OpenLLM-France/wikimedia huggingface dataset:
# MDLD/taking images

Start=taking images See=taking image MDLD/taking image Target=[[MDLD/taking image]]
# MDLD/taking image

Start=taking image Target=Mapping/Image under a mapping/Definition
# MDLD/composite functions

Start=composite functions See=composite function MDLD/composite function Target=[[MDLD/composite function]]
# MDLD/composite function

Start=composite function Target=Mapping/Composition/Definition
# MDLD/operation (binary)

Start=operation (binary) Target=Binary operation/Definition
# MDLD/intersection (2)

Start=intersection (2) Target=Two sets/Intersection/Definition
# Field/N and Z/Canonical mapping/Exercise

Let $K$ be a field. Show that, for every natural number $n\in \mathbb {N}$, there exists a field element $n_{K}$ such that $0_{K}$ is the null element in $K$, and $1_{K}$ is the unit element in $K$, and such that
    $(n+1)_{K}=n_{K}+1_{K}\,$
holds. Show that this assignment has the properties
    $(n+m)_{K}=n_{K}+m_{K}{\text{ and }}(nm)_{K}=n_{K}\cdot m_{K}.$
Extend this assignment to all integer numbers $\mathbb {Z}$, and show that the stated structural properties hold again.
# Tectonic hazards/Shake-table

Shake-table in earthquake engineering is a very effective experimental equipment to verify seismic performance associated with the direct damage to individual buildings or their models under specified ground shaking. The best way to do it is to put a building structure on a shake-table that simulates the earth shaking and to watch what may happen next (if you have no time to stand out in the field and wait for a real earthquake to strike). The earliest shake-table experiments were performed more than a century ago
Modern shake-tables typically consist of a rectangular platform that is driven in up to six degrees of freedom (DOF) by servo-hydraulic or other types of actuators. Test specimens are fixed to the platform and shaken, often to the point of failure.

### Concurrent performance testing

Concurrent testing on a shake-table is mostly used when some new building technology has to be compared against the old one; see, e.g., the images below:
Those testings might include both real (physical) and virtual (computer) experiments.
# Use of science concepts in daily lives

What is Science?
Science is a systematic and logical study towards how the universe works. Science is a dynamic subject. Science can also be defined as the systematic study of the nature and behavior of the material and physical universe, based on observation, experiment, and measurement, and the formulation of laws to describe these facts in general terms.
Science is one of the greatest blessings to the mankind. It has played a major role in improving the quality of living of the man. Science is omnipresent and omnipotent in every walk of our life. In every inch of our body, science is the protagonist.
There are different types of science:
1. Physical Science: Physics, Chemistry, Astronomy and Earth science are branches of physical sciences.
2. Life Science: Biology and Social sciences are branches of life sciences.
3. Applied Science: Engineering and Health care are branches of applied sciences.
Science is involved in cooking, eating, breathing, driving, playing, etc. The fabric we wear, the brush and paste we use, the shampoo, the talcum powder, the oil we apply, everything is the consequence of advancement of science. Life is unimaginable without all this, as it has become a necessity.
Basic science concepts behind few daily applications of science
Cooking: Heat energy is transformed to the cooking vessel in the form of radiation, conduction and convection. After the heat energy is passed the various ingredients like, salt, sugar or any edible material starts breaking bonds and forms new bonds which gives us delicious food. Therefore, physics as well as chemistry is involved. In order to prepare the perfect delicacy, the perfect proportions of the ingredients are very important. Hence math's plays a major role too.
The food which we consume, when entering into our body, undergoes many chemical reactions which is the reason for our energy. The chemically reacted food interacts with various cells present in our body and these cells transfer the proteins, carbohydrates and fats present in the food to our body through various mechanisms. Biology as well as chemistry is involved in this process.
Vehicles: The fuel in the vehicles are burnt in order for the vehicle to get energy. The concept involved in the burning of the fuel is combustion. The conversion of heat energy into electrical energy and mechanical energy takes place. Hence chemistry and physics are involved.
Household appliances:
Mixer / Juicer: It converts the electrical energy into mechanical energy. It uses the principle of centrifugal force.
Refrigerator: Electrical energy is converted to heat energy where cooling takes place.
Air Conditioner: It keeps us cool in summers. Uses electrical energy.

## See Also

* Science
# Permutation matrix/C/Diagonalizable/Fact/Proof/Exercise

Show that a permutation matrix over $\mathbb {C}$ is diagonalizable.
# Matrix/2/Characteristic polynomial/Exercise

Determine the characteristic polynomial of a matrix
    ${\begin{pmatrix}a&b\\c&d\end{pmatrix}}.$
What is the relevance of the coefficients of this polynomial?
# Matrix/3/Characteristic polynomial/Exercise

Determine the characteristic polynomial of a matrix
    ${\begin{pmatrix}a&b&c\\d&e&f\\g&h&i\end{pmatrix}}.$
What is the relevance of the coefficients of this polynomial?
# Matrix/Rational functions/Characteristic polynomial/Exercise

Compute the characteristic polynomial of the matrix
    ${\begin{pmatrix}{\frac {X^{2}-4X+7}{X-1}}&{\frac {6X-5}{X^{2}-5}}\\{\frac {4X^{2}+3X-2}{6X^{2}-3X}}&{\frac {X^{2}+8X+9}{X^{2}-3X+5}}\end{pmatrix}}$
over the field of rational functions $\mathbb {Q} (X)$.
# Characteristic polynomial/Eigenvalues/Spaces/-1 over 2, 1 over 4, -3, -1 over 2/Exercise

Determine the characteristic polynomial, the eigenvalues, and the eigenspaces of the matrix
    $M={\begin{pmatrix}-{\frac {1}{2}}&{\frac {1}{4}}\\-3&-{\frac {1}{2}}\end{pmatrix}}\,$
over $\mathbb {C}$.
# Square matrix/Characteristic polynomial splits/Trace/Exercise

Field matrix/Situation with the property, that the characteristic polynomial splits into linear factors, that is,
    $\chi _{M}=(X-\lambda _{1})^{\mu _{1}}\cdot (X-\lambda _{2})^{\mu _{2}}\cdots (X-\lambda _{k})^{\mu _{k}}\,.$
Show that
    $\operatorname {trace} {\left(M\right)}=\sum _{i=1}^{k}\mu _{i}\lambda _{i}\,.$
# Characteristic polynomial/F2/10 00/Characteristic polynomial function is 0/Exercise

Let $K$ be the field with two elements, we consider the matrix
    $M={\begin{pmatrix}1&0\\0&0\end{pmatrix}}\,$
over $K$. Show that the characteristic polynomial $\chi _{M}$ is not the zero polynomial, but that
    $\chi _{M}(\lambda )=0\,$
holds for all $\lambda \in K$.
# Characteristic polynomial/Transposed matrix/Exercise

Show that a square matrix and its transposed matrix have the same characteristic polynomial.
# Characteristic polynomial/Product rule/Wrong proof/Exercise

What is wrong in the following argumentation:
"For two $n\times n$-matrices $M,N$, the characteristic polynomials fulfill the relation
    $\chi _{M\circ N}=\chi _{M}\chi _{N}\,.$
This is because, by definition, we have
    $\chi _{M\circ N}=\det {\left(XE_{n}-M\circ N\right)}=\det {\left(XE_{n}-M\right)}\det {\left(XE_{n}-N\right)}=\chi _{M}\cdot \chi _{N}\,,$
where the equation in the middle rests on the multiplication theorem for determinants“.
# Matrix/Characteristic polynomial/Homothety/Exercise

Let $M$ be an $n\times n$-matrix, with the characteristic polynomial
    $\chi _{M}=X^{n}+c_{n-1}X^{n-1}+c_{n-2}X^{n-2}+\cdots +c_{2}X^{2}+c_{1}X+c_{0}\,.$
Determine the characteristic polynomial of the scaled matrix $sM$, $s\in K$.
# Matrix/Characteristic polynomial/Field extension/Exercise

Let $K\subseteq L$ be a field extension. Let an $n\times n$-matrix $M$ over $K$ be given. Show that the characteristic polynomial $\chi _{M}\in K[X]$ coincides with the characteristic polynomial of $M$, considered as a matrix over $L$.
# Eigenvalues and eigenspaces/Characterization of an automorphism by eigenvalues/Exercise

Let $V$ be a finite-dimensional $K$-vector space, and let $\varphi \in \operatorname {End} _{\left(V\right)}$. Show that the following statements are equivalent:
* The linear mapping $\varphi$ is an isomorphism.
* $0$ is not an eigenvalue of $\varphi$.
* The constant term of the characteristic polynomial $\chi _{\varphi }$ is $\neq 0$.
# Eigenvalue/Dual mapping/Exercise

Let
    $\varphi \colon V\longrightarrow V$
be an endomorphism on a finite-dimensional $K$-vector space $V$, and let $a\in K$
an eigenvalue of $\varphi$. Show that $a$ is also an eigenvalue of the dual mapping
    ${\varphi }^{*}\colon {V}^{*}\longrightarrow {V}^{*}.$
# Vector space/Endomorphism/Invariant linear subspace/Polynomial/Exercise

Endomorphism/Situation Let $U\subseteq V$ be a $\varphi$-invariant linear subspace of $V$. Show that, for a polynomial $P\in K[X]$, the space $U$ is also $P(\varphi )$-invariant.
# Linear subspace/Endomorphisms with invariant space/Ring/Dimension/Exercise

Let $V$ be a finite-dimensional $K$-vector space, and $U\subseteq V$ a linear subspace. Show that
    $R(U)={\left\{\varphi \in \operatorname {End} _{\left(V\right)}\mid U{\text{ is a }}\varphi -{\text{invariant linear subspace}}\right\}}\,$
is, with the natural addition and multiplication of endomorphisms, a ring, and a linear subspace of $\operatorname {End} _{\left(V\right)}$. Determine the dimension of this space.
# Endomorphism/Invariant linear subspace/Matrix/Exercise

Let $\varphi \colon V\rightarrow V$ be a linear mapping on a finite-dimensional $K$-vector space $V$. Let $k\leq n$. Show that there exists an invariant linear subspace $U\subseteq V$ of dimension $k$, if and only if there exists a basis of $V$ such that the describing matrix of $\varphi$, with respect to this basis, has the form
    ${\begin{pmatrix}a_{11}&\ldots &a_{1k}&a_{1k+1}&\ldots &a_{1n}\\\vdots &\vdots &\vdots &\vdots &\vdots &\vdots \\a_{k1}&\ldots &a_{kk}&a_{kk+1}&\ldots &a_{kn}\\0&\ldots &0&a_{k+1k+1}&\ldots &a_{k+1n}\\\vdots &\vdots &\vdots &\vdots &\vdots &\vdots \\0&\ldots &0&a_{nk+1}&\ldots &a_{nn}\end{pmatrix}}.$
# Motivation and emotion/Tutorials/Core emotions/Instructor notes
## Timing

| Topic | Time |
| Intro questions | 10-15 mins |
| Core emotion sort Select word to show others you are working on it Only classify in one column Provide 25%, 50%, and 75% encouragement in Chat | 15-20 mins depends on class size; if small, instructor helps sort in order to complete within target time frame) |
| Analysis discussion No right or wrong answers - subjective/fuzzy | 10 mins |
| Follow-up questions and discussion | 10 mins |

## Images
### Old way

For more images, see c:Category:Emotion Q-sort.
# Linear algebra (Osnabrück 2024-2025)/Field/Two elements/Example/Examplereferencenumber

Example 3.8
# Linear algebra (Osnabrück 2024-2025)/Field/Seven elements/Introduction/Example/Examplereferencenumber

Example 3.9
# Linear algebra (Osnabrück 2024-2025)/Matrix multiplication/Associativity/Exercise/Exercisereferencenumber

Exercise 4.24
# Linear algebra (Osnabrück 2024-2025)/Transposed matrix/Properties/Exercise/Exercisereferencenumber

Exercise 4.20
# MDLD/substitution method

Start=substitution method Target=Linear system/Substitution method/Remark
# MDLD/equating method

Start=equating method Target=Linear system/Equating method/Remark
# MDLD/linear systems

Start=linear systems See=linear system MDLD/linear system Target=System of linear equations/Set of variables/Homogeneous and inhomogeneous/Definition
# Linear system/Substitution method/2/Exercise

Solve the linear system
    ${\begin{matrix}5y&\,\,\,\,-z&+7&=&6\\3y&+6z&+3&=&-2\\8y&+8z&+7&=&3\,\end{matrix}}$
with the substitution method.
# MDLD/vector space axioms

Start=vector space axioms Target=Vector space/Direct/Definition
# MDLD/commutative group

Start=commutative group Target=Group/Linear algebra/Introduction/Section
# MDLD/surjective mapping

Start=surjective mapping Target=Mapping/Surjective/Definition
# MDLD/sequences

Start=sequences Target=Real numbers/Sequence/Definition
# MDLD/symmetric (matrix)

Start=symmetric (matrix) Target=Symmetric matrix/Definition
# MDLD/upper triangular matrices

Start=upper triangular matrices Target=Upper triangular matrix/Definition
# MDLD/polynomials (field 1)

Start=polynomials (field 1) See=polynomial (field 1) MDLD/polynomial (field 1) Target=[[MDLD/polynomial (field 1)]]
# MDLD/polynomial (field 1)

Start=polynomial (field 1) Target=Polynomial ring/Field/One variable/Definition
# MDLD/polynomial ring (field 1)

Start=polynomial ring (field 1) Target=Polynomial ring/Field/One variable/Definition
# Polynomials/Maximal degree/Finite dimensional linear subspace/Exercise

Let $K$ be a field, and let $K[X]$ denote the polynomial ring over $K$. Let $d\in \mathbb {N}$. Show that the set of all polynomials of degree $\leq d$ is a finite-dimensional linear subspace of $K[X]$. What is its dimension?
# MDLD/linear equations

Start=linear equations See=linear system MDLD/linear system Target=System of linear equations/Set of variables/Homogeneous and inhomogeneous/Definition
# MDLD/subspace (linear)

Start=subspace (linear) Target=Vector space/Linear subspace/Closed/Definition
# Magic square/Linear/Definition

An $n\times n$-matrix ${\left(a_{ij}\right)}_{ij}$ over a field $K$ is called a magic square (or linear-magic square over $K$), when every column sum and every row sum in the matrix equals a certain number $c\in K$.
# MDLD/linear-magic squares

Start=linear-magic squares See=linear-magic square MDLD/linear-magic square Target=[[MDLD/linear-magic square]]
# MDLD/linear-magic square

Start=linear-magic square Target=Magic square/Linear/Definition
# Magic squares/Linear/Linear subspace/Exercise

Show that the set of all linear-magic squares of length $n$ over a field $K$ is a linear subspace in the space of all $n\times n$-matrices.
# Magic squares/Linear/Dimension/Exercise

Let $K$ be a field, and $n\in \mathbb {N} _{+}$. Determine the dimension of the space of all linear-magic squares of length $n$ over $K$.
# MDLD/generated (vs)

Start=generated (vs) Target=Vector space/Spanned linear subspace/Definition
# Cayley Hamilton/Matrix/47 53/Check/Exercise

Confirm the Theorem of Cayley-Hamilton for the matrix
    ${\begin{pmatrix}4&7\\5&3\end{pmatrix}}$
by an explicit computation.
# Cayley-Hamilton/Jordan form/Direct/1/Exercise

Confirm the Theorem of Cayley-Hamilton for the matrix
    ${\begin{pmatrix}3&1&0&0\\0&3&0&0\\0&0&2&1\\0&0&0&2\end{pmatrix}}$
by an explicit computation.
# Motivation and emotion/Lectures/Implicit motives and goals

Lecture 05: Implicit motives and goals This is the fifth lecture for the motivation and emotion unit of study.

## Overview

This lecture discusses:
* implicit motives
* goal-setting and goal striving
Key questions:
* What are implicit motives? How do they arise?
* What are the key elements for successful goal setting and goal pursuit?
Take-home messages:
* Implicit (unconscious) motives are socialised rather than innate, and include achievement, affiliation/intimacy, and power motivations.
* People perform best when they have a specific plan of action to pursue a difficult, specific, and self-congruent goal.

## Outline

Implicit motives
* Explicit vs. implicit motives
* Achievement
* Affiliation
* Power
Goal setting and goal striving
* Corrective motivation
* Goal setting
* Goal striving

## Multimedia

* David McClelland and three motivational needs (Management Courses, YouTube) (8:12 mins): Explains the three psychological needs proposed by David McClelland using a practical scenario – building a sales team.
* Locke and Latham's Goal Setting Theory (MindTools, YouTube) (1:29 mins): A lot of contemporary goal setting advice is derived from Locke and Latham's (1990) goal setting theory which is summarised in this video in terms of clarity, challenge, commitment, feedback and complexity.

## Activity

* Activity: What's your implicit motivational profile?
* Watched the three motivational needs video
* Respond to this 3-question survey
* View and discuss the results

## Readings

* Chapter 07: Implicit motives (Reeve, 2018)
* Chapter 08: Goal setting and goal striving (Reeve, 2018)

## Slides

* Implicit motives (Google Slides)
* Goal setting and goal striving (Google Slides)
# Cayley Hamilton/Upper triangular matrix/Constant diagonal/Direct/Exercise

Confirm the Theorem of Cayley-Hamilton for an upper triangular matrix of the form
    ${\begin{pmatrix}a&b&c\\0&a&d\\0&0&a\end{pmatrix}}.$
# Finite field/Invertible matrix/Z mod 5/4123/Order/Exercise

Determine the order of the matrix
    ${\begin{pmatrix}4&1\\2&3\end{pmatrix}}$
over the field $\mathbb {Z} /(5)$ with $5$ elements.
# Endomorphism/Direct sum/Minimal polynomial/Ideal intersection/Exercise

Let $V$ and $W$ be finite-dimensional $K$-vector spaces, and let
    $\varphi \colon V\longrightarrow V$
and
    $\psi \colon W\longrightarrow W$
be endomorphisms, with the minimal polynomials $P$ and $Q$. Show that the minimal polynomial of
    $\varphi \oplus \psi \colon V\oplus W\longrightarrow V\oplus W$
equals the normed generator of the ideal $(P)\cap (Q)$.
# Matrix/Annihilating polynomial/Inverse matrix/Exercise

Let $M$ be an $n\times n$-matrix over a field $K$, and let
    $P=a_{0}+a_{1}X+\cdots +a_{m}X^{m}\in K[X]\,$
be a polynomial with
    $P(M)=0\,$
and with $a_{0}\neq 0$. Show that $M$ is invertible, and that its inverse matrix is given by
    $M^{-1}=-{\frac {1}{a_{0}}}{\left(a_{1}+a_{2}M+\cdots +a_{m}M^{m-1}\right)}\,.$
# Cayley Hamilton/Matrix/745 638 221/Check/Exercise

Confirm the Theorem of Cayley-Hamilton for the matrix
    ${\begin{pmatrix}7&4&5\\6&3&8\\2&2&1\end{pmatrix}}$
by an explicit computation.
# Matrix/Q/2/Order 4/Exercise

Give a matrix $M\in \operatorname {GL} _{2}\!{\left(\mathbb {Q} \right)}$ of order $4$.
# Roots of unity/Primitive root/Definition

An $n$-th root of unity is called primitive, if its order is $n$.
# Finite field/Invertible matrix/Finite order/Exercise

Let $K$ be a finite field, and let $M$ denote an invertible $n\times n$-matrix over $K$. Show that $M$ has finite order.
# Finite field/Invertible matrix/Z mod 3/0120/Order/Exercise

Determine the order of the matrix
    ${\begin{pmatrix}0&1\\2&0\end{pmatrix}}$
over the field with $3$ elements.
# Finite field/Invertible matrix/Z mod 3/0120/Order/Exercise/Solution

We have
    ${\begin{pmatrix}0&1\\2&0\end{pmatrix}}^{2}={\begin{pmatrix}0&1\\2&0\end{pmatrix}}{\begin{pmatrix}0&1\\2&0\end{pmatrix}}={\begin{pmatrix}2&0\\0&2\end{pmatrix}}\,,$
    ${\begin{pmatrix}0&1\\2&0\end{pmatrix}}^{3}={\begin{pmatrix}2&0\\0&2\end{pmatrix}}{\begin{pmatrix}0&1\\2&0\end{pmatrix}}={\begin{pmatrix}0&2\\1&0\end{pmatrix}}\,,$
and
    ${\begin{pmatrix}0&1\\2&0\end{pmatrix}}^{4}={\begin{pmatrix}2&0\\0&2\end{pmatrix}}{\begin{pmatrix}2&0\\0&2\end{pmatrix}}={\begin{pmatrix}1&0\\0&1\end{pmatrix}}\,,$
therefore, the order is $4$.
# Finite field/Unit/Finite order/Exercise

Let $K$ be a finite field. Show that every unit in $K$ is a root of unity.
# Transposition matrix/Q/Diagonalizable/Exercise

Let $M$ be the permutation matrix for a transposition. Show that $M$ is diagonalizable over $\mathbb {R}$.
# Primitive root of unity/Center formula/Exercise

Let $\zeta \in K$ be an $n$-th primitive root of unity in a field $K$. Show the formula
    $1+\zeta +\zeta ^{2}+\cdots +\zeta ^{n-1}=0\,.$
# Complex roots of unity/Unit circle/Exercise

Show that every complex root of unity lies on the unit circle.
# Roots of unity/Field/Subgroup of the unit group/Exercise

Let $K$ be a field, let $n\in \mathbb {N}$, and let $M$ be the set of all $n$-th roots of unity in $K$. Show that $M$ is a subgroup of the unit group $K^{\times }$.
# Minimal polynomial/Splits and reduced/Diagonalizable/Exercise

Let $M$ be an $n\times n$-matrix over a field $K$, and suppose that its minimal polynomial has the form
    $(X-\lambda _{1})\cdots (X-\lambda _{k})$
with different $\lambda _{i}$. Show that $M$ is diagonalizable.
# Matrix/Polynomials/Q/Coefficient matrix/1/Exercise

Express the matrix
    $M={\begin{pmatrix}4X^{2}-3X+2&X^{3}-2X+8\\3X^{4}-X^{3}-2X^{2}+7&X^{4}-6\end{pmatrix}}\,$
(whose entries are in $\mathbb {Q} [X]\subset \mathbb {Q} (X)$) in the form
    $A_{4}X^{4}+A_{3}X^{3}+A_{2}X^{2}+A_{1}X+A_{0}$
with matrices $A_{4},A_{3},A_{2},A_{1},A_{0}\in \operatorname {Mat} _{2}(\mathbb {Q} )$.
# Diagonalizable/Dual mapping/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. Let
    ${\varphi }^{*}\colon {V}^{*}\longrightarrow {V}^{*}$
denote the dual mapping. Show that $\varphi$ is diagonalizable if and only if ${\varphi }^{*}$ is diagonalizable.
# Endomorphism/Direct sum/Minimal polynomial and characteristic polynomial/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. Let
    $\psi =\varphi \oplus \cdots \oplus \varphi \colon V\oplus \cdots \oplus V\longrightarrow V\oplus \cdots \oplus V$
be the $m$-fold direct sum of $\varphi$ with itself. How is the minimal polynomial (the characteristic polynomial) of $\psi$ related to the minimal polynomial (the characteristic polynomial) of $\varphi$?
# Cayley Hamilton/Diagonalizable/Direct/Exercise

Let $M$ be a diagonalizable matrix with the characteristic polynomial $\chi _{M}$. Show directly that
    $\chi _{M}\,(M)=0\,$
holds.
# Cycle/Minimal polynomial/Exercise

Let the cycle $1\mapsto 2\mapsto 3\mapsto \ldots \mapsto n\mapsto 1$ be given, and let $M$ denote the corresponding $n\times n$-permutation matrix over a field $K$.
a) Let $P\in K[X]$ be a polynomial of degree $<n$. Establish a formula for $(P(M))(e_{1})$.
b) Determine the minimal polynomial of $M$.
c) Give an example for an endomorphism $\varphi$ on a real vector space $V$ with different vectors $v_{1},v_{2},v_{3}\in V$ such that $\varphi (v_{1})=v_{2}$, $\varphi (v_{2})=v_{3}$ and $\varphi (v_{3})=v_{1}$ holds, and such that the minimal polynomial of $\varphi$ is not $X^{3}-1$.
# Permutation matrix/Cycle/Minimal polynomial and characteristic polynomial/Exercise

Suppose that, for some permutation $\pi \in S_{n}$, its
cycle decomposition is known. Determine the minimal polynomial and the characteristic polynomial of the permutation matrix $M_{\pi }$.
# 2x2-Matrix/Trace is 0/Anti diagonal/Exercise

Let
    $M={\begin{pmatrix}a&b\\c&d\end{pmatrix}}\,$
be a matrix over $\mathbb {Q}$, whose trace is $0$. Show that there exists a matrix $N$ of the form
    $N={\begin{pmatrix}0&r\\s&0\end{pmatrix}}\,$
which is similar to $M$.
# Complex matrix/2+i 3 5i 1-i/Trigonalization/Exercise

Trigonalize the complex matrix
    ${\begin{pmatrix}2+{\mathrm {i} }&3\\5{\mathrm {i} }&1-{\mathrm {i} }\end{pmatrix}}.$
# Matrix/1011/Trigonalizable/Exercise

Show that the matrix
    ${\begin{pmatrix}1&0\\1&1\end{pmatrix}}$
is trigonalizable.
# Matrix/0110/Not trigonalizable/Exercise

Show that the matrix
    ${\begin{pmatrix}0&-1\\1&0\end{pmatrix}}$
is not trigonalizable over $\mathbb {R}$.
# Matrix/422113304/Z mod 5/Trigonalizable/Exercise

Determine whether the matrix
    ${\begin{pmatrix}4&2&2\\1&1&3\\3&0&4\end{pmatrix}}$
over the field with five elements is trigonalizable or not.
# Endomorphism/Polynomial/Trigonalizable/Exercise

Let $\varphi \colon V\rightarrow V$ be a trigonalizable endomorphism on the finite-dimensional $K$-vector space $V$, and let $P\in K[X]$ be a polynomial. Show that $P(\varphi )$ is also trigonalizable.
# Endomorphism/Dual mapping/Trigonalizable/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, and let
    ${\varphi }^{*}\colon {V}^{*}\longrightarrow {V}^{*}$
denote its dual mapping. Show that $\varphi$ is trigonalizable if and only if ${\varphi }^{*}$ is trigonalizable.
# Diagonalizable/Product not trigonalizable/Exercise

Show that the composition of two diagonalizable mappings is, in general, not trigonalizable.