definition of 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: Note
Starting Context
- The reader knows a definition of normed vectors space.
Target Context
- The reader will have a definition of 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:
\( F_1\): \(\in \{\mathbb{R}, \mathbb{C}\}\), with the canonical field structure
\( F_2\): \(\in \{\mathbb{R}, \mathbb{C}\}\), with the canonical field structure
\( V_1\): \(\in \{\text{ the normed } F_1 \text{ vectors spaces }\}\)
\( V_2\): \(\in \{\text{ the normed } F_2 \text{ vectors spaces }\}\)
\(*f\): \(: V_1 \to V_2\)
//
Conditions:
\(\forall v, v' \in V_1 (\Vert v - v' \Vert = \Vert f (v) - f (v') \Vert)\)
//
2: Natural Language Description
For any \(F_1 \in \{\mathbb{R}, \mathbb{C}\}\), with the canonical field structure, any \(F_2 \in \{\mathbb{R}, \mathbb{C}\}\), with the canonical field structure, any normed \(F_1\) vectors space, \(V_1\), and any normed \(F_2\) vectors space, \(V_2\), any map, \(f: V_1 \to V_2\), such that \(\forall v, v' \in V_1 (\Vert v - v' \Vert = \Vert f (v) - f (v') \Vert)\)
3: Note
\(f\) is not linear in general. For example, \(V_1\) and \(V_2\) are the Euclidean normed vectors spaces, \(\mathbb{R}^n\), and the translation, \(f: v \mapsto v + v_0\), where \(v_0 \neq 0\), is a motion, which is not linear, because \(f (0) = v_0\), so, \(f (r 0) = f (0) = v_0 \neq r v_0 = r f (0)\).
\(F_1\) and \(F_2\) do not need to be the same, because \(f\) is not supposed to be linear.
It is \(\Vert f (v) - f (v') \Vert\) not \(\Vert f (v - v') \Vert\): they can be different, because \(f\) is not linear in general. For example, \(V_1\) and \(V_2\) are the Euclidean normed vectors spaces, \(\mathbb{R}^n\), and for the translation, \(f: v \mapsto v + v_0\), where \(v_0 \neq 0\), \(\Vert v - v \Vert = \Vert 0 \Vert = 0 \neq \Vert v_0 \Vert = \Vert f (0) \Vert = \Vert f (v - v) \Vert\).
