745: For Vectors Space with Inner Product, Set of Nonzero Orthogonal Elements Is Linearly Independent

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description/proof of that for vectors space with inner product, set of nonzero orthogonal elements is linearly independent

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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: Natural Language Description
• 3: Proof

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Starting Context

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

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Target Context

• The reader will have a description and a proof of the proposition that for any vectors space with any inner product, any set of nonzero orthogonal elements is linearly independent.

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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, $⟨\bullet ,\bullet ⟩$
$S$: $=\{v_1,...,v_n\}\subseteq V$
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Statements:
$\mathrm{\forall }{v}_j\in S(v_j\ne 0)\wedge \mathrm{\forall }{v}_j,v_k\in S\text{ such that }{v}_j\ne v_k(⟨v_j,v_k⟩=0)$
$⟹$
$S\in \{\text{ the linearly independent subsets of }V\}$
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2: Natural Language Description

For any $F\in \{\mathbb{R},\mathbb{C}\}$ with the canonical field structure, any $F$ vectors space, $V$, with any inner product, $⟨\bullet ,\bullet ⟩$, and any subset, $S=\{v_1,...,v_n\}\subseteq V$, such that $\mathrm{\forall }{v}_j\in S(v_j\ne 0)$ and $\mathrm{\forall }{v}_j,v_k\in S\text{ such that }{v}_j\ne v_k(⟨v_j,v_k⟩=0)$, $S$ is linearly independent.

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3: Proof

Whole Strategy: Step 1: suppose that $\mathrm{\forall }{v}_j\in S(v_j\ne 0)\wedge \mathrm{\forall }{v}_j,v_k\in S\text{ such that }{v}_j\ne v_k(⟨v_j,v_k⟩=0)$ and $S$ was linearly dependent and find a contradiction.

Step 1:

Let us suppose that $\mathrm{\forall }{v}_j\in S(v_j\ne 0)\wedge \mathrm{\forall }{v}_j,v_k\in S\text{ such that }{v}_j\ne v_k(⟨v_j,v_k⟩=0)$.

Let us suppose that $S$ was linearly dependent.

There would be a not-all-zero $\{c^1,...,c^n\}\subseteq F$ such that $c^j{v}_j=0$ (with the Einstein convention). Let $c^k\ne 0$. $c^k⟨v_k,v_k⟩=⟨c^j{v}_j,v_k⟩=⟨0,v_k⟩=0$. But $⟨v_k,v_k⟩\ne 0$, so, $c^k=0$, a contradiction.

So, $S$ is linearly dependent.

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References

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