Un espacio vectorial
V
{\displaystyle V}
F
{\displaystyle F}
+
¯
{\displaystyle {\bar {+}}}
∘
{\displaystyle \circ }
Donde:
+
¯
:
V
×
V
→
V
{\displaystyle {{\bar {+}}:{V}\times {V}}\rightarrow {V}}
Es una operacion binaria en el conjunto V conocida como suma de vectores
∘
:
F
×
V
→
V
{\displaystyle {\circ :{F}\times {V}}\rightarrow {V}}
Es una operacion binaria del campo F y el conjunto V, al conjunto V conocida como multiplicacion por escalares
Propiedad 1.
∀
x
,
y
,
z
∈
V
:
x
+
¯
(
y
+
¯
z
)
=
(
x
+
¯
y
)
+
¯
z
{\displaystyle \forall x,y,z\in V:\quad x{\bar {+}}(y{\bar {+}}z)=(x{\bar {+}}y){\bar {+}}z}
Propiedad 2.
∃
!
e
∈
V
,
∀
x
∈
V
:
x
+
¯
e
=
e
+
¯
x
=
x
{\displaystyle \exists !e\in V,\quad \forall x\in V:\quad x{\bar {+}}e=e{\bar {+}}x=x}
Propiedad 3.
∀
x
∈
V
,
∃
−
x
∈
V
:
(
−
x
)
+
¯
x
=
x
+
¯
(
−
x
)
=
e
{\displaystyle \forall x\in V,\quad \exists -x\in V:\quad (-x){\bar {+}}x=x{\bar {+}}(-x)=e}
Propiedad 4.
∀
x
,
y
∈
V
:
x
+
¯
y
=
y
+
¯
x
{\displaystyle \forall x,y\in V:\quad x{\bar {+}}y=y{\bar {+}}x}
Propiedad 5.
∃
!
1
∈
F
,
∀
x
∈
V
:
1
∘
x
=
x
{\displaystyle \exists !1\in F,\quad \forall x\in V:\quad {1}\circ {x}=x}
Propiedad 6.
∀
a
∈
F
,
∀
x
,
y
∈
V
:
a
∘
(
x
+
¯
y
)
=
(
a
∘
x
)
+
¯
(
a
∘
y
)
{\displaystyle \forall a\in F,\quad \forall x,y\in V:\quad {a}\circ {(x{\bar {+}}y)}=({a}\circ {x}){\bar {+}}({a}\circ {y})}
Propiedad 7.
∀
a
,
b
∈
F
,
∀
x
∈
V
:
(
a
×
b
)
∘
x
=
a
∘
(
b
∘
x
)
{\displaystyle \forall a,b\in F,\quad \forall x\in V:\quad {({a}\times {b})}\circ {x}={a}\circ {({b}\circ {x})}}
Propiedad 8.
∀
a
,
b
∈
F
,
∀
x
∈
V
:
(
a
+
b
)
∘
x
=
(
a
∘
x
)
+
¯
(
b
∘
x
)
{\displaystyle \forall a,b\in F,\quad \forall x\in V:\quad {(a+b)}\circ {x}=({a}\circ {x}){\bar {+}}({b}\circ {x})}
+
{\displaystyle +}
×
{\displaystyle \times }
A los elementos de V se les llama Vectores y a los elementos de F se les llama escalares.
+
¯
{\displaystyle {\bar {+}}}
+
{\displaystyle +}
∘
{\displaystyle \circ }
×
{\displaystyle \times }
1.
V
=
R
2
{\displaystyle V=\mathbb {R} ^{2}}
R
{\displaystyle \mathbb {R} }
V
=
M
n
×
m
(
R
)
{\displaystyle V=M_{{n}\times {m}}(\mathbb {R} )}
n
×
m
{\displaystyle {n}\times {m}}
R
{\displaystyle \mathbb {R} }
R
{\displaystyle \mathbb {R} }
P
n
(
R
)
{\displaystyle \mathbb {P} _{n}(\mathbb {R} )}
n
{\displaystyle n}
R
{\displaystyle \mathbb {R} }
R
{\displaystyle \mathbb {R} }
Teorema En un espacio vectorial siempre se cumplen las siguientes propiedades:
0
∘
x
=
e
{\displaystyle 0\circ {x}=e}
∀
x
∈
V
{\displaystyle \forall x\in V}
donde
0
∈
F
{\displaystyle 0\in F}
(
−
1
)
∘
x
=
−
x
{\displaystyle (-1)\circ {x}=-x}
∀
x
∈
V
{\displaystyle \forall x\in V}
a
∘
e
=
e
{\displaystyle a\circ {e}=e}
∀
a
∈
F
{\displaystyle \forall a\in F}
Demostración
(
0
∘
x
)
+
¯
e
=
0
∘
x
=
(
0
+
0
)
∘
x
=
(
0
∘
x
)
+
¯
(
0
∘
x
)
{\displaystyle (0\circ {x}){\bar {+}}e=0\circ {x}=(0+0)\circ {x}\ =\ (0\circ {x}){\bar {+}}(0\circ {x})}
e
=
0
∘
x
{\displaystyle e=0\circ {x}}
x
+
¯
(
(
−
1
)
∘
x
)
=
(
1
∘
x
)
+
¯
(
(
−
1
)
∘
x
)
=
(
1
+
(
−
1
)
)
∘
x
=
0
∘
x
=
e
{\displaystyle x{\bar {+}}((-1)\circ x)=(1\circ x){\bar {+}}((-1)\circ x)=(1+(-1))\circ x=0\circ x=e}
x
{\displaystyle x}
(
−
1
)
∘
x
=
−
x
{\displaystyle (-1)\circ x=-x}
(
a
∘
e
)
+
¯
e
=
a
∘
e
=
a
∘
(
e
+
¯
e
)
=
(
a
∘
e
)
+
¯
(
a
∘
e
)
{\displaystyle (a\circ e){\bar {+}}e=a\circ e=a\circ (e{\bar {+}}e)=(a\circ e){\bar {+}}(a\circ e)}
e
=
a
∘
e
{\displaystyle e=a\circ e}
