes propiedades:
a)
(
∀
v
1
,
v
2
,
v
3
∈
V
)
,
(
v
1
+
v
2
)
+
v
3
=
v
1
+
(
v
2
+
v
3
)
{\displaystyle (\forall v_{1},v_{2},v_{3}\in V),\,(v_{1}+v_{2})+v_{3}=v_{1}+(v_{2}+v_{3})}
b)
(
∀
v
1
,
v
2
∈
V
)
,
v
1
+
v
2
=
v
2
+
v
1
{\displaystyle (\forall v_{1},v_{2}\in V),\,v_{1}+v_{2}=v_{2}+v_{1}}
c)
(
∃
0
∈
V
)
:
(
∀
v
∈
V
)
,
0
+
v
=
v
+
0
=
v
{\displaystyle (\exists 0\in V):(\forall v\in V),0+v=v+0=v}
d)
(
∀
v
∈
V
)
,
(
∃
(
−
v
)
∈
V
)
:
v
+
(
−
v
)
=
0
{\displaystyle (\forall v\in V),(\exists (-v)\in V):v+(-v)=0}
⋅
:
F
×
V
→
V
{\displaystyle \cdot :F\times V\to V}
(
λ
,
y
)
→
λ
y
{\displaystyle (\lambda ,y)\to \lambda y}
Con las siguientes propiedades:
a)
(
∀
λ
∈
F
)
(
∀
x
,
y
∈
V
)
λ
(
x
+
y
)
=
λ
x
+
λ
y
{\displaystyle (\forall \lambda \in F)(\forall x,y\in V)\lambda (x+y)=\lambda x+\lambda y}
b)
(
∀
x
∈
V
)
,
0
⋅
x
=
0
{\displaystyle (\forall x\in V),0\cdot x=0}
c)
(
∀
x
∈
V
)
,
1
⋅
x
=
x
{\displaystyle (\forall x\in V),1\cdot x=x}
