741: Orthogonal Linear Map Is Motion

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

Statements:
$f\in \{\text{ the motions }\}$
//

---

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, $v,v^{\prime }\in V_1$; Step 2: see that $\Vert v-v^{\prime }\Vert =\Vert f(v)-f(v^{\prime })\Vert$.

Step 1:

Let us choose any elements, $v,v^{\prime }\in V_1$.

Step 2:

$\Vert v-v^{\prime }\Vert =\Vert f(v-v^{\prime })\Vert =\Vert f(v)-f(v^{\prime })\Vert$.

So, $f$ is a motion.

---

References

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