description/proof of that for vectors space with inner product, set of nonzero orthogonal elements is linearly independent
Topics
About: vectors space
The table of contents of this article
- Starting Context
- Target Context
- Orientation
- Main Body
- 1: Structured Description
- 2: Natural Language Description
- 3: Proof
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.
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.
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.
Main Body
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, \(\langle \bullet, \bullet \rangle\)
\(S\): \(= \{v_1, ..., v_n\} \subseteq V\)
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Statements:
\(\forall v_j \in S (v_j \neq 0) \land \forall v_j, v_k \in S \text{ such that } v_j \neq v_k (\langle v_j, v_k \rangle = 0)\)
\(\implies\)
\(S \in \{\text{ the linearly independent subsets of } V\}\)
//
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, \(\langle \bullet, \bullet \rangle\), and any subset, \(S = \{v_1, ..., v_n\} \subseteq V\), such that \(\forall v_j \in S (v_j \neq 0)\) and \(\forall v_j, v_k \in S \text{ such that } v_j \neq v_k (\langle v_j, v_k \rangle = 0)\), \(S\) is linearly independent.
3: Proof
Whole Strategy: Step 1: suppose that \(\forall v_j \in S (v_j \neq 0) \land \forall v_j, v_k \in S \text{ such that } v_j \neq v_k (\langle v_j, v_k \rangle = 0)\) and \(S\) was linearly dependent and find a contradiction.
Step 1:
Let us suppose that \(\forall v_j \in S (v_j \neq 0) \land \forall v_j, v_k \in S \text{ such that } v_j \neq v_k (\langle v_j, v_k \rangle = 0)\).
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 \(c^k \neq 0\). \(c^k \langle v_k, v_k \rangle = \langle c^j v_j, v_k \rangle = \langle 0, v_k \rangle = 0\). But \(\langle v_k, v_k \rangle \neq 0\), so, \(c^k = 0\), a contradiction.
So, \(S\) is linearly dependent.
