1097: Orthogonal Complement of Subset of Vectors Space with Inner Product

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definition of orthogonal complement of subset of vectors space with inner product

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Topics

About: vectors space

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The table of contents of this article

• Starting Context
• Target Context
• Orientation
• Main Body
• 1: Structured Description
• 2: Note

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Starting Context

• The reader knows a definition of inner product on real or complex vectors space.

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Target Context

• The reader will have a definition of orthogonal complement of subset of vectors space with inner product.

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Orientation

There is a list of definitions discussed so far in this site.

There is a list of propositions discussed so far in this site.

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Main Body

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1: Structured Description

Here is the rules of Structured Description.

Entities:
$F$: $\in \{\mathbb{R},\mathbb{C}\}$, with the canonical field structure
$V$: $\in \{\text{ the }F\text{ vectors spaces }\}$, with any inner product, $⟨\bullet ,\bullet ⟩$
$S$: $\subseteq V$
$\ast S^{\perp }$: $=\{v\in V|\mathrm{\forall }s\in S(⟨v,s⟩=0)\}$
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Conditions:
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2: Note

$S$ does not need to be any vectors subspace.

But when $S\ne \mathrm{\varnothing }$, $S^{\perp }=(S{)}^{\perp }$, where $(S)=Span(S)$ (see Note for the sub-'vectors space' generated by $S$).

That is because for each $v\in S^{\perp }$, for each $s\in (S)$, $⟨v,s⟩=0$, because $s=r^1{s}_1+...+r^n{s}_n$ where $s_j\in S$, and $⟨v,s⟩=⟨v,r^1{s}_1+...+r^n{s}_n⟩=r^1⟨v,s_1⟩+...+r^n⟨v,s_n⟩=0+...+0=0$; for each $v\in (S{)}^{\perp }$, for each $s\in S$, $⟨v,s⟩=0$, because $s\in (S)$.

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References

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