OpenLLM-France/wikimedia huggingface dataset:
# Permutation matrix/3-cycle/R/Trigonalizable/Exercise

Determine whether the permutation matrix
    ${\begin{pmatrix}0&0&1\\1&0&0\\0&1&0\end{pmatrix}}$
is trigonalizable over $\mathbb {R}$.
# Matrix/Minimal polynomial of the restrictions/1/Exercise

Determine the minimal polynomials of the left-upper submatrices of
    ${\begin{pmatrix}1&2&0&0\\0&1&0&0\\0&0&1&3\\0&0&0&1\end{pmatrix}}.$
# Matrix/2/Trigonalizable/Eigenvalue/Exercise

Let $M$ be a $2\times 2$-matrix over a field $K$. Show that $M$ is trigonalizable if and only if $M$ has an eigenvector.
# Vector space/Chain/Flag/1/Exercise

Determine whether the following chain of linear subspaces in $\mathbb {R} ^{3}$ is a flag.
    $V_{0}=0,\,V_{1}=\mathbb {R} {\begin{pmatrix}2\\7\\-5\end{pmatrix}},\,V_{2}=\langle {\begin{pmatrix}4\\-3\\9\end{pmatrix}},{\begin{pmatrix}-6\\-4\\4\end{pmatrix}}\rangle ,\,V_{3}=\mathbb {R} ^{3}.$
# Vector space/Chain/Flag/2/Exercise

Determine whether the following chain of linear subspaces in $\mathbb {R} ^{3}$ is a flag.
    $V_{0}=0,\,V_{1}=\mathbb {R} {\begin{pmatrix}8\\-3\\1\end{pmatrix}},$
    $V_{2}=\operatorname {kern} \varphi ,{\text{ where }}\varphi :\mathbb {R} ^{3}\rightarrow \mathbb {R} {\text{ is given by the matrix}}\left(1,\,2,\,-2\right),\,V_{3}=\mathbb {R} ^{3}.$
# Vector space/Finite-dimensional/Flag/Existence/Exercise

Let $V$ be a finite-dimensional $K$-vector space. Show that there exist flags in $V$.
# Vector space/Three-dimensional/F3/Flag/Cardinality/Exercise

Let $K$ be the field with three elements, and let $V$ denote a three-dimensional $K$-vector space. Determine the number of flags in $V$.
# Vector space/Flag/Dual space/Exercise

Let $V$ be an $n$-dimensional $K$-vector space over a field $K$, and let
    $0=V_{0}\subset V_{1}\subset \ldots \subset V_{n-1}\subset V_{n}=V\,$
denote a flag in $V$. Show that the linear subspaces
    $W_{i}:=V_{n-i}^{\perp }\,$
define a flag in the dual space ${V}^{*}$.
# Vector space/C/Flag/R/Exercise

Let
    $0=V_{0}\subset V_{1}\subset \ldots \subset V_{n-1}\subset V_{n}=V\,$
be a flag in a $\mathbb {C}$-vector space $V$. We consider $V$ as a real vector space of real dimension $2n$. Show that there exist real linear subspaces
    $W_{i}\subseteq V\,$
such that
    $0\subset W_{1}\subset V_{1}\subset W_{2}\subset V_{2}\subset \ldots \subset V_{n-1}\subset W_{n}\subset V_{n}\,$
is a real flag.
# Vector space/Flag/Invariant under endomorphism/Exercise

Let $V$ be an $n$-dimensional $K$-vector space over a field $K$. Let
    $0=V_{0}\subset V_{1}\subset \ldots \subset V_{n-1}\subset V_{n}=V\,$
be a flag in $V$. Show that there exists a bijective linear mapping
    $\varphi \colon V\longrightarrow V$
such that this flag is the only $\varphi$-invariant flag.
# 2x2-Matrix/At least one 0/Not two 0/Exercise

Let
    $M={\begin{pmatrix}a&b\\c&d\end{pmatrix}}\,$
be a matrix over a field $K$.
a) Show that there exists a matrix, which is similar to $M$, and where at least one entry equals $0$.
b) Show that, in general, there does not exist a matrix, which is similar to $M$, and where at least two entries equal $0$.
# Vector space/Two-dimensional/Finite field/Flag/Cardinality/Exercise

Let $K$ be a finite field with $q$ elements, and let $V$ denote a two-dimensional $K$-vector space. Determine the number of flags in $V$.
# MDLD/standard bases

Start=standard bases See=standard basis MDLD/standard basis Target=Standard vectors/Basis from linearly independent/Example
# MDLD/describing matrices

Start=describing matrices See=describing matrix MDLD/describing matrix Target=Linear mapping/Matrix to basis and reverse/Definition
# Homothety/In every basis equal/Exercise

Let
    $\varphi \colon V\longrightarrow V$
be an endomorphism on a finite-dimensional $K$-vector space $V$. Show that $\varphi$ is a homothety if and only if the describing matrix is independent of the chosen basis.
# MDLD/commutative groups

Start=commutative groups See=commutative group MDLD/commutative group Target=Group/Linear algebra/Introduction/Section
# MDLD/real vector spaces

Start=real vector spaces See=real vector space MDLD/real vector space Target=[[MDLD/real vector space]]
# MDLD/real vector space

Start=real vector space Target=Vector space/Direct/Definition
# Matrix/Left-and right inverse/Same/Exercise

Let $M$ be an $n\times n$-matrix such that there exist $n\times n$-matrices $A,B$ satisfying $M\circ A=E_{n}$ and $B\circ M=E_{n}$. Show that $A=B$ holds, and that $M$ is invertible.
# MDLD/block matrix (2)

Start=block matrix (2) Target=Block matrix/2x2/Definition
# MDLD/homomorphism space

Start=homomorphism space Target=Linear Algebra/K/Homomorphism space/Definition
# MDLD/space of endomorphisms

Start=space of endomorphisms Target=Linear Algebra/K/Homomorphism space/Definition
# Polynomial/Bezout/X^2-1 and X^3-1/Exercise

Determine the greatest common divisor of $X^{2}-1$ and $X^{3}-1$, and a representation of it.
# Polynomial/Bezout/X-1 and X-2/Exercise

Determine the greatest common divisor of $X-1$ and $X-2$, and a representation of it.
# Upper triangular matrix/Generalized eigenspaces/1/Exercise

Determine the generalized eigenspaces of the matrix
    ${\begin{pmatrix}4&0&0&5&0&0\\0&1&3&0&6&0\\0&0&1&0&0&0\\0&0&0&5&2&7\\0&0&0&0&5&1\\0&0&0&0&0&5\end{pmatrix}}.$
# Vector space/Endomorphism/Polynomial/Kernel/Invariant linear subspace/Exercise

Let $K$ be a field, let $V$ be a $K$-vector space, and let
    $\varphi \colon V\longrightarrow V$
a linear mapping. Let $P\in K[X]$ be a polynomial. Show that
    $\operatorname {kern} {\left(P(\varphi )\right)}$
is an $\varphi$-invariant linear subspace.
# Gcd/Three numbers/Determine/1/Exercise

Determine the greatest common divisor of $4199,2431$ and $3553$, and a representation of it as a linear combination of the given numbers.
# Polynomial/Bezout/X^4-X^3+7X^2-5X+11 and X^3-6X^2+4X+6/Exercise

Determine the greatest common divisor of $X^{4}-X^{3}+7X^{2}-5X+11$ and $X^{3}-6X^{2}+4X+6$, and a representation of it.
# Polynomial ring/X^(n-1)+...+1/X-1/Representation of 1/Exercise

Let $n\geq 2$.
* Division with remainder (polynomial ring)/Exercise form
* Find a representation of $1$ as a linear combination of these polynomials.
# Polynomial ring/Field/One variable/Field extension/Divisor relation/Exercise

Let $K$ be a field, and let $K[X]$ be the polynomial ring over $K$. Let $F,G\in K[X]$ be polynomials. Let $K\subseteq L$ be a field extension. Show that $F$ is a divisor of $G$ in $K[X]$ if and only if $F$ is a divisor of $G$ in $L[X]$.
# Generated subgroup/77,91,143/Representation of 1/Bucket/Exercise

The delivery company "bucket without bugs“ is specialized in the transport of water. It has three buckets, one with $77$ liter, one with $91$ liter, and one with $143$ liter. The buckets do not have any marking. The company gets the request to transport, altogether, exactly one liter of water from the North Sea to the Baltic Sea. How can the company achieve this request?
# Euclidean algorithm (Z)/Gcd/1071 and 1029/Exercise

Euclidean algorithm (Z)/Gcd/Exercise form
# Euclidean algorithm (Z)/Gcd/Exercise form

Determine in $\mathbb {Z}$, using the Euclidean algorithm, the greatest common divisor of $a$ and $b$.
# Division with remainder (polynomial ring)/Exercise form

Perform in $[X]$ the division with remainder $P$ divided by $Q$ for the polynomials $P=P$ and $T=T$.
# Euclidean algorithm (polynomial ring)/Gcd/Exercise form

Determine in $[X]$, using the Euclidean algorithm, the greatest common divisor of the polynomials $P$ and $Q$.
# Euclidean algorithm (polynomial ring)/Gcd/C/X^3+(2-i)X^2+4 and (3-i)X^2+5X-3/Exercise

Determine in $\mathbb {C} [X]$, using the Euclidean algorithm, the greatest common divisor of the polynomials $X^{3}+(2-{\mathrm {i} })X^{2}+4$ and $(3-{\mathrm {i} })X^{2}+5X-3$.
# Euclidean algorithm (polynomial ring)/Gcd/Q/X^3+2X^2+5X+2 and X^2+4X-3/Exercise

Determine in $\mathbb {Q} [X]$, using the Euclidean algorithm, the greatest common divisor of the polynomials $P=X^{3}+2X^{2}+5X+2$ and $Q=X^{2}+4X-3$.
# Euclidean algorithm (polynomial ring)/Gcd/Q/X^9-1 and X^3-1/Exercise

Determine in $\mathbb {Q} [X]$, using the Euclidean algorithm, the greatest common divisor of the polynomials $P=X^{9}-1$ and $Q=X^{3}-1$.
# Euclidean algorithm (polynomial ring)/Gcd/F 5/X^4+3X^3+X^2+4X+2 and 2X^3+4X^2+X+3/Exercise

Determine in ${\mathbb {F} }_{5}[X]$, using the Euclidean algorithm, the greatest common divisor of the polynomials $P=X^{4}+3X^{3}+X^{2}+4X+2$ and $Q=2X^{3}+4X^{2}+X+3$.
# Euclidean algorithm (polynomial ring)/Gcd/R/X^3+pi X^2+7 and X^2+sqrt7X-sqrt2/Exercise

Determine in $\mathbb {R} [X]$, using the Euclidean algorithm, the greatest common divisor of the polynomials $P=X^{3}+\pi X^{2}+7$ and $Q=X^{2}+{\sqrt {7}}X-{\sqrt {2}}$.
# Euclidean algorithm (Z)/Gcd/2956 and 2444/Exercise

Determine in $\mathbb {Z}$, using the Euclidean algorithm, the greatest common divisor of $2956$ and $2444$.
# Euclidean algorithm (Z)/Gcd/3146 and 1515/Representation/Exercise

Determine in $\mathbb {Z}$, using the Euclidean algorithm, the greatest common divisor of $3146$ and $1515$. Find also a representation of the greatest common divisor as a linear combination with these numbers.
# Fibonacci-numbers/Euclidean algorithm/Exercise

Apply to two consecutive Fibonacci numbers the Euclidean algorithm. What can you observe?
# Matrix/Nilpotent/Jordan/Successive kernels/2/Exercise

Determine the kernels of the powers $M^{i}$ of the matrix
    $M={\begin{pmatrix}0&1&0&0&0\\0&0&1&0&0\\0&0&0&0&0\\0&0&0&0&1\\0&0&0&0&0\end{pmatrix}}\,.$
# Matrix/317035003/Successive kernels to 3Id-M/Exercise

Let
    $M={\begin{pmatrix}3&1&7\\0&3&5\\0&0&3\end{pmatrix}}\,.$
Determine the kernels of the powers
    ${\left(3E_{3}-M\right)}^{i}.$
# Matrix/1100010000230002/Generalized eigenspaces/Exercise

Determine the generalized eigenspaces of the matrix
    ${\begin{pmatrix}1&1&0&0\\0&1&0&0\\0&0&2&3\\0&0&0&2\end{pmatrix}}.$
# Permutation matrix/Cycle/Characteristic polynomial/Factorizaton/Sum decomposition/Exercise

Let $\pi$ be a cycle of length $n$, and let $M$ denote the corresponding permutation matrix, that is,
    $M={\begin{pmatrix}0&0&0&\ldots &1\\1&0&0&\ldots &0\\0&1&0&\ldots &0\\\vdots &\ddots &\ddots &\ddots &\vdots \\0&0&\ldots &1&0\end{pmatrix}}\,.$
* Determine the characteristic polynomial $\chi _{M}$ of $M$.
* Show that $P=X-1$ divides $\chi _{M}$, and compute the factorization
    $\chi _{M}=PQ\,.$
* Determine $P(M)$ and $Q(M)$.
* Determine $\operatorname {kern} P(M)$ and $\operatorname {kern} Q(M)$.
# Diagonalizable/Generalized eigenspace/Exercise

Show that for a diagonalizable mapping
    $\varphi \colon V\longrightarrow V$
and every $\lambda \in K$, the equality
    $\operatorname {Eig} _{\lambda }{\left(\varphi \right)}=\operatorname {GeEig} _{\lambda }(\varphi )\,$
holds.
# Trigonalizable/Diagonalizable/Eigen- and generalized eigenspace/Exercise

Let
    $\varphi \colon V\longrightarrow V$
be a trigonalizable mapping. Show that $\varphi$ is diagonalizable if and only if for every $\lambda \in K$, the equality
    $\operatorname {Eig} _{\lambda }{\left(\varphi \right)}=\operatorname {GeEig} _{\lambda }(\varphi )\,$
holds.
# Trigonalizable/Flag via generalized eigenspaces/Exercise

Let
    $\varphi \colon V\longrightarrow V$
be a trigonalizable endomorphism, and let
    $V=H_{1}\oplus \cdots \oplus H_{m}\,$
denote the direct sum decomposition into generalized eigenspaces in the sense of fact. Show that there exists a $\varphi$-invariant flag $V_{i}$ such that, in this flag, the linear subspaces
    $H_{1},H_{1}\oplus H_{2},\ldots ,H_{1}\oplus \cdots \oplus H_{j}$
for $j=1,\ldots ,m$ occur.
# Divisibility theory (Z)/Lemma of Bezout/Fact/Proof/Exercise

Prove the Lemma of Bezout for integers $a_{1},\ldots ,a_{n}$.
# Counting/Digital clock/Carnival steps/Exercise

We consider a digital clock, which exhibits $24$ hours, $60$ minutes and $60$ seconds. However, during carnival, it does not run in steps of seconds, but it adds, starting from the zero position, in every step $11$ hours, $11$ minutes and $11$ seconds. Does, with this way of counting, every possible digital display appear? After how many steps does the zero position reoccur for the first time?
# Nilpotent mappings/Commute/Sum/Exercise

Let $V$ be a $K$-vector space, and let
    $\varphi ,\psi \colon V\longrightarrow V$
denote nilpotent linear mappings, satisfying the relation
    $\varphi \circ \psi =\psi \circ \varphi \,.$
Show that also $\psi +\varphi$ is nilpotent.
# Nilpotent/Unipotent/Exercise

Let $V$ be a finite-dimensional $K$-vector space, and let
    $\varphi \colon V\longrightarrow V$
be nilpotent. Show that
    $\psi :=\operatorname {Id} +\varphi \,$
is bijective.
# Basis/Countably infinite/Cyclic shift/Nilpotent/Exercise

Let $V$ be a $K$-vector space with a basis $v_{n}$, $n\in \mathbb {N} _{+}$. Let
    $\varphi \colon V\longrightarrow V$
be the linear mapping given by
    $\varphi (v_{1})=0\,$
and
    $\varphi (v_{n})=v_{n-1}\,$
for all $n\geq 2$. Is $\varphi$ nilpotent?
# Nilpotent endomorphism/Basis images/2/Exercise

Let $v_{1},\ldots ,v_{7}$ be a basis of the $\mathbb {R}$-vector space $V$, and let $\varphi \colon V\rightarrow V$ denote the linear mapping given by
    $v_{3},v_{2}\mapsto v_{5},\,v_{5}\mapsto 4v_{6},\,v_{1},v_{4}\mapsto 0,\,v_{6}\mapsto 5v_{4},\,v_{7}\mapsto 3v_{2}.$
* Show that $\varphi$ is nilpotent.
* Determine the minimal $s$ satisfying $\varphi ^{s}=0$.
* Determine den kernel of $\varphi$.
* Find a basis of $V$, such that the matrix describing $\varphi$ with respect to this basis is in Jordan normal form.
# Endomorphism/2x2/Nilpotent/Determinant and trace/Exercise

Let $M$ be a $2\times 2$-matrix over a field $K$. Show that $M$ is nilpotent if and only if the determinant and the trace of $M$ are $0$.
# R/Multiplication with small a/Nilpotent/Exercise

Let $a\in \mathbb {R}$ be a real number with $\vert {a}\vert <1$. As is known from analysis, we have
    $\lim _{n\rightarrow \infty }a^{n}=0\,.$
Is the linear mapping
    $\mathbb {R} \longrightarrow \mathbb {R} ,x\longmapsto ax$
nilpotent?
# Matrix/1ii-1/Nilpotent/Exercise

Show that the complex matrix
    ${\begin{pmatrix}1&{\mathrm {i} }\\{\mathrm {i} }&-1\end{pmatrix}}$
is nilpotent.
# Matrix product/Upper triangular matrix/5/Zero diagonal/Nilpotent/Exercise

We consider the matrix
    $M={\begin{pmatrix}0&a&b&c&d\\0&0&e&f&g\\0&0&0&h&i\\0&0&0&0&j\\0&0&0&0&0\end{pmatrix}}\,$
over a field $K$. Show that the fifth power of $M$ equals $0$, that is,
    $M^{5}=MMMMM=0\,.$
# Matrix/Zero condition with diagonal/Product/Exercise

Let $A={\left(a_{ij}\right)}$ and $B={\left(b_{ij}\right)}$ be square matrices of length $n$. Suppose that $a_{ij}=0$ holds for $j\leq i+d$, and $b_{ij}=0$ holds for $j\leq i+e$ for some $d,e\in \mathbb {Z}$. Show that the entries $c_{ij}$ of the product $AB$ fulfill the condition $c_{ij}=0$ for $j\leq i+d+e+1$.
# Derivation operator/Polynomials of maximal degree/Nilpotent/Exercise

Let
    $D\colon \mathbb {R} [X]_{\geq m}\longrightarrow \mathbb {R} [X]_{\geq m}$
be the restriction of the derivation operator $P\mapsto P'$ to the space of polynomials of degree $\leq m$. Show that $D$ is nilpotent. Show also that
    $D\colon \mathbb {R} [X]\longrightarrow \mathbb {R} [X]$
is not nilpotent.
# Cyberculture/Social Media/Instagram
### History

* October 6, 2010: Instagram is launched.
* December 12, 2010: Instagram has 1 million users.
* March 21, 2011: First worldwide instameet
* August 3, 2011: 150 millionth photo uploaded.
* September 20, 2011: Introducing version 2.0 with four new filters.
* September 26, 2011: 10 million users!
* November 21, 2011: The launch of weekend hashtag project #WHP.
* April 3, 2012: Instagram launches on android too.
* April 9, 2012: Facebook buys Instagram.
* June 28, 2012: Instagram designs photo pages for the web.
* July 26, 2012: Instagram has 80 million users!
* December 11, 2012: Version 3.2 with a new filter.
* December 21, 2012: Version 3.4.1 brings 25 languages in.
* May 2, 2013: Tag photos on Instagram.
* June 20, 2013: Introducing videos on Instagram.

### Scope of Research

Social media has diffused the ideas of self and public today, to an unprecedented degree (Meadowcroft 75). Unity of time, space and action has been perpetually distorted (Meadowcraft 78). While there lies an unrelenting caveat in Micah Meadowcroft’s words, our literature review has predominantly surveyed the less wary notions of Instagram, as a blooming seat of activism, discourse and broader public engagement. Amidst the Ukrainian revolution of 2014, Sadof writes how Instagram was not only an essential tool for the diverse advocates of democracy but also key in democratizing virtual space, rendering users at the helm of first person journalism (249). Research in Australia is yet more compelling. A recent trend of Instagrammers capturing myriad urban banalities perhaps reeks of more than just quirky hobbies. Hicks propounds this newfound obsession of many Australians as a unique private activism, a mediated uncanny to revoke a keen sense of nostalgia. Not so long ago, Vice covered a cogent story of activism on Instagram. It argued that the virtue and power of imagery “against a gigantic block of text” (read Twitter) cannot go unnoticed. Instagram is the free bastion of the individual, not a brand, empowering movements and ideas across boundaries (Rose).
* ↑ Meadowcroft, Micah (2019). "The Distance Between Us". The New Atlantis (58): 74–79. ISSN 1543-1215. https://www.jstor.org/stable/26609120.

### Popularity

Instagram has become the primary social media platform for celebrities to share a glimpse of their private life with their fans. Leading the pack is Cristiano Ronaldo who is the most followed celebrity on Instagram.
* ↑ "Most Popular Instagram Stars (2021)".
# V/Injective/Not nilpotent/Exercise

Let $V$ be a $K$-vector space, and
    $\varphi \colon V\longrightarrow V$
an injective linear mapping. Show that $\varphi$ is not nilpotent.
# Endomorphism/Nilpotent/Eigenvalue/Exercise

Endomorphism/Nilpotent/Situation Show that $0$ is the only eigenvalue of $\varphi$.
# Linear mapping/Nilpotent matrix on finite-dimensional vector space/Nilexponent/Exercise

Let $K$ be a field, and let $V$ denote a finite-dimensional $K$-vector space. Let
    $\varphi \colon V\longrightarrow V$
be a nilpotent linear mapping. Show that $\varphi ^{n}=0$, where $n$ is the dimension of $V$.
# Endomorphism/Nilpotent/Diagonalizable/0/Exercise

Let $K$ be a field, and let $V$ denote a finite-dimensional $K$-vector space. Let
    $\varphi \colon V\longrightarrow V$
be a nilpotent linear mapping. Suppose that $\varphi$ is diagonalizable. Show
    $\varphi =0\,.$
# Endomorphism/Nilpotent/Trace/Exercise

Let $K$ be a field, and let $V$ denote a finite-dimensional $K$-vector space. Let
    $\varphi \colon V\longrightarrow V$
be a nilpotent linear mapping. What is the trace of $\varphi$?
# Endomorphism/Nilpotent/Determinant/Exercise

Let $K$ be a field, and let $V$ denote a finite-dimensional $K$-vector space. Let
    $\varphi \colon V\longrightarrow V$
be a nilpotent linear mapping. What is the determinant of $\varphi$?
# Nilpotent matrix/2/System of equations/Exercise

Let $K$ be a field.
a) Characterize the nilpotent $2\times 2$-matrices
    ${\begin{pmatrix}x&y\\z&w\end{pmatrix}}$
over $K$ by two equations in the variables $x,y,z,w$.
b) Are these equations linear?
# Nilpotent endomorphism/Jordan normal form/Mapping lemma/Fact/Observation/Exercise

Show that the linear subspaces $U_{i}$, constructed in the proof of fact, are not $\varphi$-invariant in general.
# Linear algebra (Osnabrück 2024-2025)/Nilpotent endomorphism/Jordan normal form/Mapping lemma/Fact/Factreferencenumber

Lemma 27.11
# Nilpotent mapping/Kernel one-dimensional/Surjectivity/Exercise

Let $K$ be a field, and let $V$ denote a finite-dimensional $K$-vector space. Let
    $\varphi \colon V\longrightarrow V$
be a nilpotent linear mapping. Suppose that the kernel of $\varphi$ is one-dimensional. Let
    $V_{i}=\operatorname {kern} \varphi ^{i}\,,$
and let $s$ be the minimal number with
    $\varphi ^{s}=0\,.$
* Show that all $V_{i}$, $1\leq i\leq s$, have a direct decomposition where $U_{i}$ is one-dimensional.
    $V_{i}=V_{i-1}\oplus U_{i}\,,$
* Show that the restrictions are bijective for $1<i<s$.
    $\varphi \colon U_{i}\longrightarrow V_{i-1}$
* Show that $s$ equals the dimension of $V$.
# Nilpotent endomorphism/nxn/Maximal possibilities/Exercise

Show that there exists a family of (up to) $2^{n-1}$ $n\times n$-matrices with the property that every nilpotent endomorphism on an $n$-dimensional vector space $V$, can be described, in a suitable basis, by one of these matrices.
# Nilpotent endomorphism/4x4/Possibilities/Exercise

Show that every nilpotent endomorphism, on a four-dimensional space, can be brought into exactly one of the following forms.
    ${\begin{pmatrix}0&0&0&0\\0&0&0&0\\0&0&0&0\\0&0&0&0\end{pmatrix}},\,{\begin{pmatrix}0&0&0&0\\0&0&0&0\\0&0&0&1\\0&0&0&0\end{pmatrix}},\,{\begin{pmatrix}0&1&0&0\\0&0&0&0\\0&0&0&1\\0&0&0&0\end{pmatrix}},\,{\begin{pmatrix}0&0&0&0\\0&0&1&0\\0&0&0&1\\0&0&0&0\end{pmatrix}},\,{\begin{pmatrix}0&1&0&0\\0&0&1&0\\0&0&0&1\\0&0&0&0\end{pmatrix}}.$
# Nilpotent endomorphism/Basis images/1/Exercise

Let $v_{1},\ldots ,v_{6}$ be a basis of the $K$-vector space $V$, and let $\varphi \colon V\rightarrow V$ denote the linear mapping, given by
    $v_{1},v_{2}\mapsto v_{4},\,v_{3},v_{5}\mapsto v_{6},\,v_{4}\mapsto 0,\,v_{6}\mapsto v_{2}.$
* Show that $\varphi$ is nilpotent.
* Determine the minimal $s$ satisfying $\varphi ^{s}=0$.
* Determine the kernel of $\varphi$.
* Find a basis of $V$ such that the describing matrix of $\varphi$, with respect to this basis, is in Jordan normal form.
# Set mapping/Pointed/Corresponding linear mapping/Exercise

We consider, on the set
    $S=\{1,\ldots ,n,*\}\,,$
the set of mappings
    $B={\left\{\pi :S\rightarrow S\mid \pi (*)=*\right\}}\,.$
For $\pi \in B$, we assign (for a fixed field $K$) the linear mapping
    $\varphi _{\pi }\colon K^{n}\longrightarrow K^{n}$
given by
    $\varphi _{\pi }(e_{i})={\begin{cases}e_{\pi (i)},{\text{ if }}\pi (i)\neq *\,,\\0,{\text{ if }}\pi (i)=*\,.\end{cases}}\,$
We denote by $M_{\pi }$ the corresponding matrix with respect to the standard basis.
a) Establish the matrix $M_{\pi }$, in case $n=4$, for the following $\pi$:
(1)
(2)
(3)
(4)
b) What properties hold for the columns and for the rows of $M_{\pi }$?
c) For what $\pi$ is $M_{\pi }$ bijective?
d) For what $\pi$ is $M_{\pi }$ nilpotent?
e) What is the dimension of the kernel of $M_{\pi }$?
f) Show
    $M_{\pi \circ \rho }=M_{\pi }\circ M_{\rho }\,.$
g) Show that every nilpotent $n\times n$-matrix $M$ is similar to a matrix of the form $M_{\pi }$.
# Nilpotent mapping/Commute/Nilpotent/Exercise

Let $V$ be a $K$-vector space, and let
    $\varphi \colon V\longrightarrow V$
denote a nilpotent linear mapping. Let
    $\psi \colon V\longrightarrow V$
be another linear mapping satisfying
    $\psi \circ \varphi =\varphi \circ \psi \,.$
Show that $\psi \circ \varphi$ is also nilpotent.
# Nilpotent mapping/Sum and product/Exercise

Give an example of two nilpotent linear mappings
    $\varphi ,\psi \colon K^{2}\longrightarrow K^{2}$
such that neither $\varphi \circ \psi$ nor $\varphi +\psi$ are nilpotent.
# Nilpotent mapping/Sum and product/Exercise/Solution

Let
    $\varphi ={\begin{pmatrix}0&1\\0&0\end{pmatrix}}\,$
and
    $\psi ={\begin{pmatrix}0&0\\1&0\end{pmatrix}}\,.$
Because of
    ${\begin{pmatrix}0&1\\0&0\end{pmatrix}}^{2}={\begin{pmatrix}0&1\\0&0\end{pmatrix}}{\begin{pmatrix}0&1\\0&0\end{pmatrix}}={\begin{pmatrix}0&0\\0&0\end{pmatrix}}\,$
and
    ${\begin{pmatrix}0&0\\1&0\end{pmatrix}}^{2}={\begin{pmatrix}0&0\\1&0\end{pmatrix}}{\begin{pmatrix}0&0\\1&0\end{pmatrix}}={\begin{pmatrix}0&0\\0&0\end{pmatrix}}\,,$
these matrices are nilpotent. We have
    $\varphi \circ \psi ={\begin{pmatrix}0&1\\0&0\end{pmatrix}}{\begin{pmatrix}0&0\\1&0\end{pmatrix}}={\begin{pmatrix}1&0\\0&0\end{pmatrix}}\,,$
which is not nilpotent, due to wegen
    ${\begin{pmatrix}1&0\\0&0\end{pmatrix}}^{2}={\begin{pmatrix}1&0\\0&0\end{pmatrix}}\,.$
Moreover,
    $\varphi +\psi ={\begin{pmatrix}0&1\\0&0\end{pmatrix}}+{\begin{pmatrix}0&0\\1&0\end{pmatrix}}={\begin{pmatrix}0&1\\1&0\end{pmatrix}}\,$
is bijective, and, hence, not nilpotent.