OpenLLM-France/wikimedia huggingface dataset:
# Ithkuil/Lexicon

In other languages: العربية | Deutsch | Español | esperanto | français | 日本語 | русский | 汉语

## 1.0 ROOTS ASSOCIATED WITH GRAMMATICAL FUNCTIONS
### 1.1 The Carrier Root

| Root | Stem | BSC | CTE | CSV | OBJ |
| -S- | S0 | [the “carrier” root] | [the “carrier” root] | [the “carrier” root] | [the “carrier” root] |
| -S- | S1 | [carrier stem representing an animate being named by the succeeding foreign or proper name/phrase] | [carrier stem representing the identity of the animate being named by the succeeding foreign or proper name/phrase] | [carrier stem representing the physical presence/body of the animate being named by the succeeding foreign or proper name/phrase] | that which is impacted/targeted or a patient of the animate being named named by the succeeding foreign or proper name/phrase |
| -S- | S2 | [stem representing inanimate object or abstract entity named by the succeeding foreign or proper name/phrase] | [stem representing the identity of the inanimate object or abstract entity named by the succeeding foreign or proper name/phrase] | [stem representing the physical presence or body of the inanimate entity or tangible manifestation/effect of the abstract entity named by the succeeding foreign or proper name/phrase] | that which is impacted/targeted or a patient of the inanimate or abstract entity named by the succeeding foreign or proper name/phrase |
| -S- | S3 | [stem representing geographical or geo-demographic place/location named by the succeeding foreign or proper name/phrase] | [stem representing the identity of the geographical or geo-demographic place/location named by the succeeding foreign or proper name/phrase] | [stem representing the physical/tangible praxes and/or tangible infrastructure of the geographical or geo-demographic place/location named by the succeeding foreign or proper name/phrase] | that which is impacted/targeted or a patient of the geographical or geo-demographic place/location named by the succeeding foreign or proper name/phrase |
# Ithkuil/Resources

This page is intended to house external materials and tools relating to the Ithkuil languages.

### Ithkuil IV
#### Official documentation

Design For The New Revision of Ithkuil 1.3
Lexical Roots For The Ithkuil Revision 1.0
VxCs Affixes For Ithkuil IV 1.0
Phonotactic Rules For The Ithkuil Successor Language 0.5.4
New Ithkuil Script 1.0

### Ithkuil III
#### Official documentation

A Grammar of Ithkuil, a Constructed Philosophical Language

### Ithkuil I & Ithkuil II (Ilaksh)
#### Official documentation

A Philosophical Grammar of Ithkuil, a Constructed Language
# KOIIA/Reflections on Hjörne & Säljö/Några tankar från Ingela

Det är intressant att man inom skolan väldigt gärna vill diagnostisera de barn som på ett eller annat sätt inte passar in och att det tycks finnas en uppfattning om att bara barnet har en diagnos så är det bra och man kan "gå vidare". Så enkelt är det naturligtvis inte, samtidigt som det ju faktiskt finns barn som har speciella svårigheter. Att se diagnoser som en enkel utväg att förklara alla möjliga anpassningssvårigheter hos barn tycker jag är något som är problematiskt, men jag anser inte att man ska förkasta allt vad diagnoser heter heller, för t.ex. då det gäller elever som får diagnosen Aspergers syndrom, så faller många bitar på plats och elevens behov av struktur, ordning och tydlighet kan bättre tillgodoses, liksom att man får större förståelse för sociala problem som kan finnas i relationen till klasskamraterna. Samtidigt är det ofta så, som Annaliina skriver utifrån Hjörne och Säljö, att hela problemet läggs på eleven medan läraren kan fortsätta driva sin undervisning på samma sätt som förut. För att diagnoser ska ha någon funktion och bli till en hjälp för de elever som fått dem, så måste man gå flera steg längre och också titta på undervisningen, miljön i klassrummet och liknande. Där finns det mycket att göra!

## Google Translation

Some thoughts from Ingela
It is interesting that within the school very much like to diagnose the children in one way or another do not fit in and there seems to be a perception that only the child has a diagnosis , it's good and you can "move on" . It's that simple , of course not , while it actually has children who have special difficulties. Seeing diagnoses as an easy way to explain all the possible adjustment difficulties in children, I think, is something that is problematic , but I do not think you should dismiss all of what diagnoses named either, for example, in the case of students who are diagnosed with Asperger's syndrome, so many pieces fall into place and the student's need for structure , order and clarity can better accommodated, as well as to obtain a greater understanding of social problems that may exist in relation to classmates. Meanwhile, it is often the case , as Annaliina writes based Hjörne and Säljö, the whole problem is put on the student while the teacher may continue to pursue their education in the same way as before. To diagnoses must have some function and become a help to the students who received them , so you have to go several steps further and also look at education , the environment in the classroom and the like. Where there is much to do!
# KVM/Live Migration

Live Migration is available in KVM since 2007 but high memory write load can impact migration process..

## Activities

* Configure an environment with two physical machines and execute a live migration between them, if your application load is memory write intensive use post-copy method if you can risk reboot machines if there is a network error.
# KVM/libvirt/Review libvirt error messages in journald

journalctl -r | grep libvirt | grep error
Errors:
* Cannot allocate memory
* No space left on device
* Failed to get shared "write" lock
* Operation failed: Storage source conflict with pool
* internal error:
** qemu unexpectedly closed the monitor
** End of file from qemu monitor
** connection closed due to keepalive timeout
** process exited while connecting to monitor
* End of file while reading data: Input/output error
* Failed to acquire pid file
* argument unsupported
* Operation not supported
* unknown procedure
* Unable to read from monitor: Connection reset by peer
# KVM/qemu-img

qemu-img allows you to create, convert and modify images offline, It can handle all image formats supported by QEMU.
* qemu-img info
* qemu-img bench
          lazy_refcounts=on Reference count updates are postponed with the goal of avoiding metadata I/O and improving performance. Particularly interesting with cache=writethrough.
* qemu-img convert
* qemu-img check
** qemu-img check -r all

## Activities

* KVM/Extend LVM on qcow2 images
# KVM/virsh pool-list

$ virsh pool-list --all
# KVM/virsh dominfo

virsh dominfo MACHINE_NAME
# Localization/Glossary
## Glossary

* Also see: http://www.gala-global.org/view/terminology
| Acronym | Explanation | Comment |
| CAT | Computer Assisted Translation | Also Computer Aided Translation |
| CHM | Compiled HTML Help | |
| CSS | Cascading Style Sheet | Describe the presentation semantics of markup documents(html, xml) |
| DTD | Document Type Definition | Markup declarations that define a document typ |
| DTP | Desk Top Publishing | |
| ERP | Enterprise Resource Planning | |
| FL | Freelance Localizer | Independent Localizers |
| GMS | Globalization management System | |
| HT | Human Translation | |
| HTML | Hypertext Markup Language | |
| IPE | International Project Engineer | Also referred to as SLEor Loc Engineer |
| IPM | International Project Manager | |
| LPM | Localization Project Manager | Also referred to as IPM |
| LQA | Localization Quality Assurance | Also Linguistic Quality Assurance - to assess the quality of localized content. |
| LSP | Language Service Provider | Also Localization Vendor |
| MLV | Multi Language Vendor | |
| MT | Machine Translation | |
| RC | Release Candidate | |
| RFP | Request For Proposal | |
| RFI | Request for Information | |
| ROM | Rough Order of Magnitude (ROM) Estimate | A rough order of magnitude (ROM) estimate is the least accurate estimate. ROMs are -50% to +50% accurate, or potentially even larger. ROMs are used very early in the project when there is limited information from which to develop an accurate estimate. |
| RTF | Rich Text Format | |
| RTM | Release to Manufacturing | |
| RTO | Release to Operations | |
| RTW | Release to Web | |
| SGML | Standard Generalized Markup Language | ISO-standard technology for defining generalized markup languages |
| SL | Source Language | |
| SLE | Software Localization Engineer | Also referred to as IPE |
| SLV | Single Language Vendor | |
| SRX | Segmentation Rule Exchange | |
| SWAG | Scientific Wild-Ass Guess | A rough estimate made by an expert in the field, based on experience and intuition. |
| SWOT | Strength, Weakness, Opportunity, Threat | http://en.wikipedia.org/wiki/SWOT_analysis |
| TEP | Translation, Editing, Proofreading | Standard tasks included in localization |
| TL | Target Language | |
| TM | Translation Memory | |
| TMS | Translation Management System | aka GMS |
| TMX | Translation Memory Exchange | |
| UA | User Assistance | Also referred to as UE, content, help, documentation |
| UE | User Education | Also referred to as UA, content, help, documentation |
| UI | User Interface | Also referred to as SW (Software) |
| XML | Extende Markup Language | |
| XSL | Extensibel Stylesheet | Stylesheet for rendering and transforming XML data |
# Localization/Internationalization

(Internationalization is called Globalization at Microsoft)

### Regional formats
#### Handling regional formats in general

In order to prepare a product for easy localization it is necessary to enable code to support local, regional, or language preferences. Usually this involves incorporating predefined localization data and features derived from existing libraries or user preferences. Examples include date and time formats, local calendars, number formats and numeral systems, sorting and presentation of lists, handling of personal names and forms of address, etc.
| | Russian | English US | French | Italian | German | Spanish |
| Case | Yes | Yes | Yes | Yes | Yes | Yes |
| Date | dd.mm.yyyy | mm/dd/yyyy | dd-mm-yyyy | dd/mm/yyyy | yyyy-mm-dd | dd/mm/yyyy |
| Time | hh:mm:ss (24-h) | hh:mm:ss am/pm (12-h) | hh:mm:ss (24-h) | hh:mm:ss (24-h) | hh:mm:ss (24-h) | hh:mm:ss (24-h) |
| Decimal Separator | Comma (,) | Period (.) | Comma (,) | Comma (,) | Comma (,) | Comma (,) |
| Thousand Separator | Space ( ) | Comma (,) | Space ( ) | Space ( ) | Space ( ) | Space ( ) |
| Number Example | 15 631,87 | 15,631.87 | 15 631,87 | 15 631,87 | 15 631,87 | 15 631,87 |
| Ordinals | 1-ый, 2-ой, 3-й… | 1st, 2nd, 3rd… | 1er, 2e, 3e | 1°, 2°, 3°… | 1. 2. 3. … | 1°, 2°, 3°… |
| First day of the week | Monday | Sunday | Monday | Monday | Monday | Monday |

#### Units of Measurement

Km vs. miles, F vs. C, lb. vs. kg

### Text support

* ASCII
* Unicode

#### Text input

* Keyboard layouts
* Input Method Editors

#### GB18030 certification

As per the Wikipedia article on GB18030, 'GB 18030 is a Chinese government standard, described as Information technology — Chinese coded character set and defines the required language and character support necessary for software in China'. It is a multi-byte encoding using 1-byte, 2-byte, and 4-byte codes. The Chinese Government has mandated that all applications released on or after 2001-Sep-01 must support GB 18030.
China has confirmed that it is sufficient to be able to
* Input and output text in GB 18030 encoding
* Process the full character repertoire of GB 18030

### Internationalization testing

Testing of regional formats support and text support

## Marketization/market customization
### Payment systems

* ↑ https://www.w3.org/International/questions/qa-i18n
* ↑ https://en.wikipedia.org/wiki/GB_18030
* ↑ http://icu-project.org/docs/papers/unicode-gb18030-faq.html
# Localization/Localization Vendors
### There are different models for resourcing localization services such as translating, editing, proofreading, and QA.

* SLV: Single Language Vendor
* MLV: Multiple Language Vendor
* Freelance Translator
SLVs Companies offering translation services into one language.
MLVs Companies offering translation services into multiple languages.
Freelance Translator Independent translators.
# Linear mapping/K^3 to K^2/125/411/On hyperplane given by 234/Several bases/Exercise

We consider the linear map
    $\varphi \colon K^{3}\longrightarrow K^{2},{\begin{pmatrix}x\\y\\z\end{pmatrix}}\longmapsto {\begin{pmatrix}1&2&5\\4&1&1\end{pmatrix}}{\begin{pmatrix}x\\y\\z\end{pmatrix}}.$
Let $U\subseteq K^{3}$ be the subspace of $K^{3}$, defined by the linear equation $2x+3y+4z=0$, and let $\psi$ be the restriction of $\varphi$ on $U$. On $U$, there are given vectors of the form
    $u=(0,1,a),\,v=(1,0,b){\text{ and }}w=(1,c,0).$
Compute the "change of basis" matrix between the bases
    ${\mathfrak {b}}_{1}=v,w,\,{\mathfrak {b}}_{2}=u,w{\text{ and }}{\mathfrak {b}}_{3}=u,v$
of $U$, and the transformation matrix of $\psi$ with respect to these three bases (and the standard basis of $K^{2}$).
# Matrix/44/Depending on k/Inverse to itself/Exercise

Prove that the matrix
    ${\begin{pmatrix}0&0&k+2&k+1\\0&0&k+1&k\\-k&k+1&0&0\\k+1&-(k+2)&0&0\end{pmatrix}}$
for all $k$ is the inverse of itself.
# Field/Idempotent element (explained)/Projection/Exercise

Determine, for a field $K$, the idempotent elements, that is, elements $e\in K$ with $e^{2}=e$. Determine the linear projections $\varphi \colon K\rightarrow K$.
# Linear projection/Basis/2 -5/4 3/Exercise

We consider the basis
    ${\begin{pmatrix}2\\-5\end{pmatrix}},{\begin{pmatrix}4\\3\end{pmatrix}}$
of $\mathbb {R} ^{2}$. Let $\varphi \colon \mathbb {R} ^{2}\rightarrow \mathbb {R} ^{2}$ denote the projection from $\mathbb {R} ^{2}$ to
    $\mathbb {R} {\begin{pmatrix}2\\-5\end{pmatrix}}\subseteq \mathbb {R} ^{2}\,$
with respect to this basis. Determine the matrix of $\varphi$ with respect to the standard basis.
# Linear projection/R^3/Space for 3x-4y+6z is 0 and (2,0,5)/Exercise

Let $L\subseteq \mathbb {R} ^{3}$ be the solution space for the linear equation
    $3x+4y+6z=0\,,$
and set $W=\mathbb {R} {\begin{pmatrix}2\\0\\5\end{pmatrix}}$. Show that
    $\mathbb {R} ^{3}=L\oplus W\,$
holds, and describe the projections onto $L$ and onto $W$ with respect to the standard basis.
# Linear projection/Description with matrix/Exercise

Let $\varphi \colon V\rightarrow V$ denote a linear projection on a finite-dimensional $K$-vector space $V$. Show that $\varphi$ is described, with respect to a suitable
basis $v_{1},\ldots ,v_{n}$ of $V$, by a matrix of the form
    ${\begin{pmatrix}1&0&\cdots &\cdots &\cdots &0\\0&\ddots &\ddots &\ddots &\vdots &\vdots \\\vdots &\ddots &1&\ddots &\ddots &\vdots \\\vdots &\ddots &\ddots &0&\ddots &\vdots \\\vdots &\vdots &\ddots &\ddots &\ddots &0\\0&\cdots &\cdots &\cdots &0&0\end{pmatrix}}.$
# Linear projections/t t^2 t^3/Sketch/Exercise

Let $C\subseteq \mathbb {R} ^{3}$ denote the image of the mapping
    $\mathbb {R} \longrightarrow \mathbb {R} ^{3},t\longmapsto (t,t^{2},t^{3})=(x,y,z).$
Sketch the images of $C$ under the projections onto the different coordinate plans.
# Linear projection/Idempotent/Sum is not/Exercise

Show that the sum of two linear projections
    $\varphi ,\psi \colon V\longrightarrow V$
is in general not a projection.
# Linear projection/Linear subspace/Idempotent/Finite-dimensional with dimension formula/Exercise

Simplify the proof of fact with the help of the dimension formula.
# Homomorphism space/Vector space/Exercise

Let $K$ be a field, and let $V$ and $W$ be $K$-vector spaces. Show that the homomorphism space
    $\operatorname {Hom} _{K}{\left(V,W\right)}$
is a vector space.
# Homomorphism space/Linear subspace/Mapping space/Exercise

Let $K$ be a field, and let $V$ and $W$ be $K$-vector spaces. Show that the homomorphism space
    $\operatorname {Hom} _{K}{\left(V,W\right)}$
is a linear subspace of the mapping space $\operatorname {Map} \,{\left(V,W\right)}$.
# Homomorphism space/K to W/Example/Exercise

Let $W$ be a $K$-vector space over the field $K$. Show that the mapping
    $\operatorname {Hom} _{K}{\left(K,W\right)}\longrightarrow W,\varphi \longmapsto \varphi (1),$
is an isomorphism of vector spaces.
# Homomorphism space/Functorial properties/Fact/Proof/Exercise

Let $K$ be a field, and let $V$ and $W$ be $K$-vector spaces. Show the following statements.
* A linear mapping with another vector space $U$ induces a linear mapping
    $\varphi \colon U\longrightarrow V$
    $\operatorname {Hom} _{K}{\left(V,W\right)}\longrightarrow \operatorname {Hom} _{K}{\left(U,W\right)},f\longmapsto f\circ \varphi .$
* A linear mapping with another vector space $T$ induces a linear mapping
    $\psi \colon W\longrightarrow T$
    $\operatorname {Hom} _{K}{\left(V,W\right)}\longrightarrow \operatorname {Hom} _{K}{\left(V,T\right)},f\longmapsto \psi \circ f.$
# Endomorphism/Subspace condition/Matrices/Exercise

Let the vectors $u={\begin{pmatrix}6\\9\\-2\end{pmatrix}}$ and $v={\begin{pmatrix}4\\-4\\7\end{pmatrix}}$ in $\mathbb {R} ^{3}$ be given. We consider the linear subspace
    $U\subseteq \operatorname {Hom} _{K}{\left(\mathbb {R} ^{3},\mathbb {R} ^{3}\right)}\,$
consisting of all linear mappings $\varphi$ satisfying simultaneously the following conditions:
a) $\varphi (u)\in \langle {\begin{pmatrix}2\\8\\3\end{pmatrix}},{\begin{pmatrix}7\\0\\5\end{pmatrix}}\rangle$.
b) $\varphi (v)\in \langle {\begin{pmatrix}6\\-5\\4\end{pmatrix}}\rangle$.
* Determine the dimension of $U$.
* Describe the corresponding linear subspace of the space of matrices with respect to suitably chosen bases by linear equations.
* Describe the corresponding linear subspace of the space of matrices with respect to suitably chosen bases by a basis.
# K^3/Two lines/To themselves/Exercise

Let $G_{1}\neq G_{2}$ and $H_{1}\neq H_{2}$ denote lines in $K^{3}$. What is the dimension of the space
    $W={\left\{\varphi \in \operatorname {Hom} _{K}{\left(K^{3},K^{3}\right)}\mid \varphi (G_{1})\subseteq H_{1}{\text{ and }}\varphi (G_{2})\subseteq H_{2}\right\}}\,?$
# Hom(V,W)/Several morphisms/Rank estimate/Exercise

Let $V$ and $W$ denote finite-dimensional $K$-vector spaces, and let $\varphi _{1},\ldots ,\varphi _{k}\in \operatorname {Hom} _{K}{\left(V,W\right)}$. Show
    $\operatorname {rk} \,{\left(\varphi _{1}+\cdots +\varphi _{k}\right)}\leq \sum _{i=1}^{k}\operatorname {rk} \,\varphi _{i}\,.$
# Linear projection/Basis/3 7/4 -9/Exercise

We consider the basis
    ${\begin{pmatrix}3\\7\end{pmatrix}},{\begin{pmatrix}4\\-9\end{pmatrix}}$
in $\mathbb {R} ^{2}$. Let $\varphi \colon \mathbb {R} ^{2}\rightarrow \mathbb {R} ^{2}$ be the projection of $\mathbb {R} ^{2}$ onto $\mathbb {R} {\begin{pmatrix}3\\7\end{pmatrix}}\subseteq \mathbb {R} ^{2}$ with respect to this basis. Determine the matrix of $\varphi$ with respect to the standard basis.
# Trace/V over K/Independent of basis/Exercise

Show that the definition of the trace of a linear mapping is independent of the chosen matrix.
# Hyperplane/Kernel of a linear form/Fact/Proof/Exercise

Let $V$ be an $n$-dimensional $K$-vector space, and let $U\subseteq V$ denote an $n-1$-dimensional linear subspace. Show that there exists a linear form $f\colon V\rightarrow K$ such that $U=\operatorname {kern} f$.
# Linear independence/Test with linear forms/Fact/Proof/Exercise

Let $K$ be a field, and let $V$ be a $K$-vector space. Let $v_{1},\ldots ,v_{n}\in V$ be vectors. Suppose that for every $k$, there exists a linear form
    $\varphi _{k}\colon V\longrightarrow K$
such that
    $\varphi _{k}(v_{k})\neq 0{\text{ and }}\varphi _{k}(v_{i})=0{\text{ for }}i\neq k.$
Show that the $v_{1},\ldots ,v_{n}$ are linearly independent.
# Determinant/Compute/1 3 5/2 1 3/8 7 4/Exercise

Compute the determinant of the matrix
    ${\begin{pmatrix}1&3&5\\2&1&3\\8&7&4\end{pmatrix}}.$
# Determinant/Recursive/Upper triangular matrix/Diagonal product/Exercise

Prove by induction that the determinant of an upper triangular matrix is equal to the product of the diagonal elements.
# Determinant/Recursive/Lower triangular matrix/Diagonal product/Exercise

Prove by induction that the determinant of a lower triangular matrix is equal to the product of the diagonal elements.
# Determinant/3x3/Multilinarity/Direct/Exercise

Check the multilinearity and the property to be alternating, directly for the determinant of a $3\times 3$-matrix.
# Determinant/2x2/Multilinarity/Direct/Exercise

Check the multilinearity and the property to be alternating, directly for the determinant of a $2\times 2$-matrix.
# Homothety/Determinant/Exercise

What is the determinant of a homothety?
# Permutation/Cross-free arrow diagram/Exercise

Show that one can represent every finite permutation by an arrow diagram without crossings.
# Characteristic polynomial/Companion matrix/Exercise

Show that the characteristic polynomial of the so-called companion matrix
    $M={\begin{pmatrix}0&1&0&\ldots &0\\0&0&1&\ldots &0\\\vdots &\vdots &\ddots &\ddots &\vdots \\0&0&\ldots &0&1\\-a_{0}&-a_{1}&\ldots &-a_{n-2}&-a_{n-1}\end{pmatrix}}\,$
equals
    $\chi _{M}=X^{n}+a_{n-1}X^{n-1}+\cdots +a_{1}X+a_{0}\,.$
# Endomorphism/Characteristic polynomial/Different linear factors/Diagonalizable/Fact/Proof/Exercise

Let $K$ denote a field, and let $V$ denote a $K$-vector space of finite dimension. Let
    $\varphi \colon V\longrightarrow V$
be a linear mapping. Suppose that the characteristic polynomial $\chi _{\varphi }$ factors into different linear factors. Show that $\varphi$ is diagonalizable.
# Linear algebra (Osnabrück 2024-2025)/Part I/Exercise sheets

* List of definitions
* Important theorems
# Linear algebra (Osnabrück 2024-2025)/Part I/Lectures

* List of definitions
* Important theorems
# Linear algebra (Osnabrück 2024-2025)/Part I/Link on portal

Linear algebra (Osnabrück 2024-2025)/Part I
Linear algebra is a fundamental course in mathematics. It deals with "linear structures“: linear systems of equations, vectors and their representation, bases and base change, linear mappings and matrices, determinants, eigenvalues and eigenvectors, optimal descriptions of linear mappings, affine mappings, normed vector spaces, lengths and angles, analytic geometry, isometries and symmetries, bilinear forms, more general algebraic structures like equivalence relations, groups, residue class construction, multilinear algebra.
To the course.
# Affine space/Finite point family/Affinely independent/Characterization/Fact/Proof/Exercise

Let $E$ be an affine space over a $K$-vector space $V$, and let
    $P_{1},\ldots ,P_{n}$
denote a finite family of points in $E$. Show that the following statements are equivalent.
* The points $P_{1},\ldots ,P_{n}$ are affinely independent.
* For every $i\in \{1,\ldots ,n\}$, the family of vectors is linearly independent.
    ${\overrightarrow {P_{i}P_{1}}},\ldots ,{\overrightarrow {P_{i}P_{i-1}}},\,{\overrightarrow {P_{i}P_{i+1}}},\ldots ,{\overrightarrow {P_{i}P_{n}}}$
* There exists some $i\in \{1,\ldots ,n\}$ such that the family of vectors is linearly independent.
    ${\overrightarrow {P_{i}P_{1}}},\ldots ,{\overrightarrow {P_{i}P_{i-1}}},\,{\overrightarrow {P_{i}P_{i+1}}},\ldots ,{\overrightarrow {P_{i}P_{n}}}$
* The points $P_{1},\ldots ,P_{n}$ form an affine basis in the affine subspace generated by them.
# Affine space/Finite point set/Barycentric combinations/Barycentric combination/Exercise

Let $E$ be an affine space over a $K$-vector space $V$, and let $P_{1},\ldots ,P_{n}$ denote a finite family of points in $E$. For $j=1,\ldots ,k$, let
    $Q_{j}=\sum _{i=1}^{n}a_{ij}P_{i}\,,$
with $\sum _{i=1}^{n}a_{ij}=1$ for each $j$, be a family of barycentric combinations of the $P_{i}$. Let $b_{1},\ldots ,b_{k}\in K$ fulfilling $\sum _{j=1}^{k}b_{j}=1$. Show that one can write
    $\sum _{j=1}^{k}b_{j}Q_{j}$
as a barycentric combination of the $P_{i}$.
# Linear and affine spaces/Compare/Exercise

Compare several concepts for vector spaces and affine spaces, including their mappings.
# Mathematics for Applied Sciences (Osnabrück 2023-2024)/Part I/Links

Course page on Stud.IP (restricted access)