OpenLLM-France/wikimedia huggingface dataset:
# Plane in R^3/Linear form/2/Exercise

Let
    $U=\langle {\begin{pmatrix}9\\2\\-9\end{pmatrix}},\,{\begin{pmatrix}13\\23\\33\end{pmatrix}}\rangle \subseteq \mathbb {R} ^{3}\,.$
Find a linear form $f\colon \mathbb {R} ^{3}\rightarrow \mathbb {R}$ such that $U=\operatorname {kern} f$.
# Linear projection/Trace/Exercise

Determine the trace of a linear projection
    $\varphi \colon V\longrightarrow V$
on a finite-dimensional $K$-vector space $V$.
# Trace/V over K/Linearity/Exercise

Let $K$ be a field, and let $V$ be a finite-dimensional $K$-vector space. Show that the assignment
    $\operatorname {End} _{\left(V\right)}\longrightarrow K,\varphi \longmapsto \operatorname {trace} {\left(\varphi \right)},$
is $K$-linear.
# Trace/Product of matrices/Exercise

Let $K$ be a field, and let $A$ denote an $m\times n$-matrix and let $B$ denote an $n\times m$-matrix over $K$. Show
    $\operatorname {trace} {\left(A\circ B\right)}=\operatorname {trace} {\left(B\circ A\right)}\,.$
# Plane in R^3/Linear form/1/Exercise

Let
    $U=\langle {\begin{pmatrix}4\\6\\7\end{pmatrix}},\,{\begin{pmatrix}3\\-3\\8\end{pmatrix}}\rangle \subseteq \mathbb {R} ^{3}\,.$
Find a linear form $f\colon \mathbb {R} ^{3}\rightarrow \mathbb {R}$ such that $U=\operatorname {kern} f$ holds.
# Linear form/Linear system/4x+7y-3z+6u+5v is 0/Exercise

Solve the linear system
    $4x+7y-3z+6u+5v=0\,.$
# C/Real- and imaginary part/Linear form/Exercise

Show that the real part and the imaginary part define real linear forms on $\mathbb {C}$, where $\mathbb {C}$ is considered as a real vector space.
Is the modulus of a complex number a real linear form?
# Linear subspace/Separating linear form/Exercise

Let $K$ denote a field, let $V$ be a $K$-vector space, and $U\subseteq V$ a linear subspace. Let $v\in V$ with $v\notin U$. Show that there exists a linear form $\varphi \colon V\rightarrow K$ satisfying $\varphi (U)=0$ and $\varphi (v)=1$.
# Linear function/Not 0/No extrema/Exercise

Let $V$ be a finite-dimensional real vector space. Show that a linear mapping
    $f\colon V\longrightarrow \mathbb {R} ,$
different from $0$, does not have a local extrema. Does this also hold for infinite-dimensional vector spaces? Does this require analysis?
# Linear forms/Relation between kernels/Linear subspaces/Exercise

Let $V$ be a finite-dimensional $K$-vector space over a field $K$, and let $L,L_{1},\ldots ,L_{m}$ denote linear forms on $V$. Show that the relation
    $\bigcap _{i=1}^{m}\operatorname {kern} L_{i}\subseteq \operatorname {kern} L\,$
holds if and only if $L$ belongs to the linear subspace (in the dual space) generated by the $L_{1},\ldots ,L_{m}$.
# R^2/Dual basis to (1,3),(2,-5)/Standard dual basis/Exercise

Express the vectors $u_{1}^{*},u_{2}^{*}$ of the dual basis of the basis $u_{1}={\begin{pmatrix}1\\3\end{pmatrix}},\,u_{2}={\begin{pmatrix}2\\-5\end{pmatrix}}$ in $\mathbb {R} ^{2}$ as linear combinations with respect to the standard dual basis $e_{1}^{*},e_{2}^{*}$.
# R^2/Dual basis to (1,4),(-2,2)/Exercise

Express the vectors $e_{1}^{*},e_{2}^{*}$ of the standard dual basis as linear combinations with respect to the dual basis $u_{1}^{*},u_{2}^{*}$ to the basis $u_{1}={\begin{pmatrix}1\\4\end{pmatrix}}\,,u_{2}={\begin{pmatrix}-2\\2\end{pmatrix}}$.
# Homomorphism space/Finite bases/Basis/Exercise

Let $V$ and $W$ be vector spaces over a field $K$, with a basis $v_{1},\ldots ,v_{n}$ of $V$, and a basis $w_{1},\ldots ,w_{m}$ of $W$. Show that Template:Mathcon2/disp is a basis of the space of homomorphisms $\operatorname {Hom} _{K}{\left(V,W\right)}$.
# Psychology/Keywords

abnormal behavior
applied psychology
behavior
behaviorism
clinical psychology
cognitive psychology
consumer psychology
counseling psychology
developmental psychology
educational psychology
environment
environmental psychology
experimental psychology
forensic psychology
functionalism
gestalt psychology
heredity
industrial/organizational psychology
introspection
psychiatry
psychoanalysis
psychoanalyst
psychodynamic psychology
psychology
sociocultural perspective
sport and excercise psychology
structuralism
system of psychology
theory
unconscious
# Vector space/Dual space/Duality/Not linear/Exercise

Let $V$ be a $K$-vector space, together with its dual space ${V}^{*}$. Show that the natural mapping
    $V\times {V}^{*}\longrightarrow K,(v,f)\longmapsto f(v),$
is not linear.
# Hyperplane/Kernel of a linear form/Fact/Proof

Due to fact, there exists a one-dimensional linear subspace $W\subseteq V$ such that
    $V=U\oplus W\,.$
This decomposition determines a projection
    $p_{W}\colon V\longrightarrow W,$
which is linear and has $U$ as its kernel. Since $W$ is one-dimensional, there exists an isomorphism
    $\psi \colon W\longrightarrow K.$
Therefore, $\psi \circ p_{W}$ is a linear form, whose kernel is just $U$.
# Hyperplane/Kernel of a linear form/Fact/Proof/Exercise/Solution

Due to fact, there exists a one-dimensional linear subspace $W\subseteq V$ such that
    $V=U\oplus W\,.$
This decomposition determines a projection
    $p_{W}\colon V\longrightarrow W,$
which is linear and has $U$ as its kernel. Since $W$ is one-dimensional, there exists an isomorphism
    $\psi \colon W\longrightarrow K.$
Therefore, $\psi \circ p_{W}$ is a linear form, whose kernel is just $U$.
# Linear algebra (Osnabrück 2024-2025)/Space/Plane equation/Example intersection/R/Example/Examplereferencenumber

Example 1.2
# Depressive disorder

This psychology resource is a "stub". Learn how you can help Wikiversity to develop it.
Depressive disorders are disorders that are characterized by the presence of irrationally sad, empty, and irritable moods that cause a significant decrease in human function. They are not combined into a singular diagnosis because they differ in duration, timing, or presumed cause. According to international guidelines, a depressive disorder should not be diagnosed in anyone with a history of manic episodes because then their condition would fall under a bipolar disorder.

## Etiology

Depression stems from complex interactions between genetic, biological and environmental factors. Genetic predisposition plays a crucial role, with individuals who have a family history of depression being at higher risk. Biologically, it is associated with neurotransmitter imbalances (serotonin, dopamine) in the brain, affecting mood and emotional regulation. Hormonal changes, such as thyroid dysfunction or perinatal hormone fluctuations, can also trigger depressive symptoms.

## Disorders (DSM-V classification)

Disruptive Mood Dysregulation Disorder
Major Depressive Disorder
Persistent Depressive Disorder
Premenstrual Dysphoric Disorder
Substance/Medication-Induced Depressive Disorder
Depressive Disorder Due to Another Medical Condition
Other Specified Depressive Disorder
Depressive disorders, unspecified

## Disorders (ICD-11 classification)

Single episode depressive disorder
Recurrent depressive disorder
Dysthymic disorder
Mixed depressive and anxiety disorder
Premenstrual dysphoric disorder
Other Specified Depressive Disorder
Depressive disorders, unspecified

## Resources

https://www.psychiatry.org/psychiatrists/practice/dsm
https://abcsveikata.lt/az/depresija-didysis-depresinis-sutrikimas/
https://icd.who.int/en
https://www.thelancet.com
https://psychiatryonline.org/journal/ajp
# The Spring Creek Project

The Spring Creek Project Mission
The challenge of the Spring Creek Project is to bring together the practical wisdom of the environmental sciences, the clarity of philosophical analysis, and the creative, expressive power of the written word, to find new ways to understand and re-imagine our relation to the natural world.
Projects:
Writers’ Residency Programs
The Collaborative Retreat at the Cabin at Shotpouch Creek is a two-week-long retreat for two participants who wish to pursue a collaborative project, or two participants who each have individual projects and who anticipate a synergistic benefit from each other's presence. At least one of the applicants must be a writer who takes inspiration from the natural world. The second applicant may also be a writer, or he or she may work in any other field (e.g. science, philosophy, music, art, crafts, etc.).
Andrews Forest Writers' Residencies
Creative writers whose work in any genre reflects a keen awareness of the natural world and an appreciation for both scientific and literary ways of knowing are invited to apply for one-week residencies at the H.J.Andrews Experimental Forest. The mission of the Long-Term Ecological Reflections program is to bring together writers, humanists and scientists to create a living, growing record of how we understand the forest and the relation of people to the forest, as that understanding and that forest both change over time. The residencies are intended to provide concentrated time for personal creative work that promises to further the exploration of the relationship of humans to the rest of the natural world.
The Blue River Gathering
the Spring Creek Project website
# KnitR/Templates
## Introduction

This page about KnitR/Templates can be displayed as Wiki2Reveal slides. Single sections are regarded as slides and modifications on the slides will immediately affect the content of the slides. The following aspects of using, definition of Templates for KnitR are considered in detail:
* (1) How do templates reduce workload for generation of R-Markdown documents
* (2) Main Workflow - Load, Process, Display, Save
* (3) Sharing of templates, template engines, template generator

## Objective

This learning resource about KnitR Templates in Wikiversity has the objective to analyze template management under the FAIR Principle and as Open Educational Resources.

## Learning Tasks / Activities

Learning activies focus on
* What is a template and what is benefit and drawback of using templates?
* What are main use-cases for templates?
* How can templates support the generation of dynamic document generation? What are the requirements and constraints for learning resources?
* How can template management and in conjunction with Open Educational Resources and Machine Learning be used for Intelligent Tutoring Systems?

## Workflow Templates

* Loads a string from a file with filename mytemplate.txt.
* The template may contain identfiers that are replaced with a specific content.
* Now we have specific CSV file with 3 columns "firstname", "lastname" and "age".
* for each row of the CSV file the raw template is used and replaced by the values in the CSV file.
* A possible output with two data sets in the CSV file could be

## Technical Implementation

Realize the workflow about in R.
# Wikiphilosophers/Foundationalism/S. Perquin

The only thing I know is that my human mind exists because I know that I think. This essentially corresponds to René Descartes' cogito, ergo sum ("I think, therefore I am"). From that perspective, I also know for sure that I experience a world through my senses. I see what I see, I hear what I hear, I feel what I feel, I smell what I smell and I taste what I taste. I see this as the only world that I, as a human, can experience and understand. However, I do not know if other people consciously experience the world, because I cannot experience it from someone else's body. It could just as well be that all people are NPCs, like in a video game, and that they only say they can think consciously, but this is actually not the case. Maybe I have imagined everything and everyone in this world in my own mind?
This solipsistic thought might sound somewhat egocentric, but it is the only conclusion that can be drawn from the idea that the existence of my mind is the only thing I can be certain of. Others will say this is not true, because they will claim that they are also aware of their mind and can think, but since I have no proof that they also experience the world through their mind, as I cannot experience it myself, I can only continue to doubt.
I must mention that the longer you think about it, the lonelier this thought becomes. Since I do not know whether people have a mind or not, I assume that they do, because otherwise, it becomes difficult to live.
# Open questions

This is a place on this wiki for potentially unanswered questions that may have some educational value. These can be spun off into unique educational resources if appropriate.

## After life

Can science prove or disprove an after life?

## Spirit

Where does spirituality and science intersect? How?

## Miscellaneous

* What are some important unanswered questions within the field of cosmology?

## Questions

* Why do plants store food?
* Should suicide be legal?
# Multilinear alternating/V^3/u+v+w/2u+3z/4w-5z/Compute/Exercise

Let $K$ be a field, and let $V$ denote a $K$-vector space. Let
    $\triangle \colon V\times V\times V\longrightarrow K$
be a multilinear and alternating mapping. Let $u,v,w\in V$. Simplify the term
    $\triangle {\begin{pmatrix}u+v+w\\2u+3z\\4w-5z\end{pmatrix}}.$
# Determinant/Q,R,C/Exercise

Let $M\in \operatorname {Mat} _{n}(\mathbb {Q} )$. Show that it does not make a difference, whether we compute the determinant in $\mathbb {Q}$, in $\mathbb {R}$, or in $\mathbb {C}$.
# Linear forms/Product/Multilinear/Exercise

Let $K$ be a field, and let $V_{1},\ldots ,V_{n}$ vector spaces over $K$. Let
    $\varphi _{i}\colon V_{i}\longrightarrow K$
($i=1,\ldots ,n$), denote linear mappings. Show that the mapping
    $\varphi \colon V_{1}\times \cdots \times V_{n}\longrightarrow K,{\left(v_{1},\ldots ,v_{n}\right)}\longmapsto \varphi _{1}(v_{1})\cdots \varphi _{n}(v_{n}),$
is multilinear.
# Vector space/Dual space/Bilinear/Exercise

Show that, for a $K$-vector space $V$ with dual space ${V}^{*}$, the evaluation mapping
    $V\times {V}^{*}\longrightarrow K,(v,f)\longmapsto f(v),$
is bilinear.
# Linear mapping/Multilinear and alternating/Exercise

Linear mapping/Situation Show that $\varphi$ is multilinear and alternating.
# Field/Multiplication/Multilinear/Exercise

Field/Situation Show that the multiplication
    $K\times K=K^{2}\longrightarrow K,(a,b)\longmapsto a\cdot b,$
is multilinear. Is it also alternating?
# K^n/Inner product/Bilinear/Exercise

Let $K$ be a field, and let $n\in \mathbb {N}$. Show that the mapping
    $K^{n}\times K^{n}\longrightarrow K,({\begin{pmatrix}u_{1}\\\vdots \\u_{n}\end{pmatrix}},{\begin{pmatrix}v_{1}\\\vdots \\v_{n}\end{pmatrix}})\longmapsto (u_{1},\ldots ,u_{n})\circ {\begin{pmatrix}v_{1}\\\vdots \\v_{n}\end{pmatrix}},$
is multilinear.
# Tensor product/Mapping spaces/Product set/Exercise

Let $K$ be a field, and let $I$ and $J$ denote finite index sets. Show that the mapping
    $\operatorname {Map} \,{\left(I,K\right)}\times \operatorname {Map} \,{\left(J,K\right)}\longrightarrow \operatorname {Map} \,{\left(I\times J,K\right)},(f,g)\longmapsto f[[:Template:Tensor]]g,$
given by
    $(f[[:Template:Tensor]]g)(i,j):=f(i)\cdot g(j)\,,$
is multilinear.
# Determinant/Transposed of a matrix/Elementary matrices/Exercise

Show that, for every elementary matrix $E$, the relation
    $\det E=\det {E^{\text{tr}}}\,$
holds.
# 2x2-matrix/Trace/Determinant sum/Exercise

Let $M$ be a $2\times 2$-matrix. Show that
    $\operatorname {trace} {\left(M\right)}=1+\det M-\det {\left(E_{2}-M\right)}\,$
holds.
# Multilinear mapping/Generating system/Exercise

Let $K$ be a field, and let $V_{1},\ldots ,V_{n}$ and $W$ denote vector spaces over $K$. Let $v_{i_{j}}$, $i_{j}\in I_{j}$, be generating systems of $V_{j}$, $j=1,\ldots ,n$. Show that a multilinear mapping
    $\triangle \colon V_{1}\times \cdots \times V_{n}\longrightarrow W$
is determined by
    $\triangle (v_{i_{1}},\ldots ,v_{i_{n}}).$
# Multilinear alternating/V^2/u+2v/v+3w/Compute/Exercise

Let $K$ be a field, and let $V$ denote a $K$-vector space. Let
    $\triangle \colon V\times V\longrightarrow K$
be a multilinear and alternating mapping. Let $u,v,w\in V$. Simplify
    $\triangle {\begin{pmatrix}u+2v\\v+3w\end{pmatrix}}.$
# Matrices/2x2/Column determinant/Multilinear?/Exercise

Let $K$ be a field. Is the mapping
    $\operatorname {Mat} _{2}(K)\longrightarrow K,{\begin{pmatrix}a&b\\c&d\end{pmatrix}}\longmapsto ac-bd,$
multilinear in den rows? In the columns?
# Determinant/Diagonal block matrix/No cross formula/Exercise

Let $M$ be a square matrix of the form
    $M={\begin{pmatrix}A&B\\C&D\end{pmatrix}}\,$
with square matrices $A,B,C$ and $D$. Show by an example that the equality
    $\det M=\det A\cdot \det D-\det B\cdot \det C\,$
does not hold in general.
# Vector spaces/Evaluation mapping/Multilinearity/Exercise

Vector spaces/Situation Determine whether the mapping
    $\operatorname {Hom} _{K}{\left(V,W\right)}\times V\longrightarrow W,(\varphi ,v)\longmapsto \varphi (v),$
is multilinear.
# Multilinear mapping/Vector space/Exercise

Let $K$ be a field, and let $V_{1},\ldots ,V_{n}$ and $W$ denote vector spaces over $K$. Show that the set of all multilinear mappings is, in a natural way, a vector space, denoted by $[[:Template:Op:Mult]]$.
# Multilinear mapping/Alternating/Linear subspace/Exercise

Let $K$ be a field, let $V$ and $W$ be vector spaces over $K$, and $n\in \mathbb {N}$. Show that the set of all alternating mappings
(denoted by $[[:Template:Op:Alt]]$) is a linear subspace of $\operatorname {Mult} _{K}{\left(V,\ldots ,V,W\right)}$ (where the vector space $V$ appears $n$-fold).
# Linear mapping/Composed with multilinear/Multilinear/Exercise

Let $K$ be a field, and let $V_{1},\ldots ,V_{n},W_{1},\ldots ,W_{n}$ and $Z$ be vector spaces over $K$. Let
    $\varphi _{i}\colon V_{i}\longrightarrow W_{i}$
denote linear mappings, and let
    $\pi \colon W_{1}\times \cdots \times W_{n}\longrightarrow Z$
be a multilinear mapping. Show that the mapping
    $\pi \circ (\varphi _{1}\times \cdots \times \varphi _{n})\colon V_{1}\times \cdots \times V_{n}\longrightarrow Z,(v_{1},\ldots ,v_{n})\longmapsto \pi (\varphi _{1}(v_{1}),\ldots ,\varphi _{n}(v_{n})),$
is also multilinear.
# Matrices/2x2/Antideterminant/Multilinear not alternating/Exercise

Let $K$ be a field. Show that the mapping
    $\operatorname {Mat} _{2}(K)\longrightarrow K,{\begin{pmatrix}a&b\\c&d\end{pmatrix}}\longmapsto ad+cb,$
is multilinear, but not alternating.
# Defamation law in Czechia

This article explores the codification of defamation in the Czech legal system. It is not written by lawyers, and does not replace a professional legal advice.
Relevant Wikidata terminology: natural person, defamation, defamation, libel, slander.

## English defamation terminology

First, let us clarify related terminology in English:
* defamation
** subclass: slander: spoken
** subclass: libel: written
** superclass: crime
** superclass: tort
The above is likely to be an oversimplification.

## Czech defamation law

About Czech term pomluva, which is informally well translated as defamation:
* In Wikidata, the term is here: defamation (Q20057997).
* A semi-implicit definition of pomluva is here: . This is 40/2009 Sb. Zákon trestní zákoník.
* A pomluva (defamation) is a trestný čin (crime) since it is specified in trestní zákoník (penal code).
* The penalty is 1 year or 2 years in prison (odnětí svobody), depending on the medium of publication of the defamatory statements.
* The definition requires the statement made to be untrue.
* The definition stays silent about the intent or belief of the author of the potentially defamatory statement.
* The definition stays silent about demonstration or proof of truth or untruth of the statement; it merely requires the statement to be untrue.
* The definition does not make it clear who has the burden of proof as for truth.
* There is a contrasting term křivé obvinění (false accusation) that has a separate codification and penalty.
* Another contrasting term is urážka na cti (insult of honor) that has a separate codification and penalty.
* According to one source--but not explicitly the law--the object of defamation has to be a natural person in order for the title to apply.
* According to one source--but not explicitly the law--the person doing the defamation must have a certain intent.

## The case of Milan Knížák vs. David Černý

Milan Knížák was vulgarly insulted by David Černý in a program of Czech Television. The judgment was that David Černý was to pay 100 000 CZK of compensation. It is not perfectly clear which title was applied; perhaps urážka na cti. However, the maximum fine for urážka na cti seems to be 30 000 CZK, surpassed by the awarded 100 000 CZK.

## The case of Tomio Okamura and the label Pitomio

Japanese-Czech politician Tomio Okamura sued on account of having been labeled Pitomio, apparently from the Czech word pitomec. Which law was invoked is not perfectly clear. It seems Charter of Fundamental Rights and Freedoms was invoked, as part of constitution, but this would need a more careful (or qualified) look. The case seems to have gone through multiple courts; it could be worthwhile to find out about the decisions.
Further reading:
* Wikipedia:cs: Tomio Okamura, wikipedia.org
# Determinant/Invertible matrix/Inverse/Exercise

Let $M$ be an invertible $n\times n$-matrix. Show
    $\det {\left(M^{-1}\right)}={\left(\det M\right)}^{-1}\,.$
# Matrix multiplication/Linear dependent columns/Exercise

Let $A$ be an $m\times n$-matrix, and $B$ be an $n\times k$-matrix, where the columns of $B$ are linearly dependent. Show that the columns of $A\circ B$ are also linearly dependent.
# Determinant/Multiplication/3x3/Example 2/Exercise

Confirm the multiplication theorem for determinants for the matrices
    $A={\begin{pmatrix}1&4&0\\2&0&5\\0&2&-1\end{pmatrix}}\,\,{\text{ and }}\,\,B={\begin{pmatrix}1&2&7\\0&3&6\\0&-2&-3\end{pmatrix}}.$
# Determinant/Surjective group homomorphism to K^x/Exercise

Let $K$ be a field, and $n\in \mathbb {N} _{+}$. Show that the determinant
    $\operatorname {GL} _{n}\!{\left(K\right)}\longrightarrow (K\setminus \{0\},\cdot ,1),M\longmapsto \det M,$
is a surjective group homomorphism.
# General linear group/Injective group homomorphism for n \leq m/Exercise

Let $K$ be a field, and let $n,m\in \mathbb {N} _{+}$ with $n\leq m$. Define an injective group homomorphism
    $\operatorname {GL} _{n}\!{\left(K\right)}\longrightarrow \operatorname {GL} _{m}\!{\left(K\right)}.$
# Square matrix/Z/Bijection with determinant/Exercise

Let $M$ be an $n\times n$-matrix with entries in $\mathbb {Z}$, and let
    $\varphi _{M}\colon \mathbb {Z} ^{n}\longrightarrow \mathbb {Z} ^{n}$
denote the corresponding group homomorphism. Show that $\varphi _{M}$ is bijective if and only if the determinant of $M$ equals $1$ or equals $-1$.
# Endomorphisms/Determinant multiplication theorem/Exercise

Let $V$ be a finite-dimensional $K$-vector space, and let $\varphi ,\psi \colon V\rightarrow V$ denote linear mappings. Show $\det {\left(\varphi \circ \psi \right)}=\det \varphi \det \psi$.
# Cramer's rule/2x2/2/Exercise

Solve the linear system
    ${\begin{pmatrix}6&4\\5&2\end{pmatrix}}{\begin{pmatrix}x_{1}\\x_{2}\end{pmatrix}}={\begin{pmatrix}8\\3\end{pmatrix}}\,$
with Cramer's rule.
# Complex endomorphism/Complex and real determinant/Exercise

Let $V$ be a finite-dimensional vector space over the complex numbers $\mathbb {C}$, and let
    $\varphi \colon V\longrightarrow V$
be a $\mathbb {C}$-linear mapping. We consider $V$ also as a real vector space of double dimension. $\varphi$ is also a real-linear mapping, which we denote by $\psi$. Show that between the complex determinant and the real determinant, the relation
    $\vert {\det \varphi }\vert ^{2}=\det \psi \,$
holds.
# Cramer's rule/Linear system/2 4 3/1 5 7/3 5 2/Is 3 3 4/Exercise

Solve the linear system (over $\mathbb {Q}$)
    ${\begin{matrix}2y&+4z&+3&=&3\\y&+5z&+7&=&3\\3y&+5z&+2&=&4\,.\end{matrix}}$
using Cramer's rule.
# Group homomorphism/Matrix 3 4 1 2/Bijective for Q, not for Z/Exercise

We consider the matrix
    ${\begin{pmatrix}3&4\\1&2\end{pmatrix}}.$
Show that this matrix defines a group homomorphism from $\mathbb {Q} ^{2}$ to $\mathbb {Q} ^{2}$, and from $\mathbb {Z} ^{2}$ to $\mathbb {Z} ^{2}$ as well. Study this group homomorphism with respect to injectivity and surjectivity.
# Determinant/Leibniz-formula/631/682/754/Exercise

Determine with the Leibniz-formula the determinant of the matrix
    ${\begin{pmatrix}6&3&1\\6&8&2\\7&5&4\end{pmatrix}}.$