b3.4_5.1_1.1____2.1. [entry] title: Conditions for inner products that generate a real number
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b3.4_5.1_1.1____2.1. [entry] title: Conditions for inner products that generate a real number
•
User can define a custom inner product.
◦
b3.4_5.1_1.1____2. title: A user can define a custom inner product. If a new inner product is defined, the definition of the length of a vector will be changed. But there are certain conditions must be met for the inner product to be considered valid.
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Which meet conditions below. We are used to using the standard inner product, which was defined by physicists, but it can also be considered a custom inner product.
1.
⟨
u
,
v
⟩
↦
R
{\lang}u,v{\rang}\mapsto\mathbb{R}
⟨
u
,
v
⟩
↦
R
2.
⟨
u
,
v
⟩
=
⟨
v
,
u
⟩
{\lang}u,v{\rang}= {\lang}v,u{\rang}
⟨
u
,
v
⟩
=
⟨
v
,
u
⟩
3.
⟨
u
+
v
,
w
⟩
=
⟨
u
,
w
⟩
+
⟨
v
,
w
⟩
{\lang}u+v,w{\rang}={\lang}u,w{\rang}+{\lang}v,w{\rang}
⟨
u
+
v
,
w
⟩
=
⟨
u
,
w
⟩
+
⟨
v
,
w
⟩
4.
k
⟨
u
,
v
⟩
=
⟨
k
u
,
v
⟩
k{\lang}u,v{\rang} = {\lang}ku,v{\rang}
k
⟨
u
,
v
⟩
=
⟨
k
u
,
v
⟩
5.
v
≠
0
⇒
⟨
v
,
v
⟩
>
0
\mathrm{v \neq 0 \Rightarrow \lang{v,v}\rang > 0}
v
=
0
⇒
⟨
v
,
v
⟩
>
0
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b3.4_5.1_1.1____2.1.1. title: The result of the inner product must be greater than zero if the vector is not zero. This can be related to the length of a vector and is very useful when trying to understand abstract vector spaces.
