OpenLLM-France/wikimedia huggingface dataset:
# Sequence Orchid

| | diagonal | triangle | row sums |
| ♀ | Violet | Robinia | Primula |
| ♂ | HalfGrass | HalfCedar | HalfCrocus |
| all | Lotus | Willow | Grass |
| ♂ − ♀ | Orchid | Hickory | Hyacinth |
| | valency gravity | strength depth |
| triangle | Willow | Poplar |
| diagonal | Lotus | Lily |
| difference | Orchid | Oxalis |

## Sequences Orchid$_{(Drop)}$

These are the row sums of TurnedWillow$_{(Drop)}$ without column 0.
| | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 |
| 🌊 Orchid (A218384) | 0 | 2 | 6 | 38 | 942 | 325262 | 25768825638 | 129127208425774833206 |
| 💧 OrchidDrop | 0 | 2 | 4 | 32 | 904 | 324320 | 25768500376 | 129127208400006007568 |
Orchid is the difference of Grass and Lotus.   (Also of HalfGrass and Violet.) :
| | Orchid | Grass − Lotus | HalfGrass − Violet |
| 0 | 0 | 2 − 2 | 1 − 1 |
| 1 | 2 | 4 − 2 | 2 − 0 |
| 2 | 6 | 16 − 10 | 8 − 2 |
| 3 | 38 | 256 − 218 | 128 − 90 |
| 4 | 942 | 65536 − 64594 | 32768 − 31826 |
| 5 | 325262 | 4294967296 − 4294642034 | 2147483648 − 2147158386 |
| 6 | 25768825638 | 18446744073709551616 − 18446744047940725978 | 9223372036854775808 − 9223372011085950170 |
| 7 | 129127208425774833206 | 340282366920938463463374607431768211456 − 340282366920938463334247399005993378250 | 170141183460469231731687303715884105728 − 170141183460469231602560095290109272522 |
half :
| | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 |
| 🌊 HalfOrchid (A051381) | 0 | 1 | 3 | 19 | 471 | 162631 | 12884412819 | 64563604212887416603 |
| 💧 HalfOrchidDrop | 0 | 1 | 2 | 16 | 452 | 162160 | 12884250188 | 64563604200003003784 |

## Sequences SmallOrchid$_{(Drop)}$

These are the row sums of SmallTurnedWillow$_{(Drop)}$ without column 0.
| | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 |
| 🌊 SmallOrchid | 0 | 1 | 5 | 37 | 941 | 325261 | 25768825637 | 129127208425774833205 |
| 💧 SmallOrchidDrop | 0 | 1 | 4 | 32 | 904 | 324320 | 25768500376 | 129127208400006007568 |
# Triangle Robinia

| | diagonal | triangle | row sums |
| ♀ | Violet | Robinia | Primula |
| ♂ | HalfGrass | HalfCedar | HalfCrocus |
| all | Lotus | Willow | Grass |
| ♂ − ♀ | Orchid | Hickory | Hyacinth |
* Pascal's triangle with multiplied columns

## Triangles Robinia$_{(Drop)}$

The diagonal of both triangles is Violet.
🌊    triangle Robinia    row sums Primula (A246537) :
| kn | 0 | 1 | 2 | 3 | 4 | 5 | sums |
| 0 | 1 · 1 1 | | | | | | 1 |
| 1 | 1 · 1 1 | 1 · 0 0 | | | | | 1 |
| 2 | 1 · 1 1 | 2 · 0 0 | 1 · 2 2 | | | | 3 |
| 3 | 1 · 1 1 | 3 · 0 0 | 3 · 2 6 | 1 · 90 90 | | | 97 |
| 4 | 1 · 1 1 | 4 · 0 0 | 6 · 2 12 | 4 · 90 360 | 1 · 31826 31826 | | 32199 |
| 5 | 1 · 1 1 | 5 · 0 0 | 10 · 2 20 | 10 · 90 900 | 5 · 31826 159130 | 1 · 2147158386 2147158386 | 2147318437 |
💧    triangle RobiniaDrop    row sums PrimulaDrop :
| kn | 0 | 1 | 2 | 3 | 4 | 5 | sums |
| 0 | 1 · 1 1 | | | | | | 1 |
| 1 | 0 | 1 · 0 0 | | | | | 0 |
| 2 | 0 | 1 · 0 0 | 1 · 2 2 | | | | 2 |
| 3 | 0 | 1 · 0 0 | 2 · 2 4 | 1 · 90 90 | | | 94 |
| 4 | 0 | 1 · 0 0 | 3 · 2 6 | 3 · 90 270 | 1 · 31826 31826 | | 32102 |
| 5 | 0 | 1 · 0 0 | 4 · 2 8 | 6 · 90 540 | 4 · 31826 127304 | 1 · 2147158386 2147158386 | 2147286238 |
# Sequences Pansy and Peony

| | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 |
| Lotus | 2 | 2 | 10 | 218 | 64594 | 4294642034 | 18446744047940725978 | |
| Pansy (A342286) | 2 | 0 | 2 | 8 | 210 | 64384 | 4294577650 | 18446744043646148328 |
| Peony (A342287) | 0 | 2 | 0 | 10 | 208 | 64386 | 4294577648 | 18446744043646148330 |
The entries of both sequences differ by 2. Pansy is bigger for even n. (Peony for odd n.)
Both sequences are related to Lotus, the number of all dense BF: S(n) = Lotus(n−1) − S(n−1)
These are the diagonals of triangles Chestnut and Chinkapin.
half :
| | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 |
| HalfLotus | 1 | 1 | 5 | 109 | 32297 | 2147321017 | 9223372023970362989 | |
| HalfPansy | 1 | 0 | 1 | 4 | 105 | 32192 | 2147288825 | 9223372021823074164 |
| HalfPeony | 0 | 1 | 0 | 5 | 104 | 32193 | 2147288824 | 9223372021823074165 |
# Should Russia retract its territorial claims, retreat its armies, and recognize Ukrainian independence and sovereignty?

The European Union–Ukraine Association Agreement was initialed in 2012. After prime minister Victor Yanukovych refused to sign the agreement, the Euromaidan uprising ousted him, culminating in Ukraine's w:Revolution of Dignity. Russia then in 2014 launched the w:War in Donbas and w:Annexation of Crimea. For primary sources on the Russia Ukraine war see s:Russia Ukraine war. After the Minsk agreements failed to deter, Russia invaded Ukraine in 2022.

## Russia should retract its territorial claims, retreat its armies, and recognize Ukrainian independence and sovereignty.
### Pro

* Pro the Ukrainian people a have a right to national self-determination in their homeland, they have decisively chosen association with Europe, and Russia has no good reason aggressively invade and absorb this independent State.

### Con

* Con Russia should invade and seize as much territory as possible in order to "Make Russia Great Again". Russians should keep fighting, regardless of the mass loss of human life and potential, and despite the vast expenditure of wealth, until the Ukrainians die or surrender to Russian sovereignty.
** Objection This line of thinking implicitly assumes a "might makes right" doctrine, which has been criticized as inhumane and barbaric.
* ↑ Temam, Edgar I. (2014). The "might makes right" fallacy: On a tacit justification for violence (in en). University of Oregon. https://books.google.com.eg/books/about/The_might_makes_right_fallacy.html?id=tgJF0AEACAAJ&redir_esc=y.
# Is postmodernism a pseudo-philosophy?

Some claim postmodernist philosophical writing (e.g. Derrida, Foucault, Lyotard, and Lacan) to be pseudo-philosophy. Are they right?

## Postmodernism is a pseudo-philosophy
### Arguments for

* Pro Postmodernism was exposed by the Sokal hoax. See also Wikipedia: Sokal hoax.
** Objection This is far from conclusive, being a mere single incident, as suggestive it may be.
* Pro Postmodernism was exposed by Sokal and Bricmont book Fashionable Nonsense (published as Intellectual Impostures in the U.K.).
** Objection Since the content of the book is not available online, the online reader has no way of putting this argument under stress by reviewing the book.
*** Objection The contents of the book are available online.
* Pro It is so by the duck test: if something looks like incomprehensible likely-meaningless text after first, second and third reading, there is a good chance it is meaningless text. The quotations from postmodernists given by Dawkins are suggestive.
** Objection The problematic passages are not entirely void of semantics, and therefore, they are probably not entirely meanigless.
*** Objection The quoted passages are at least largely (if not fully) meaningless.
* Pro According to Britannica, postmodernism states the following: "Reality, knowledge, and value are constructed by discourses; hence they can vary with them. This means that the discourse of modern science, when considered apart from the evidential standards internal to it, has no greater purchase on the truth than do alternative perspectives, including (for example) astrology and witchcraft." More does not need to be said.
** Objection Britannica is an Anglophone encyclopedia and it is not certain it gets postmodernism right.
*** Objection However, this debate is Anglophone (all the objections and counterobjections are made in English). So it makes sense to appeal to a prominent Anglophone encyclopedia.

### Arguments against

* Con Postmodernism is not a single monolith and therefore, it is unlikely to be possible to show it to be pseudo-philosophy without analyzing its different strands one at a time.
** Comment One could respond to this by adding "largely" into the motion.
* Con One could argue that for something to be pseudo, it needs to have at least an appearance of the thing, here philosophy. It is questionable that these author create at least an appearance of philosophy.
** Objection In so far as enough lay people and even professionals seem convinced they are dealing with philosophy, the requirement given by the prefix pseudo- seems to be met.
# Wikidebate

Wikidebate is a collaborative learning project to gather and organize all arguments on controversial issues. It's not a place to defend your preferred points of view, but original arguments are allowed and welcome.
Browse wikidebates New wikidebate

## Recent changes

Special:RecentChangesLinked/Category:Wikidebates
# Wikiphilosophers/Extraterrestrial life

Extraterrestrial life is life which does not originate from Earth. The question is whether extraterrestrial life exists and if so, where it is to be found. And the question is also whether intelligent extraterrestrial life exists. What if we have already been found by them but cannot communicate with them?
Read users' philosophical views below.
* S. Perquin suggests that the vast number of potentially habitable planets in the universe makes the existence of extraterrestrial life highly probable, particularly primitive life, while advanced intelligent life, if it exists, could have comprehensive knowledge and control over the universe. Read more...
* Mars Sterling Turner proves the existence of advanced extraterrestrial life is likely given that he proves the destruction of Sodom and Gomorrah required advanced weapon technology that humans don't have.
# Triangle Hickory

gender :
| | diagonal | triangle | row sums |
| ♀ | Violet | Robinia | Primula |
| ♂ | HalfGrass | HalfCedar | HalfCrocus |
| all | Lotus | Willow | Grass |
| ♂ − ♀ | Orchid | Hickory | Hyacinth |
* Pascal's triangle with multiplied columns
Hickory$_{(Drop)}$ = HalfCedar$_{(Drop)}$ − Robinia$_{(Drop)}$

## Triangles Hickory$_{(Drop)}$

The diagonal of both triangles is Orchid.
🌊    triangle Hickory    row sums Hyacinth (2·A059301) :
| kn | 0 | 1 | 2 | 3 | 4 | 5 | sums |
| 0 | 1 · 0 0 | | | | | | 0 |
| 1 | 1 · 0 0 | 1 · 2 2 | | | | | 2 |
| 2 | 1 · 0 0 | 2 · 2 4 | 1 · 6 6 | | | | 10 |
| 3 | 1 · 0 0 | 3 · 2 6 | 3 · 6 18 | 1 · 38 38 | | | 62 |
| 4 | 1 · 0 0 | 4 · 2 8 | 6 · 6 36 | 4 · 38 152 | 1 · 942 942 | | 1138 |
| 5 | 1 · 0 0 | 5 · 2 10 | 10 · 6 60 | 10 · 38 380 | 5 · 942 4710 | 1 · 325262 325262 | 330422 |
💧    triangle HickoryDrop    row sums HyacinthDrop :
| kn | 0 | 1 | 2 | 3 | 4 | 5 | sums |
| 0 | 1 · 0 0 | | | | | | 0 |
| 1 | 0 | 1 · 2 2 | | | | | 2 |
| 2 | 0 | 1 · 2 2 | 1 · 6 6 | | | | 8 |
| 3 | 0 | 1 · 2 2 | 2 · 6 12 | 1 · 38 38 | | | 52 |
| 4 | 0 | 1 · 2 2 | 3 · 6 18 | 3 · 38 114 | 1 · 942 942 | | 1076 |
| 5 | 0 | 1 · 2 2 | 4 · 6 24 | 6 · 38 228 | 4 · 942 3768 | 1 · 325262 325262 | 329284 |
# Sine and cosine/Real/Addition theorem/Rotation/Fact/Proof/Exercise

Prove the addition theorems for sine and cosine, using the rotation matrices.
# Clock/Minute- and second hand/Angle formula/Exercise

We look at a clock with minute and second hands, both moving continuously. Determine a formula which calculates the angular position of the second hand from the angular position of the minute hand (each starting from the 12-clock-position measured in the clockwise direction).
# Angle/Point on unit circle/Exercise

Show that
    $\angle ({\begin{pmatrix}1\\0\end{pmatrix}},{\begin{pmatrix}\cos \alpha \\\sin \alpha \end{pmatrix}})={\begin{cases}\alpha \,,{\text{ falls }}\alpha \leq \pi \,,\\2\pi -\alpha \,,{\text{ falls }}\alpha >\pi \end{cases}}\,$
holds.
# Angle/Inner product/Positive homogeneity/Exercise

Let $u,v\in V$ be vectors different from $0$ in a real vector space $V$, endowed with an inner product. Show that the angle between $u$ and $v$ equals the angle betwee $su$ and $tv$, where $s,t$ are positive real numbers.
# Linear algebra (Osnabrück 2024-2025)/Sine and cosine/Real/Addition theorem/Rotation/Fact/Proof/Exercise/Exercisereferencenumber

Exercise 34.11
# Linear algebra (Osnabrück 2024-2025)/Axis reflection/Angle/Eigenvectors and axes/Exercise/Exercisereferencenumber

Exercise 34.15
# Angle/Only plane/Exercise

Let $V$ be a real vector space, endowed with an inner product. Show that the angle
    $\angle (u,v)$
does only depend on the restriction of the inner product to the linear subspace generated by $u$ and $v$.
# Angle/Additivity/Exercise

Let $u,v,w\in V$ vectors different from $0$ in a real vector space $V$, endowed with an inner product. Show that
    $\angle (u,w)\leq \angle (u,v)+\angle (v,w)\,$
holds.
# Angle/Line/Exercise

What angles exist on a line?
# Angle/Standard vector and space diagonal/Limit/Exercise

Let $\alpha _{n}$ be the angle between the first standard vector $e_{1}$ and the vector $v_{n}=\sum _{i=1}^{n}e_{i}$ in $\mathbb {R} ^{n}$. Determine the limit
    $\lim _{n\rightarrow \infty }\alpha _{n}.$
# Plane rotation/Pythagorean/5,12,13/Complex eigenvalues/Exercise

Let
    $M={\begin{pmatrix}{\frac {5}{13}}&-{\frac {12}{13}}\\{\frac {12}{13}}&{\frac {5}{13}}\end{pmatrix}}\,.$
a) Show that $M$ defines an isometry on $\mathbb {R} ^{2}$ and also on $\mathbb {C} ^{2}$.
b) Determine the complex eigenvalues of $M$.
c) Determine an orthonormal basis of $\mathbb {C} ^{2}$ consisting of eigenvectors of $M$.
# Plane rotation/Pythagorean/3,4,5/Complex eigenvalues/Exercise

Let
    $M={\begin{pmatrix}{\frac {3}{5}}&-{\frac {4}{5}}\\{\frac {4}{5}}&{\frac {3}{5}}\end{pmatrix}}\,.$
a) Show that $M$ defines an isometry on $\mathbb {R} ^{2}$ and also on $\mathbb {C} ^{2}$.
b) Determine the complex eigenvalues of $M$.
c) Determine an orthonormal basis of $\mathbb {C} ^{2}$ consisting of eigenvectors of $M$.
# R^2/Ellipse to itself/No rotation/Exercise

Give an example of a (symmetric to the axes) ellipse $E$ in $\mathbb {R} ^{2}$ and a bijective linear mapping $\varphi \colon \mathbb {R} ^{2}\rightarrow \mathbb {R} ^{2}$ with $\varphi (E)=E$ that is not an isometry.
# Unit circle/Metric by positive angle/Exercise

Let
    $K={\left\{(x,y)\in \mathbb {R} ^{2}\mid x^{2}+y^{2}=1\right\}}\,$
be the unit circle. Show that we can obtain a metric on $K$ by defining $d(P,Q)$ ($P,Q\in K$) to be the positive angle at the origin $(0,0)$ between the corresponding rays.
# R^2/Sphere to sphere/Continuous not linear/Exercise

Give an example of a bijective, continuous mapping $\varphi \colon \mathbb {R} ^{2}\rightarrow \mathbb {R} ^{2}$, with $\varphi (S^{1})=S^{1}$ and with $\varphi (sv)=s\varphi (v)$ (for all $s\in \mathbb {R}$ and $v\in \mathbb {R} ^{2}$) that is not an isometry.
# Rotation with angle/Images of standard vectors/Linear mapping/Matrix/Exercise

Determine, in an elementary- geometric way, the image vectors of the standard vectors $e_{1}$ and $e_{2}$ under a rotation about the origin counter-clockwise with an angle $\alpha$.
# Axis reflection/Angle/Eigenvectors and axes/Exercise

Let
    $M={\begin{pmatrix}\cos \alpha &\sin \alpha \\\sin \alpha &-\cos \alpha \end{pmatrix}}\,$
be a plane reflection at an axis. Show that ${\begin{pmatrix}-\sin \alpha \\\cos \alpha -1\end{pmatrix}}$ is an eigenvector for the eigenvalue $1$, and that ${\begin{pmatrix}\cos \alpha -1\\\sin \alpha \end{pmatrix}}$ is an eigenvector for the eigenvalue $-1$ of $M$.
# Plane isometry/C/Special/Explicit form/Exercise

Let $M$ be a complex $2\times 2$-matrix such that the columns form an orthonormal basis of $\mathbb {C} ^{2}$ and such that the determinant equals $1$. Show that $M$ has the form
    $M={\begin{pmatrix}u&-{\overline {v}}\\v&{\overline {u}}\end{pmatrix}}\,,$
with $u,v\in \mathbb {C}$ and $\Vert {\begin{pmatrix}u\\v\end{pmatrix}}\Vert =1$.
# Space isometry/Matrix/From 4 numbers/Explicit/Exercise

Show that for $a,b,c,d\in \mathbb {R}$ satisfying $a^{2}+b^{2}+c^{2}+d^{2}=1$, the matrix
    ${\begin{pmatrix}a^{2}+b^{2}-c^{2}-d^{2}&2(-ad+bc)&2(ac+bd)\\2(ad+bc)&a^{2}-b^{2}+c^{2}-d^{2}&2(-ab+cd)\\2(-ac+bd)&2(ab+cd)&a^{2}-b^{2}-c^{2}+d^{2}\end{pmatrix}}$
defines an isometry on $\mathbb {R} ^{3}$.
# Space isometry/Rotation about coordinate axes/Exercise

Show that every proper linear isometry of $\mathbb {R} ^{3}$ can be realized as a composition of rotations around the three coordinate axes.
# Angle/Normed vectors/Bisection/Exercise

Let $u,v\in V$ be normed vectors in a real vector space $V$, endowed with an inner product. Show that the vector $u+v$ bisects the two vectors in equal angles.
# Space rotation/z-axis/45 degree/(1,2,4), (3,3,-1), (5,0,7)/Exercise

Let
    $\varphi \colon \mathbb {R} ^{3}\longrightarrow \mathbb {R} ^{3}$
be the rotation of the $3$-space around the $z$-axis by $45$ degree counter-clockwise. How does the describing matrix look like with respect to the basis
    ${\begin{pmatrix}1\\2\\4\end{pmatrix}},\,{\begin{pmatrix}3\\3\\-1\end{pmatrix}},\,{\begin{pmatrix}5\\0\\7\end{pmatrix}}?$
# Hyperbolic cosine/R/Power series/Odd coefficients 0/Exercise

Prove that in the power series $\sum _{n=0}^{\infty }c_{n}x^{n}$ of the hyperbolic cosine the coefficients $c_{n}$ are $0$ if $n$ is odd.
# Permutation matrix/Isometry/Exercise

Let $\pi \in S_{n}$ be a permutation, and let
    $M_{\pi }\colon \mathbb {R} ^{n}\longrightarrow \mathbb {R} ^{n}$
denote the corresponding permutation matrix and the corresponding linear mapping. Show that $M_{\pi }$ is an isometry When is it a proper isometry?
# R^3/Orthonormal basis/Permutation/Rotation axis/Exercise

Determine, for every permutation $\pi \in S_{3}$, the eigenline of the corresponding permutation matrix.
# R^3/Proper isometry/Not commutative/Exercise

Show that the group of the spatial rotations is not commutative.
# Space rotation/Example no entry 0,1/Exercise

Give an example of a spatial rotation where all matrix entries are $\neq 0,1$.
# R^3/Isometry/Given flag/Exercise

Let $U_{1}=\langle {\begin{pmatrix}4\\-8\\7\end{pmatrix}}\rangle$ and $U_{2}=\langle {\begin{pmatrix}4\\-8\\7\end{pmatrix}},\,{\begin{pmatrix}3\\-5\\2\end{pmatrix}}\rangle$ be linear subspaces in $\mathbb {R} ^{3}$. Find an isometry $\varphi \colon \mathbb {R} ^{3}\rightarrow \mathbb {R} ^{3}$ fulfilling $U_{1}=\langle \varphi (e_{1})\rangle$ and $U_{2}=\langle \varphi (e_{1}),\,\varphi (e_{2})\rangle$.
# Isometry/R^4/Two axes reflections/Exercise

The matrix
    ${\begin{pmatrix}\cos \alpha &\sin \alpha &0&0\\\sin \alpha &-\cos \alpha &0&0\\0&0&\cos \beta &\sin \beta \\0&0&\sin \beta &-\cos \beta \end{pmatrix}}$
defines a linear mapping $\varphi \colon \mathbb {R} ^{4}\rightarrow \mathbb {R} ^{4}$ ($\alpha ,\beta \in [0,\pi [$). Determine the eigenvalues of $\varphi$ and their algebraic and geometric multiplicities.
# Linear algebra (Osnabrück 2024-2025)/Part II/Lectures

* List of definitions
* Important theorems
# MDLD/angles

Start=angles See=angle MDLD/angle Target=[[MDLD/angle]]
# MDLD/angle

Start=angle Target=Inner product/Cauchy-Schwarz/Angle/Remark
# Euclidean spaces/Angle-preserving mappings/Section
## Definition

A linear mapping
    $\varphi \colon V\longrightarrow W$
between Euclidean vector spaces $V$ and $W$ is called angle-preserving, if for any two vectors $u,v\in V$, the relation
    $\angle (\varphi (u),\varphi (v))=\angle (u,v)\,$
Angles are only defined for vectors different from $0$. Therefore, angle-preserving mappings are injective. An isometry is always angle-preserving, because the norm and the angles are defined with respect to an inner product, and are not changed under an isometry. Further examples of angle-preserving mappings are homotheties by a scalar different from $0$, see exercise. An angle-preserving mapping maps orthogonal vectors to orthogonal vectors.

## Example

Let
    $\varphi \colon \mathbb {C} \longrightarrow \mathbb {C}$
be a $\mathbb {C}$-linear mapping that is given by the multiplication with the complex number
    $w=a+b{\mathrm {i} }\neq 0\,.$
With respect to the real basis $1,{\mathrm {i} }$ of $\mathbb {C} =\mathbb {R} ^{2}$, this mapping is described by the real $2\times 2$-matrix
    ${\begin{pmatrix}a&-b\\b&a\end{pmatrix}}.$
We write this matrix as
    ${\begin{pmatrix}a&-b\\b&a\end{pmatrix}}={\begin{pmatrix}{\sqrt {a^{2}+b^{2}}}&0\\0&{\sqrt {a^{2}+b^{2}}}\end{pmatrix}}{\begin{pmatrix}{\frac {a}{\sqrt {a^{2}+b^{2}}}}&-{\frac {b}{\sqrt {a^{2}+b^{2}}}}\\{\frac {b}{\sqrt {a^{2}+b^{2}}}}&{\frac {a}{\sqrt {a^{2}+b^{2}}}}\end{pmatrix}}\,.$
Therefore, we have a composition of an isometry (a plane rotation) and of a homothety with the scalar factor
    $\vert {w}\vert ={\sqrt {a^{2}+b^{2}}}\,;$
in particular, this is an angle-preserving mapping.

## Theorem

Let
    $\varphi \colon V\longrightarrow V$
denote an angle-preserving linear mapping on the Euclidean vector space $V$. Then there exists an isometry
    $\psi \colon V\longrightarrow V$
and a homothety
    $\sigma \colon V\longrightarrow V$
such that
    $\varphi =\sigma \circ \psi \,.$

### Proof

Let
    $r=\det \varphi \,,$
and set
    $s:={\sqrt[{n}]{\vert {r}\vert }}\,,$
where $n$ denotes the dimension of $V$. Let $\sigma$ be the homothety with factor $s$. We consider the mapping
    $\psi :=\sigma ^{-1}\circ \varphi \,.$
This mapping is still angle-preserving and its determinant is $1$ or $-1$. Because of exercise, $\psi$ is an isometry.
$\Box$

## Remark

For an angle-preserving mapping
    $\varphi \colon V\longrightarrow W$
between Euclidean vector spaces $V$ and $W$, not only the angles at the origin, but all angles are preserved. For points $P,Q,R\in V$, the angle of the triangle at $Q$ coincides with the angle at $\varphi (Q)$ of the image triangle $\varphi (P),\varphi (Q),\varphi (R)$, because of
    $\angle (P,Q,R)=\angle ({\overrightarrow {QP}},{\overrightarrow {QR}})=\angle (\varphi ({\overrightarrow {QP}}),\varphi ({\overrightarrow {QR}}))=\angle ({\overrightarrow {\varphi (Q)\varphi (P)}}),{\overrightarrow {\varphi (Q)\varphi (R)}})=\angle (\varphi (P),\varphi (Q),\varphi (R))\,.$
# Euclidean vector space/Linear mapping/Angle-preserving/Structure/Fact/Proof

Let
    $r=\det \varphi \,,$
and set
    $s:={\sqrt[{n}]{\vert {r}\vert }}\,,$
where $n$ denotes the dimension of $V$. Let $\sigma$ be the homothety with factor $s$. We consider the mapping
    $\psi :=\sigma ^{-1}\circ \varphi \,.$
This mapping is still angle-preserving and its determinant is $1$ or $-1$. Because of exercise, $\psi$ is an isometry.
# Linear algebra (Osnabrück 2024-2025)/Euclidean vector space/Homotheties/Angle-preserving/Exercise/Exercisereferencenumber

Exercise 35.1
# Linear algebra (Osnabrück 2024-2025)/Isometry/Orthogonal bijective determinant/Exercise/Exercisereferencenumber

Exercise 33.18
# Matrices/Isometric with respect to maximum norm/Exercise

We consider $\mathbb {R} ^{n}$ with the maximum norm
    $\Vert {v}\Vert =\Vert {v}\Vert _{\rm {max}}\,.$
We want to understand real matrices
    $M={\left(a_{ij}\right)}_{1\leq i,j\leq n}\,$
with the property
    $\Vert {Mv}\Vert =\Vert {v}\Vert \,$
for all $v\in \mathbb {R} ^{n}$. We call such a matrix $\Vert {-}\Vert _{\rm {max}}$-isometric.
a) Show that a $\Vert {-}\Vert _{\rm {max}}$-isometric matrix is invertible.
b) Show that the set of $\Vert {-}\Vert _{\rm {max}}$-isometric matrices is a subgroup of the general linear group.
c) Show that a permutation matrix is $\Vert {-}\Vert _{\rm {max}}$-isometric.
d) Under of a sign-permutation matrix, we mean a matrix that arises from a permutation matrix by adding in front of some of the entries a $-$-sign. Give an example of a $3\times 3$-sign-permutation matrix that is not a permutation matrix, that is not an upper triangular matrix, and such that its determinant is $1$.
e) Show that a sign-permutation matrix is $\Vert {-}\Vert _{\rm {max}}$-isometric.
f) Show that every $\Vert {-}\Vert _{\rm {max}}$-isometric matrix is a sign-permutation matrix.
# Linear algebra (Osnabrück 2024-2025)/Part II/Exercise sheets

* List of definitions
* Important theorems
# Euclidean vector space/Homotheties/Angle-preserving/Exercise

Let $V$ be a Euclidean vector space, and let $\varphi \colon V\rightarrow V$ denote a homothety with factor $s\neq 0$. Show that $\varphi$ is angle-preserving.