744: Orthogonal Linear Map Between Same-Finite-Dimensional Normed Vectors Spaces Is 'Vectors Spaces - Linear Morphisms' Isomorphism and Inverse Is Orthogonal Linear Map

<The previous article in this series | The table of contents of this series | The next article in this series>

description/proof of that orthogonal linear map between same-finite-dimensional normed vectors spaces is 'vectors spaces - linear morphisms' isomorphism and inverse is orthogonal linear map

---

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 dimension of vectors space.
• The reader admits the proposition that any orthogonal linear map is a motion.
• 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.

---

Target Context

• The reader will have a description and a proof of the proposition that any orthogonal linear map between any same-finite-dimensional normed vectors spaces is a 'vectors spaces - linear morphisms' isomorphism and the inverse is an orthogonal linear map.

---

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
$V_1$: $\in \{\text{ the }d\text{ -dimensional normed }F\text{ vectors spaces }\}$
$V_2$: $\in \{\text{ the }d\text{ -dimensional normed }F\text{ vectors spaces }\}$
$f$: $:V_1\to V_2$, $\in \{\text{ the orthogonal linear maps }\}$
//

Statements:
$f\in \{\text{ the 'vectors spaces - linear morphisms' isomorphisms }\}$
$\wedge$
$f^{-1}\in \{\text{ the orthogonal linear maps }\}$
//

---

2: Natural Language Description

For any $F\in \{\mathbb{R},\mathbb{C}\}$, with the canonical field structure, any $d$-dimensional normed $F$ vectors spaces, $V_1,V_2$, and any orthogonal linear map, $f:V_1\to V_2$, $f$ is a 'vectors spaces - linear morphisms' isomorphism, and $f^{-1}$ is an orthogonal linear map.

---

3: Proof

Whole Strategy: Step 1: see that $f$ is a motion, so, $f$ is injective, and so, $f$ is a 'vectors spaces - linear morphisms' isomorphism; Step 2: see that for each $v\in V_2$, $\Vert v\Vert =\Vert f^{-1}(v)\Vert$.

Step 1:

$f$ is a motion, by the proposition that any orthogonal linear map is a motion.

$f$ is injective, by the proposition that any motion is injective.

$f$ 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.

So, $f^{-1}$ exists.

Step 2:

Let $v\in V_2$ be any. $\Vert v\Vert =\Vert f^{-1}(v)\Vert$?

$\Vert v\Vert =\Vert f\circ f^{-1}(v)\Vert =\Vert f^{-1}(v)\Vert$. So, yes.

---

References

<The previous article in this series | The table of contents of this series | The next article in this series>