description/proof of that for vectors space with inner product, orthonormal subset is linearly independent
Topics
About: vectors space
The table of contents of this article
Starting Context
- The reader knows a definition of orthonormal subset of vectors space with inner product.
- 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 orthonormal subset 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
\(S\): \(\in \{\text{ the orthonormal subsets of } V\}\)
//
Statements:
\(S \in \{\text{ the linearly independent subsets of } V\}\)
//
2: Proof
Whole Strategy: Step 1: take any finite subset of \(S\), \(S^` = \{s_1, ..., s_n\} \subseteq S\), and take \(r^1 s_1 + ... + r^n s_n = 0\), and see that \(r^j = 0\).
Step 1:
Let \(S^` = \{s_1, ..., s_n\} \subseteq S\) be any finite subset.
Let \(r^1 s_1 + ... + r^n s_n = 0\), where \(r^j \in F\).
Let \(j \in \{1, ..., n\}\) be any.
\(r^j = r^j \langle s_j, s_j \rangle = \langle r^1 s_1 + ... + r^n s_n, s_j \rangle = \langle 0, s_j \rangle = 0\).
So, \(S\) is linearly independent.
