750: For Motion Between Same-Finite-Dimensional Real Vectors Spaces with Norms Induced by Inner Products, Motion Is Bijective

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description/proof of that for motion between same-finite-dimensional real vectors spaces with norms induced by inner products, motion is bijective

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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.
• The reader knows a definition of dimension of vectors space.
• The reader knows a definition of norm induced by inner product on real or complex vectors space.
• The reader knows a definition of bijection.
• The reader admits the proposition that any finite composition of motions is a motion.
• The reader admits the proposition that for any motion between any same-finite-dimensional real vectors spaces with the norms induced by any inner products that (the motion) fixes 0, the motion is an orthogonal linear map.
• The reader admits the proposition that any motion is injective.
• The reader admits the proposition that any linear injection between any same-finite-dimensional vectors spaces is a 'vectors spaces - linear morphisms' isomorphism.
• The reader admits the proposition that any finite composition of bijections is a bijection, if the codomains of the constituent bijections equal the domains of the succeeding bijections.

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Target Context

• The reader will have a description and a proof of the proposition that for any motion between any same-finite-dimensional real vectors spaces with the norms induced by any inner products, the motion is bijective.

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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:
$V_1$: $\in \{\text{ the }d\text{ -dimensional normed real vectors spaces }\}$ with the norm, $\Vert \bullet {\Vert }_1$, induced by any inner product, $⟨\bullet ,\bullet {⟩}_1$
$V_2$: $\in \{\text{ the }d\text{ -dimensional normed real vectors spaces }\}$ with the norm, $\Vert \bullet {\Vert }_2$, induced by any inner product, $⟨\bullet ,\bullet {⟩}_2$
$f$: $:V_1\to V_2$, $\in \{\text{ the motions }\}$
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Statements:
$f\in \{\text{ the bijections }\}$
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2: Natural Language Description

For any $d$-dimensional normed real vectors space, $V_1$, with the norm, $\Vert \bullet {\Vert }_1$, induced by any inner product, $⟨\bullet ,\bullet {⟩}_1$, any $d$-dimensional normed real vectors space, $V_2$, with the norm, $\Vert \bullet {\Vert }_2$, induced by any inner product, $⟨\bullet ,\bullet {⟩}_2$, and any motion, $f:V_1\to V_2$, $f$ is a bijection.

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3: Proof

Whole Strategy: Step 1: for $v_0=f(0)$, take the map, $f^{\prime }:V_2\to V_2,v\mapsto v-v_0$, and see that $f^{\prime }$ is a bijective motion; Step 2: see that $f^{\prime }\circ f:V_1\to V_2$ is a motion such that $f^{\prime }\circ f(0)=0$ and see that $f^{\prime }\circ f$ is a bijection; Step 3: see that $f=f^{\prime -1}\circ f^{\prime }\circ f$ is a bijection.

Step 1:

Let $v_0=f(0)$.

Let us take the map, $f^{\prime }:V_2\to V_2,v\mapsto v-v_0$.

Let us see that $f^{\prime }$ is bijective.

For each $v,v^{\prime }\in V_2$ such that $v\ne v^{\prime }$, $f^{\prime }(v)=v-v_0\ne v^{\prime }-v_0=f^{\prime }(v^{\prime })$. For each $v\in V_2$, $f^{\prime }(v+v_0)=v+v_0-v_0=v$.

Let us see that $f^{\prime }$ is a motion.

For each $v,v^{\prime }\in V_2$, $\Vert v-v^{\prime }{\Vert }_2=\Vert v-v_0-(v^{\prime }-v_0){\Vert }_2=\Vert f^{\prime }(v)-f^{\prime }(v^{\prime }){\Vert }_2$.

Step 2:

$f^{\prime }\circ f:V_1\to V_2$ is a motion, by the proposition that any finite composition of motions is a motion.

$f^{\prime }\circ f$ is injective, by the proposition that any motion is injective.

$f^{\prime }\circ f(0)=f^{\prime }(v_0)=v_0-v_0=0$.

$f^{\prime }\circ f$ is an orthogonal linear map, by the proposition that for any motion between any same-finite-dimensional real vectors spaces with the norms induced by any inner products that (the motion) fixes 0, the motion is an orthogonal linear map.

$f^{\prime }\circ f$ is a 'vectors spaces - linear morphisms' isomorphism, by the proposition that any linear injection between any same-finite-dimensional vectors spaces is a 'vectors spaces - linear morphisms' isomorphism, especially a bijection.

Step 3:

$f=f^{\prime -1}\circ f^{\prime }\circ f$ is a bijection, by the proposition that any finite composition of bijections is a bijection, if the codomains of the constituent bijections equal the domains of the succeeding bijections.

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References

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