description/proof of that orthogonal linear map is motion
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 orthogonal linear map.
- The reader knows a definition of motion.
Target Context
- The reader will have a description and a proof of the proposition that any orthogonal linear map is a motion.
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
\(V_1\): \(\in \{\text{ the normed } F \text{ vectors spaces }\}\)
\(V_2\): \(\in \{\text{ the normed } F \text{ vectors spaces }\}\)
\(f\): \(: V_1 \to V_2\), \(\in \{\text{ the orthogonal linear maps }\}\)
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Statements:
\(f \in \{\text{ the motions }\}\)
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2: Natural Language Description
For any \(F \in \{\mathbb{R}, \mathbb{C}\}\), with the canonical field structure, any normed \(F\) vectors spaces, \(V_1, V_2\), and any orthogonal linear map, \(f: V_1 \to V_2\), \(f\) is a motion.
3: Proof
Whole Strategy: Step 1: choose any elements, \(v, v' \in V_1\); Step 2: see that \(\Vert v - v' \Vert = \Vert f (v) - f (v') \Vert\).
Step 1:
Let us choose any elements, \(v, v' \in V_1\).
Step 2:
\(\Vert v - v' \Vert = \Vert f (v - v') \Vert = \Vert f (v) - f (v') \Vert\).
So, \(f\) is a motion.
