La Importancia de los espacios vectoriales en Física
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Definición de Espacio Vectorial
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Un espacio vectorial
V
{\displaystyle {\mathcal {V}}}
C
{\displaystyle \mathbb {C} }
|
a
i
⟩
{\displaystyle |a_{i}\rangle }
+
{\displaystyle +}
+
:
V
×
V
→
V
{\displaystyle +:{\mathcal {V}}\times {\mathcal {V}}\rightarrow {\mathcal {V}}}
|
a
⟩
+
|
b
⟩
=
|
b
⟩
+
|
a
⟩
,
∀
|
a
⟩
,
|
b
⟩
∈
V
{\displaystyle |a\rangle +|b\rangle =|b\rangle +|a\rangle ,\forall \ |a\rangle ,|b\rangle \in {\mathcal {V}}}
|
a
⟩
+
(
|
b
⟩
+
|
c
⟩
)
=
(
|
a
⟩
+
|
b
⟩
)
+
|
c
⟩
,
∀
|
a
⟩
,
|
b
⟩
,
|
c
⟩
∈
V
{\displaystyle |a\rangle +(|b\rangle +|c\rangle )=(|a\rangle +|b\rangle )+|c\rangle ,\forall \ |a\rangle ,|b\rangle ,|c\rangle \in {\mathcal {V}}}
Existe un único vector
|
0
⟩
∈
V
{\displaystyle |0\rangle \in {\mathcal {V}}}
|
a
⟩
+
|
0
⟩
=
|
a
⟩
,
∀
|
a
⟩
∈
V
{\displaystyle |a\rangle +|0\rangle =|a\rangle ,\forall |a\rangle \in {\mathcal {V}}}
Para todo vector
|
a
⟩
∈
V
{\displaystyle |a\rangle \in {\mathcal {V}}}
−
|
a
⟩
∈
V
{\displaystyle -|a\rangle \in {\mathcal {V}}}
|
a
⟩
+
(
−
|
a
⟩
)
=
|
0
⟩
{\displaystyle |a\rangle +(-|a\rangle )=|0\rangle }
