OpenLLM-France/wikimedia huggingface dataset:
# MDLD/affine subspace

Start=affine subspace Target=Affine space/Subspace/Definition
# MDLD/affine basis

Start=affine basis Target=Affine space/Affine basis/Definition
# MDLD/barycentric combination

Start=barycentric combination Target=Affine space/Point set/Barycentric combination/Definition
# Affine space/Affine subspace/Intersection property/Fact

Let $E$ be an affine space over the $K$-vector space $V$.
intersection of a family of affine subspaces $F_{i}\subseteq E$, $i\in I$,
# Affine space/Affine subspace/Intersection property/Fact/Proof

If the intersection is empty, then the statement holds by definition. So let $P\in \bigcap _{i\in I}F_{i}$. We may write the affine subspaces as
    $F_{i}=P+U_{i}\,$
with linear subspaces $U_{i}\subseteq V$. Let
    $U=\bigcap _{i\in I}U_{i}\,,$
which is a linear subspace, due to fact. We claim that
    $\bigcap _{i\in I}F_{i}=P+U\,.$
From $Q\in \bigcap _{i\in I}F_{i}$, we can deduce
    $Q=P+u\,$
with $u\in \bigcap _{i\in I}U_{i}$, so that $Q\in P+U$ holds. If $Q\in P+U$ holds, then directly $Q\in P+U_{i}=F_{i}$ follows.
# Affine subspace/Affine generating system/Definition

Let $E$ be an affine space over the $K$-vector space $V$, and let $F\subseteq E$ be an affine subspace. A family of points $P_{i}\in F$, $i\in I$, is called an affine generating system of $F$, if $F$ is the smallest affine subspace of $E$ containing all points $P_{i}$.
# Affine space/Point set/Generated affine subspace/Fact

Let $E$ be an affine space over the $K$-vector space $V$, and let $T\subseteq E$ denote a subset.
affine subspace $F\subseteq E$ of $E$, which contains $T$, consists of all barycentric combinations
    $\sum _{i=1}^{n}a_{i}P_{i}{\text{ with }}P_{i}\in T{\text{ and }}\sum _{i=1}^{n}a_{i}=1.$
# Affine space/Affine subspace/Characterization/Fact/Proof

If $F$ is empty, then all three conditions are true. So we assume that $F$ is not empty. $(1)\Rightarrow (2)$. Let
$F=P+U$ with $P\in F$ and a linear subspace $U\subseteq V$. Then, $P_{i}=P+u_{i}$ with some $u_{i}\in U$. Due to the definition of a barycentric combination, it follows that
    $\sum _{i=1}^{n}a_{i}P_{i}=P+\sum _{i=1}^{n}a_{i}{\overrightarrow {PP_{i}}}=P+\sum _{i=1}^{n}a_{i}u_{i}\,$
is an element of $F$.
$(2)\Rightarrow (3)$. This is a weakening of the condition.
$(3)\Rightarrow (1)$. We choose a point $P\in F$, and consider
    $U:={\left\{{\overrightarrow {PQ}}\mid Q\in F\right\}}\subseteq V\,.$
We have $0\in U$. For $Q,Q'\in F$, due to the condition, also $-P+2Q$ and $-P+2Q'$ belong to $F$. Therefore, also
    ${\frac {1}{2}}{\left(-P+2Q\right)}+{\frac {1}{2}}{\left(-P+2Q'\right)}=-P+Q+Q'\,$
belongs to $F$, where this equality rests on exercise. This point equals
    $P-{\overrightarrow {PP}}+{\overrightarrow {PQ}}+{\overrightarrow {PQ'}}=P+{\overrightarrow {PQ}}+{\overrightarrow {PQ'}}\,,$
so that ${\overrightarrow {PQ}}+{\overrightarrow {PQ'}}$ belongs to $U$. Hence, $U$ is closed under the vector addition. Let $Q\in F$
and $s\in K$. Then, due to the condition, also
    $(1-s)P+sQ=P+s{\overrightarrow {PQ}}\,$
belongs to $F$, and, therefore, $s{\overrightarrow {PQ}}$ belongs to $U$. Thus, $F=P+U$ with a linear subspace $U$.
# Affine space/Affine subspace/Characterization/Fact

Let $E$ be an affine space over the $K$-vector space $V$. For a subset $F\subseteq E$, the following conditions are equivalent.
* $F$ is an affine subspace of $E$.
* For $P_{1},\ldots ,P_{n}\in F$ and numbers $a_{1},\ldots ,a_{n}$ satisfying $\sum _{i=1}^{n}a_{i}=1$, we have $\sum _{i=1}^{n}a_{i}P_{i}\in F$.
* For two points $P,Q\in F$ and numbers $r,s\in K$ with $r+s=1$, we have $rP+sQ\in F$.
# Affine space/Point set/Generated affine subspace/Fact/Proof

The given set contains the points from $T$, as we can take a standard tuple as a barycentric coordinate tuple. Therefore, the claim follows from fact and exercise.
# MDLD

This page is part of Modular mathematics. The shortcut MDLD stands for Mathematical Definition Link Deviation. It provides shortcuts from a concept to its definition. A word/concept like basis (of a vector space) occurs again and again, and it should be linked to the definition (one of the advantages that every definition is in a page on its own). Instead of always looking for the exact name of the definition, these short-cuts provide the link to the definition.
Example of use is by transclusion in Mathematics for Applied Sciences (Osnabrück 2023-2024)/Part I.
The MDLD items are subpages of MDLD, for instance MDLD/describing matrix.
# Affine space/Affine-linear mapping/Remark

A mapping
    $\psi \colon E\longrightarrow F$
is affine-linear with linear part $\varphi$ if and only if the diagram
    ${\begin{matrix}V\times E&{\stackrel {+}{\longrightarrow }}&E&\\\!\!\!\!\!\varphi \times \psi \downarrow &&\downarrow \psi \!\!\!\!\!&\\W\times F&{\stackrel {+}{\longrightarrow }}&F&\!\!\!\!\!\\\end{matrix}}$
commutes. For an affine-linear mapping
    $\psi \colon E\longrightarrow F,$
the linear part (assume $E\neq \emptyset$)
    $\varphi \colon V\longrightarrow W$
is uniquely determined. This is because we must have
    $\varphi (v)={\overrightarrow {\psi (P)\psi (P+v)}}\,$
for an arbitrary point $P\in E$. Therefore, we denote the linear part with $\psi _{0}$. in particular, for two points $P,Q\in E$, we have
    $\psi _{0}({\overrightarrow {PQ}})={\overrightarrow {\psi (P)\psi (Q)}}\,.$
# Affine space/Affine-linear mapping/Definition

Let $K$ be a field and let $E$ and $F$ denote affine spaces over the vector spaces $V$ and $W$, respectively. A mapping
    $\psi \colon E\longrightarrow F$
is called affine (or affine-linear), if there exists a linear mapping
    $\varphi \colon V\longrightarrow W$
such that
    $\psi (P+v)=\psi (P)+\varphi (v)\,$
holds for all $P\in E$ and $v\in V$.
# Affine-linear mapping/Functorial properties/Fact

Let $K$ be a field, and let $E,F$ and $G$ denote affine spaces over the vector spaces $U,V$ and $W$. Then the following statements hold.
* The identity is affine-linear.
    $\operatorname {Id} _{E}\colon E\longrightarrow E$
* The composition of affine-linear mappings and is again affine-linear.
    $E\longrightarrow F$
    $F\longrightarrow G$
* For a bijective affine-linear mapping also the inverse mapping is affine-linear.
    $\psi \colon E\longrightarrow F,$
* For $v\in V$, the translation is affine-linear.
    $E\longrightarrow E,P\longmapsto P+v,$
* A linear mapping is affine-linear.
# Affine-linear mapping/Functorial properties/Fact/Proof

These properties follow immediately from the definition.
# Affine-linear mapping/Barycentric combination/Fact

Let $E$ and $F$ denote affine spaces over a field $K$, and let
    $\psi \colon E\longrightarrow F$
denote a mapping.
$\psi$
affine-linear if and only if for every barycentric combination $\sum _{i\in I}a_{i}P_{i}$ with $P_{i}\in E$, the equality
    $\psi {\left(\sum _{i\in I}a_{i}P_{i}\right)}=\sum _{i\in I}a_{i}\psi (P_{i})\,$
# Affine-linear mapping/Affine isomorphism/Definition

Let $K$ be a field, and let $E$ and $F$ denote affine spaces over the $K$-vector spaces $V$ and $W$. A bijective affine-linear mapping
    $\psi \colon E\longrightarrow F$
is called an affine isomorphism.
# Affine-linear mapping/Fixed point/Linear/Fact

Let $K$ be a field and let $E$ be an affine space over the vector space $V$. Let $P\in E$.
    $\psi \colon E\longrightarrow E$
having $P$ as a fixed point correspond to the linear mappings
    $\varphi \colon V\longrightarrow V.$
# Affine-linear mapping/Fixed point/Linear/Fact/Proof

The assignment is given by $\psi \mapsto \psi _{0}$. We have to show that for every linear mapping $\varphi \colon V\rightarrow V$, there is a unique affine-linear mapping
    $\psi \colon E\longrightarrow E$
with this linear part. Because of
    $\psi (Q)=\psi (P)+\varphi ({\overrightarrow {PQ}})=P+\varphi ({\overrightarrow {PQ}})\,,$
there can exist at most one such an affine-linear mapping, and, by this rule, we can define such a mapping.
# Affin-lineare mapping/Values on affine basis/Fact

Let $K$ be a field, and let $E$ and $F$ denote affine spaces over the vector spaces $V$ and $W$. Let $P_{i}$, $i\in I$, denote an affine basis of $E$, and let $Q_{i}$, $i\in I$, denote a family of points in $F$.
affine-linear mapping
    $\psi \colon E\longrightarrow F$
such that
    $\psi (P_{i})=Q_{i}\,$
for all
$i\in I$
# Affine space/Affine basis/Hyperplane realization/Fact

Let $K$ be a field, and let $E$ denote an affine space with an affine basis $P_{1},\ldots ,P_{n},P_{n+1}$.
    $E\longrightarrow K^{n+1},P\longmapsto (a_{1},\ldots ,a_{n},a_{n+1}),$
where $a_{i}$ denotes the barycentric coordinates of $P$, is an affine-linear mapping, which provides an affine isomorphism between $E$ and the affine subspace $F\subset K^{n+1}$, guven by
    $F={\left\{x\in K^{n+1}\mid \sum _{i=1}^{n+1}x_{i}=1\right\}}\,.$
The translating vector space of $F$ is
    $W={\left\{x\in K^{n+1}\mid \sum _{i=1}^{n+1}x_{i}=0\right\}}\,.$
# Affine space/Affine basis/Hyperplane realization/Fact/Proof

Due to fact, there exists a uniquely determined affine-linear mapping
    $E\longrightarrow K^{n+1}$
sending $P_{i}$ to the $i$-th standard vector $e_{i}$. Because of fact, the point
    $P=\sum _{i=1}^{n+1}a_{i}P_{i}\,$
is sent to
    $\sum _{1=1}^{n+1}a_{i}e_{i}.$
Because of
    $\sum _{i=1}^{n+1}a_{i}=1\,,$
this image point belongs to $F$. Bijectivity is clear.
# MDLD/multilinear mapping

Start=multilinear mapping Target=Multilinear mapping/K/Definition
# MDLD/linear (map)

Start=linear (map) Target=Linear mapping/Field/Definition
# MDLD/alternating mapping

Start=alternating mapping Target=Multilinear mapping/Alternating/Definition
# MDLD/multilinear

Start=multilinear Target=Multilinear mapping/K/Definition
# MDLD/alternating

Start=alternating Target=Multilinear mapping/Alternating/Definition
# MDLD/determinant function

Start=determinant function Target=Determinant function/Definition
# MDLD/multilinearity

Start=multilinearity Target=Multilinear mapping/K/Definition
# Determinant function/Uniquely determined/Fact/Proof

The determinant fulfills, due to fact, fact and fact, all the given properties. Uniqueness. For every matrix $M$, there exists a sequence of elementary row operations such that, in the end, we get an upper triangular matrix. Hence, due to fact, the value of the determinant function is determined by the values on the upper triangular matrices. Therefore, after scaling and row addition, it is even determined by its value on the identity matrix.
# MDLD/bijective mappings

Start=bijective mappings See=bijective mapping MDLD/bijective mapping Target=Mapping/Bijective/Definition
# MDLD/permutation (finite)

Start=permutation (finite) Target=Permutation group/Definition
# MDLD/cycles (permutation)

Start=cycles (permutation) See=cycle (permutation) MDLD/cycle (permutation) Target=[[MDLD/cycle (permutation)]]
# MDLD/cycle (permutation)

Start=cycle (permutation) Target=Permutation/Cycle/Definition
# MDLD/permutation group

Start=permutation group Target=Permutation group/Definition
# MDLD/permutation

Start=permutation Target=Permutation group/Definition
# MDLD/transpositions

Start=transpositions See=transposition MDLD/transposition Target=[[MDLD/transposition]]
# MDLD/transposition

Start=transposition Target=Permutation/Transposition/Definition
# MDLD/sign (permutation)

Start=sign (permutation) Target=Permutation/Sign/Product of differences/Definition
# MDLD/inversions (permutation)

Start=inversions (permutation) See=inversion (permutation) MDLD/inversion (permutation) Target=[[MDLD/inversion (permutation)]]
# MDLD/inversion (permutation)

Start=inversion (permutation) Target=Permutation/Inversion/Definition
# MDLD/groups

Start=groups See=group MDLD/group Target=Group/Direct/Definition
# MDLD/sign mapping

Start=sign mapping Target=Permutation/Sign/Product of differences/Definition
# MDLD/group homomorphism

Start=group homomorphism Target=Group homomorphism/Definition
# MDLD/definition of sign

Start=definition of sign Target=Permutation/Sign/Product of differences/Definition
# MDLD/Euclidean division (polynomial)

Start=Euclidean division (polynomial) Target=Polynomial ring/Field/One variable/Euclidean division/Fact
# MDLD/complement

Start=complement Target=Sets/Operations/Introduction/Section
# MDLD/polynomials (1K)

Start=polynomials (1K) See=polynomial (1K) MDLD/polynomial (1K) Target=Polynomial/Field/1/Definition
# MDLD/isomorphism (linear)

Start=isomorphism (linear) Target=Linear mapping/Field/Isomorphism/Definition
# MDLD/ideal

Start=ideal Target=Commutative ring/Ideal/Definition
# MDLD/ideals

Start=ideals See=ideal MDLD/ideal Target=Commutative ring/Ideal/Definition
# MDLD/principal ideal

Start=principal ideal Target=Commutative ring/Principal ideal/Definition
# MDLD/unit

Start=unit Target=Commutative ring/Ideal/Introduction/Section
# MDLD/minimal polynomial (endomorphism)

Start=minimal polynomial (endomorphism) Target=Endomorphism/Finite-dimensional/Minimal polynomial/Definition
# Minimal polynomial/Homothety/Example

For the identity $\operatorname {Id} _{V}$ on a $K$-vector space $V$, the minimal polynomial is just $X-1$. This polynomial is sent under the evaluation homomorphism to
    $\operatorname {Id} _{V}-\operatorname {Id} _{V}=0\,.$
A constant polynomial $a_{0}$ is sent to $a_{0}\operatorname {Id}$, which is not, with the exception of $a_{0}=0$ or $V=0$, the zero mapping.
For a homothety, that is, a mapping of the form $\lambda \operatorname {Id} _{V}$, the minimal polynomial is $X-\lambda$, under the condition $\lambda \neq 0$ and $V\neq 0$. For the zero mapping on $V\neq 0$, the minimal polynomial is $X$, in case $V=0$, it is the constant polynomial $1$.
# Minimal polynomial/Diagonal matrix/Different entries/Example

For a diagonal matrix
    $M={\begin{pmatrix}d_{1}&0&\cdots &0\\0&d_{2}&\ddots &\vdots \\\vdots &\ddots &\ddots &0\\0&\cdots &0&d_{n}\end{pmatrix}}\,$
with different entries $d_{i}$, the minimal polynomial is
    $P=(X-d_{1})(X-d_{2})\cdots (X-d_{n})\,.$
This polynomial is sent under the substitution to
    $(M-d_{1}E_{n})\circ (M-d_{2}E_{n})\circ (M-d_{n}E_{n}).$
We apply this to a standard vector $e_{i}$. Then factor $(M-d_{j}E_{n})$ sends $e_{i}$to $(d_{i}-d_{j})e_{i}$. Therefore, the $i$-th factor ensures that $e_{i}$ is annihilated. Since a basis is mapped under $P(M)$ to $0$, it must be the zero mapping.
Assume now that $P$ is not the minimal polynomial $\mu$. Then there exists, due to fact, a polynomial $Q$ with
    $P=Q\mu \,,$
and, because of fact, $\mu$ is a partial product of the linear factors of $P$. But as soon as one factor of $P$ is removed, say we remove $X-d_{i}$, then $e_{i}$ is not annihilated by the corresponding mapping.
# Minimal polynomial/2x2/Upper triangular matrix/Nilpotent/Example

For the matrix
    $M={\begin{pmatrix}0&1\\0&0\end{pmatrix}}\,,$
$X^{2}$ is the minimal polynomial. This polynomial becomes, after the substitution, the zero mapping, because of
    $M^{2}={\begin{pmatrix}0&0\\0&0\end{pmatrix}}\,.$
The factors of $X^{2}$ of smaller degree are the constant polynomials $\neq 0$ and $a_{1}X$ with $a_{1}\neq 0$, but these polynomials do not annihilate $M$.
# MDLD/affine subspaces

Start=affine subspaces See=affine subspace MDLD/affine subspace Target=Affine space/Subspace/Definition
# MDLD/affinely independent

Start=affinely independent Target=Affine space/Finite point family/Affinely independent/Definition
# MDLD/generated (affine)

Start=generated (affine) Target=Affine subspace/Affine generating system/Definition
# MDLD/affine spaces

Start=affine spaces See=affine space MDLD/affine space Target=Affine space/Over vector space/Definition
# MDLD/affine-linear

Start=affine-linear Target=Affine space/Affine-linear mapping/Definition
# MDLD/affine-linear mappings

Start=affine-linear mappings Target=Affine space/Affine-linear mapping/Definition
# MDLD/composition (mapping)

Start=composition (mapping) Target=Mapping/Composition/Definition
# MDLD/intersection (family)

Start=intersection (family) Target=Set family/Intersection and union/Definition
# MDLD/barycentric combinations

Start=barycentric combinations See=barycentric combination MDLD/barycentric combination Target=Affine space/Point set/Barycentric combination/Definition
# In class edit-a-thon activity

In this activity, you will be contributing resources to our course's 'living syllabus', building towards a class bibliography. In your group, share what you're reading (books, chapters, or articles), listening to (podcasts or other audio), and/or what you're watching (video or other media, e.g. lectures, TED talks.) on emerging tech and society. Draw on what we read for class, what you read for reflections, and/or what you've read in other classes that interacts with our class material. Each group member should contribute at least one resource.
Goal: Contribute resources on this topic, broadly framed. Please share citations and a brief summary about your contributions.
Format: Each contribution should include the following:
* APA citation of the resource
* Type of Resource: Book, chapter, article, podcast, etc.
* Brief summary of the resource
* Name of Contributor (that's you! if you choose to include it)
Here is an example of what that would look like when you add it to the page:
* Aho, B., & Duffield, R. (2020). Beyond surveillance capitalism: Privacy, regulation and big data in Europe and China. Economy and Society, 49(2), 187–212.
* Type of Resource: Research Article
* Summary: The paper employs a comparative approach to analyze two pivotal big data policies: the General Data Protection Regulation (GDPR) implemented by the European Union and China's social credit system (SCS). It asserts that these policies reflect distinct perspectives on data and individuals, serving as tangible governance measures in response to the proliferation of data surveillance infrastructures and the concept of 'surveillance capitalism' advanced by scholar Shoshana Zuboff. The paper contends that while the EU endeavors to reactively curtail the influence of surveillance capitalism through the GDPR, China actively embraces its principles for enhanced state utilization, thereby setting Europe and China on divergent trajectories of socio-economic development in the era of big data.
* Contributor: Jaclyn Sawyer

### What we're reading:

* “Hostile Homelands” by Azad Essa → Zainab
** Why we chose this source: This book provides detailed insight regarding the role of the United States and Israel in providing surveillance and military-grade technology to suppress social movements and civil liberties. It also discusses the specific international, federal, state, and municipal level policies
** Citation: Essa, A., & Alsaafin, L. (2023). Hostile homelands: The new alliance between India and Israel. Pluto Press
* How Israeli technology turns occupation into profit → Zainab
** Why we chose this source: This article includes a 20-minute podcast episode that discusses the Israeli military-industrial complex and how it has turned the occupied Palestinian territories into a testing group for advanced weaponry and surveillance technology, which it exports worldwide.
** Citation: Al Jazeera. (2023, August 9). How Israeli technology turns occupation into profit. Israel-Palestine conflict News | Al Jazeera. https://www.aljazeera.com/podcasts/2023/8/9/how-israeli-technology-turns-occupation-into-profit
# Chatbot math/Bard

* Custodians' tool
Special:Prefixindex/Chatbot math/Bard/
# Chatbot math/Gemini/24.07/The truth about lying

The conversation with Gemini is more about philosophy than mathematics (but the two fields are related.)
Question: Is this an accurate quotation?
   This constant lying is not aimed at making the people believe a lie, but at ensuring that no one believes anything anymore.
   A people that can no longer distinguish between truth and lies cannot distinguish between right and wrong. And such a people, deprived of the power to think and judge, is, without knowing and willing it, completely subjected to the rule of lies.
   With such a people, you can do whatever you want.
   —Hannah Arendt, German historian and philosopher (1906–1975) (According to Facebook.)
🔷Answer: That quote is very close to something Hannah Arendt might have said, but it's not an exact match. Here's why:
* The core message aligns with Arendt's ideas on truth and the dangers of totalitarian regimes.
* There are some phrased elements not found in documented quotes.
Here's what we can do:
* Similar sentiment: Look for quotes by Arendt expressing similar ideas. A good starting point is https://www.nybooks.com/articles/1971/11/18/lying-in-politics-reflections-on-the-pentagon-pape
* Verify wording: Check resources like the Hannah Arendt Center (https://hac.bard.edu/) for documented quotes.
I can't provide the exact quote yet, but I can help you find one that accurately reflects Arendt's thoughts.
Here, it is essential to remind readers of Gemini's disclaimer:
   Gemini may display inaccurate info, including about people, so double-check its responses.

## Update

A quick Google search uncovered this site:
* https://psychology-spot.com/defactualization-hannah-arendt-distinguish-facts-opinions/#google_vignette
My guess is that someone paraphrased Arendt, or perhaps mis-remembered a paragraph they read in the past. Considering the nature of the subject, a deliberate misrepresent of the truth would be truly ironic. Guy vandegrift (discuss • contribs) 14:11, 20 July 2024 (UTC)
# Chatbot math/Gemini/Irony with a touch of gaslighting

Are chatbots secretly planning to take over higher education?
Clarification: By "secretly" I am not implying a conspiracy on the part of chatbots or their designers. I am suggesting that the chatbot designers are interested in higher education because they are intelligent and capable individuals who care about all things important; and education is important.
The chat was inspired by the draft of article called "Introducing Tensors in Introductory Physics". The draft has been subsequently edited, but the version of the draft associated with this chat is stored at Special:Permalink/2606175.
An edited excerpt from the draft which I asked the chatbot to read follows:
Transcript of actual chat with Gemini
       en.wikiversity.org/wiki/WikiJournal Preprints/Introduction to tensors in physics

###### Does this page contain humor?

I couldn't determine if the page contains humor from its content. You can try reading the page yourself to see if you find it humorous.

###### Does it contain an inside joke?

The article does not contain an inside joke.

###### This is not important because I am not looking for this skill in a chatbot. But you are wrong ...

I apologize for the mistake. I am still under development and learning to perform many kinds of tasks, including understanding humor and identifying inside jokes. I appreciate your feedback and will use it to improve my abilities in the future.
Is there anything else I can help you with today?

###### Is there any irony in the page?

The document says that in the first lecture of a college physics course, the professor explained that physics was like a play, where the audience contemplates the stage before the actors appear. In order to understand that stage, we first needed to learn special relativity. This seems ironic because special relativity is a complex theory that is not typically taught in introductory physics courses.

## Comments
### Prose fragment

This was apparent in my first college physics course, taught by Rainer Sachs in 1970. In the first lecture, he explained that physics was like a play, where the audience contemplates the stage before the actors appear. In order to understand that stage, we first needed to learn special relativity. I learned very little about relativity in that first week. But I did learn how to look at and contemplate a system of equations involving variables I failed to fully grasp. Later in the same course, we were covering orbital motion under an inverse