748: Finite Composition of Motions Is Motion

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description/proof of that finite composition of motions is motion

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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 motion.

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Target Context

• The reader will have a description and a proof of the proposition that any finite composition of motions is a motion.

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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:
$\{F_1,...,F_n\}$: $F_j\in \{\mathbb{R},\mathbb{C}\}$, with the canonical field structure
$\{F_1^{\prime },...,F_n^{\prime }\}$: $F_j^{\prime }\in \{\mathbb{R},\mathbb{C}\}$, with the canonical field structure, such that $\mathrm{\forall }j\in \{1,...,n-1\}(F_j^{\prime }=F_{j+1})$
$\{V_1,...,V_n\}$: $V_j\in \{\text{ the normed }{F}_j\text{ vectors spaces }\}$, with the norm, $\Vert \bullet {\Vert }_j$
$\{V_1^{\prime },...,V_n^{\prime }\}$: $V_j^{\prime }\in \{\text{ the normed }{F}_j^{\prime }\text{ vectors spaces }\}$ such that $\mathrm{\forall }j\in \{1,...,n-1\}(V_j^{\prime }\subseteq V_{j+1})$, with the norm, $\Vert \bullet {\Vert }_{j+1}$
$\{f_1,...,f_n\}$: $f_j:V_j\to V_j^{\prime }$, $\in \{\text{ the motions }\}$
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Statements:
$f_n\circ ...\circ f_1:V_1\to V_n^{\prime }\in \{\text{ the motions }\}$
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2: Natural Language Description

For any $\{F_1,...,F_n\}$ where $F_j\in \{\mathbb{R},\mathbb{C}\}$ with the canonical field structure, any $\{F_1^{\prime },...,F_n^{\prime }\}$ where $F_j^{\prime }\in \{\mathbb{R},\mathbb{C}\}$ with the canonical field structure, such that for each $j\in \{1,...,n-1\}$, $F_j^{\prime }=F_{j+1}$, any $\{V_1,...,V_n\}$ where $V_j\in \{\text{ the normed }{F}_j\text{ vectors spaces }\}$ with the norm, $\Vert \bullet {\Vert }_j$, any $\{V_1^{\prime },...,V_n^{\prime }\}$ where $V_j^{\prime }\in \{\text{ the normed }{F}_j^{\prime }\text{ vectors spaces }\}$ such that for each $j\in \{1,...,n-1\}$, $V_j^{\prime }\subseteq V_{j+1}$, with the norm, $\Vert \bullet {\Vert }_{j+1}$, and any $\{f_1,...,f_n\}$ where $f_j:V_j\to V_j^{\prime }$ is any motion, $f_n\circ ...\circ f_1:V_1\to V_n^{\prime }$ is a motion.

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3: Proof

Whole Strategy: prove it inductively with respect to $n$; Step 1: prove it for the $n=1$ case; Step 2: prove it for the $n=2$ case; Step 3: suppose it for the $n=1,...,n^{\prime }-1$ cases, and prove it for the $n=n^{\prime }$ case.

Step 1:

Let us suppose that $n=1$.

$f_1$ is a motion.

Step 2:

Let us suppose that $n=2$.

For each $v,v^{\prime }\in V_1$, $\Vert v-v^{\prime }{\Vert }_1=\Vert f_1(v)-f_1(v^{\prime }){\Vert }_2=\Vert f_2\circ f_1(v)-f_2\circ f_1(v^{\prime }){\Vert }_3$. So, $f_2\circ f_1$ is a motion.

Step 3:

Let us suppose that the proposition holds for the $n=1,...,n^{\prime }-1$ cases.

Let us suppose that $n=n^{\prime }$.

$f_{n^{\prime }}\circ ...\circ f_1=f_{n^{\prime }}\circ (f_{n^{\prime }-1}\circ ...\circ f_1)$. $f_{n^{\prime }-1}\circ ...\circ f_1$ is a motion, by the induction hypothesis for the $n=n^{\prime }-1$ case. $f_{n^{\prime }}\circ (f_{n^{\prime }-1}\circ ...\circ f_1)$ is a motion, by the induction hypothesis for the $n=2$ case.

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References

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