OpenLLM-France/wikimedia huggingface dataset:
# Permutation/Cycle/Minimal transposition number/Exercise

Let $\sigma$ be a cycle of length $n$. Show that $\sigma$ can be written as a product of $n-1$ transpositions, but not with a smaller number of transpositions.
# Permutation/Partition of ground set/Subgroup/Exercise

Let $M$ be a set, and let $M=\biguplus _{i\in I}M_{i}$ be a partition of $M$, that is, every $M_{i}$ is a subset of $M$, and $M$ is the disjoint union of the $M_{i}$. Show that the product group
    $\prod _{i\in I}\operatorname {Perm} \,(M_{i})$
is a subgroup of $\operatorname {Perm} \,(M)$.
# Permutation/1357642/Several properties/Exercise

We consider the permutation $\tau \in S_{7}$, given by the value table
* Determine the cycle representation of $\tau$, and the range of action.
* Compute $\tau ^{3}$ and the order of $\tau ^{3}$.
* Determine the inversions of $\tau$ and the sign of $\tau$.
* Express $\tau$ as a product of transpositions, and determine again the sign of $\tau$
# Fixed point/M to M/Intersection of graph and diagonal/Exercise

Let $M$ denote a set, and let
    $F\colon M\longrightarrow M$
be a mapping. Show that $F$ has an fixed point if and only if the intersection of the graph of $F$ with the diagonal $\triangle ={\left\{(x,x)\in M\times M\mid x\in M\right\}}$ is not empty.
# Complex Numbers

In mathematics, a complex number is an element of a number system that contains the real numbers $\mathbb {R}$ and a specific element denoted $i$, called the imaginary unit, satisfying $i^{2}=-1$. Moreover, every complex number can be expressed in the form $a+bi$, where $a$ and $b$ are real numbers. Because no real number satisfies the above equation, $i$ was called an imaginary number by René Descartes. For the complex number $a+bi$, $a$ is called the real part and $b$ is called the imaginary part. The set of complex numbers is typically denoted by the symbol $\mathbb {C}$. Despite the historical nomenclature "imaginary", complex numbers are regarded in the mathematical sciences as just as "real" as the real numbers and are fundamental in many aspects of the scientific description of the natural world.

## Lessons

* Introduction
* From real to complex numbers - (Wiki2Reveal slides)

## See Also

* Complex Analysis
* Wikipedia: Complex number
* Wikibooks: Algebra/Complex Numbers
* OpenStax: College Algebra 2e 2.4 - Complex Numbers
# Finite permutation/Cycle length/Elements/Exercise

We consider the permutation $F$ on
    $M=\{1,2,\ldots ,8\}\,$
given by
* Establish a value table for $F^{2}=F\circ F$.
* Establish a value table for $F^{3}=F\circ F\circ F$.
* Show that all iterated compositions $F^{n}$ are bijective.
* Determine, for every $x\in M$, the minimal $n\in \mathbb {N} _{+}$ such that holds.
    $F^{n}(x)=x\,$
* Determine the minimal $n\in \mathbb {N} _{+}$ such that holds for all $x\in M$.
    $F^{n}(x)=x\,$
# Permutations/n and n+1/Bijection/Exercise

Show that the assignment
    $S_{n}\times \{1,\ldots ,n+1\}\longrightarrow S_{n+1},(\varphi ,x)\longmapsto {\tilde {\varphi }},$
given by
    ${\tilde {\varphi }}(k)={\begin{cases}\varphi (k){\text{ for }}k\leq n{\text{ and }}\varphi (k)<x\,,\\\varphi (k)+1{\text{ for }}k\leq n{\text{ and }}\varphi (k)\geq x\,,\\x{\text{ for }}k=n+1\,,\end{cases}}\,$
is well-defined and bijective.
# Secret Santa/4 people/Fixed point free/Exercise

Gabi Hochster, Heinz Ngolo, Lucy Sonnenschein and Mustafa Müller want to play Secret Santa. That is, every child gets a gift from exactly one of the other (!) children. How many possibilities are there?
# Permutation matrix/Determinant is sign/Exercise

Let $M=\{1,\ldots ,n\}$ and let $\pi$ be a permutation on $M$. The corresponding permutation matrix $M_{\pi }$ is given by
    $a_{\pi (i),i}=1\,,$
all other entries being $0$. Show that
    $\det M_{\pi }=\operatorname {sgn} (\pi )\,.$
# Permutation matrix/4x4/Queen problem/Exercise

a) Give an example of an $4\times 4$-permutation matrix such that in every diagonal (diagonal and antidiagonal, including all parallel diagonals), there is at most one $1$.
b) Show that there is no solution for a) where $a_{11}=1$ holds.
# 2x2-matrices/Determinant/Direct/Group homomorphism/Exercise

Let $K$ a field, and let
    $M={\left\{{\begin{pmatrix}a&b\\c&d\end{pmatrix}}\mid a,b,c,d\in K,\,ad-bc\neq 0\right\}}\,$
the set of all invertible $2\times 2$-matrices.
a) Show that (without referring to the determinant), $M$ is, with
matrix multiplication as operation, a group.
b) Show that (without referring to the determinant), the mapping
    $M\longrightarrow K^{\times },{\begin{pmatrix}a&b\\c&d\end{pmatrix}}\longmapsto ad-bc,$
is a group homomorphism.
# Determinant/Leibniz-formula/345/987/123/Exercise

Determine with the Leibniz-formula the determinant of the matrix
    ${\begin{pmatrix}3&4&5\\9&8&7\\1&2&3\end{pmatrix}}.$
# Group/Subgroup/Definition

Let $(G,e,\circ )$ be a group. A subset $H\subseteq G$ is called a subgroup of $G$, when the following conditions hold.
* $e\in H$.
* If $g,h\in H$, then also $g\circ h\in H$.
* If $g\in H$, then also $g^{-1}\in H$.
# Determinant/3x3/Permutation matrices/Exercise

Compute the determinant of all the $3\times 3$-matrices, such that in each column and in each row, there are exactly one $1$ and two $0$s.
# Division with remainder/Z/Fact

$\mathbb {Z}$
Let $d$ be a fixed positive natural number.
$n$
$q$
$r$, $0\leq r\leq d-1$, such that
    $n=qd+r\,$
# Division with remainder/Z/Fact/Proof/Exercise

Let $d$ be a fixed positive natural number. Show that, for every integer number $n$, there exists a uniquely determined integer number $q$ and a uniquely determined natural number $r$, $0\leq r\leq d-1$, such that
    $n=qd+r\,$
holds.
# Polynomials/N/Guessing/Exercise

Two people, $A$ and $B$, play polynomial-guessing. In this game, $A$ imagines a polynomial $P(x)$, where all coefficients are in $\mathbb {N}$. Person $B$ is allowed to ask for the values $P(n_{1}),P(n_{2}),\ldots ,P(n_{r})$ for certain natural numbers $n_{1},n_{2},\ldots ,n_{r}$. Here, $B$ may choose these numbers arbitrarily, taking the previous answers into account. The goal is to find the polynomial.
Describe a strategy for $B$ to find always the polynomial, where the number of questions is (independent of the polynomial) bounded.
# Polynomial ring/1/Field/Commutative ring/Fact/Proof/Exercise

Show that the polynomial ring $K[X]$ over a field $K$ is a commutative ring.
# Polynomials/Q/Computation/1/Exercise

Compute the polynomial
    $(3X^{2}-5X+4)\cdot (X^{3}-6X^{2}+1)-(4X^{3}+2X^{2}-2X+3)\cdot (-2X^{2}-5X)$
in the polynomial ring $\mathbb {Q} [X]$.
# Polynomial/Expansion/x^3+2x^2-3x+4/u is x+2/Exercise

Write the polynomial
    $X^{3}+2X^{2}-3X+4$
in the new variable $U=X+2$.
# Division with remainder (polynomials)/Z mod 7/5x^4+3x^3+5x^2+3x+6/3x^2+6x+4/Exercise

Perform, in the polynomial ring $\mathbb {Z} /(7)[X]$, the division with remainder ${\frac {P}{T}}$, where $P=5X^{4}+3X^{3}+5X^{2}+3X+6$ and $T=3X^{2}+6X+4$.
# Quadratic equation/R/Solve/1/Exercise

Solve the quadratic equation $3x^{2}-6x+2=0$ over $\mathbb {R}$.
# Quadratic equation/R/Completion/2/Exercise

Solve the real-quadratic equation ${\frac {3}{4}}x^{2}+{\frac {2}{7}}x-{\frac {4}{5}}=0$ by completing the square.
# Quadratic equation/R/Solution formula/Fact/Proof/Exercise

Prove the solution formula for real quadratic equations.
# Division with remainder (polynomials)/Z mod 7/6X^4+2X^3+4X^2+2X+5/5X^2+3X+2/Exercise

Perform, in the polynomial ring $\mathbb {Z} /(7)[X]$, the division with remainder ${\frac {P}{T}}$, where $P=6X^{4}+2X^{3}+4X^{2}+2X+5$ and $T=5X^{2}+3X+2$.
# Quadratic polynomial/Minimum/1/Exercise

Determine the minimal value of the real function
    $f(x)=x^{2}-3x+{\frac {4}{3}}\,.$
# Biquadratic equation/R/Solve/1/Exercise

Solve the biquadratic equation $x^{4}+7x^{2}-11=0$ over $\mathbb {R}$.
# Normed cubic polynomials/Intersection points/1/Exercise

Determine the $x$-coordinates of the intersection points of the graphs of the two real polynomials
    $P=X^{3}+4X^{2}-7X+1\,$
and
    $Q=X^{3}-2X^{2}+5X+3\,.$
# Finite geometric series/Polynomial/Exercise

Prove the formula
    $X^{n}-1=(X-1)(X^{n-1}+X^{n-2}+X^{n-3}+\cdots +X^{2}+X+1)\,.$
# Polynomial/Complex zeroes of X^3-1/Exercise

Determine all complex zeroes of the polynomial
    $X^{3}-1,$
and describe the factorization of this polynomial in $\mathbb {R} [X]$ and in $\mathbb {C} [X]$.
# Polynomial ring/Power of X/Factors/Exercise

Field polynomial ring 1/Situation Let $P=X^{n}\in K[X]$ with $n\geq 1$. Show that all normed divisors of $P$ have the form $X^{k}$, $1\leq k\leq n$.
# Complex polynomial/Surjective/Exercise

Let $P\in \mathbb {C} [X]$ denote a nonconstant polynomial. Show that the mapping
    $\mathbb {C} \longrightarrow \mathbb {C} ,z\longmapsto P(z),$
is surjective.
# Functions/R/Composition and multiplication/Exercise

Let
    $f,g,h\colon \mathbb {R} \longrightarrow \mathbb {R}$
denote functions.
a) Show the equality
    ${\left(h\cdot g\right)}\circ f={\left(h\circ f\right)}\cdot {\left(g\circ f\right)}\,.$
b) Show by an example that the equality
    ${\left(h\circ g\right)}\cdot f={\left(h\cdot f\right)}\circ {\left(g\cdot f\right)}\,$
does not hold in general.
# Polynomials/Formula for X^odd+1/Exercise

Prove the formula
    $X^{u}+1=(X+1){\left(X^{u-1}-X^{u-2}+X^{u-3}-\cdots +X^{2}-X+1\right)}\,$
for $u$ odd.
# Normed cubic polynomials/Intersection points/2/Exercise

Determine the $x$-coordinates of the intersection points of the graphs of the two real polynomials
    $P=X^{3}+8X^{2}-8X+3\,$
and
    $Q=X^{3}-5X^{2}+4X+7\,.$
# Polynomial/Substitution/iX-3+2i in X^3-2iX^2+4X-1/Exercise

Let the complex polynomials
    $P=X^{3}-2{\mathrm {i} }X^{2}+4X-1\,\,{\text{ and }}\,\,Q={\mathrm {i} }X-3+2{\mathrm {i} }$
be given. Compute $P(Q)$ (that is, plug in $Q$ into $P$).
# Polynomial ring/Division with remainder/By X^m/Exercise

Let $K$ be a field, and let $K[X]$ be the polynomial ring over $K$. What is the result when we divide (with remainder) a polynomial $P$ by $X^{m}$?
# Quadratic equation/Side lengths from perimeter and area of a rectangle/Exercise

Suppose that, of a rectangle, its perimeter $U$ and its area $A$ are known. Determine the lengths of the edges of the rectangle.
# Division with remainder (polynomials)/C/(5+i)x^4+ix^2+(3-2i)x-1/x^2+ix+3-i/Exercise

Perform, in the polynomial ring $\mathbb {C} [X]$, the division with remainder ${\frac {P}{T}}$, where
    $P=(5+X^{2}+{\mathrm {i} }X+3-{\mathrm {i} })X^{4}+X^{2}+{\mathrm {i} }X+3-{\mathrm {i} }X^{2}+(3-2X^{2}+{\mathrm {i} }X+3-{\mathrm {i} })X-1\,$
and
    $T=X^{2}+{\mathrm {i} }X+3-{\mathrm {i} }\,.$
# Quadratic equation/R/Solution formula/Fact

Let
    $aX^{2}+bX+c=0\,$
denote a real quadratic equation.
* For there is no real solution.
    $b^{2}-4ac<0\,,$
* For the only solution is
    $b^{2}-4ac=0\,,$
    $x={\frac {-b}{2a}}\,$
* For there exist the two solutions
    $b^{2}-4ac>0\,,$
    $x_{1,2}={\frac {\pm {\sqrt {b^{2}-4ac}}-b}{2a}}\,.$
# Polynomial ring/Substitution/R to Mat2R/x^2-5x+3/67/-45/Exercise

Compute the result, when we replace in the polynomial
    $X^{2}-5X+3$
the Variable $X$ by the $2\times 2$-matrix
    ${\begin{pmatrix}6&7\\-4&5\end{pmatrix}}.$
# Set theory/Listing of numbers/Polynomial law/Exercise

Find, for each of the following three sets
    $\{1,3,5,7,9,\ldots \},\,\{1,2,4,8,16,\ldots \},\,\{9,99,999,9999,99999,\ldots \}$
(of the form $\{a_{1},a_{2},a_{3},a_{4},a_{5},\ldots \}$) a polynomial
    $P(k)=c_{0}+c_{1}k+c_{2}k^{2}+c_{3}k^{3}+c_{4}k^{4}\,$
(with coefficients $c_{j}\in \mathbb {Q}$) such that
    $P(1)=a_{1},\,P(2)=a_{2},\,P(3)=a_{3},\,P(4)=a_{4},\,P(5)=a_{5}$
holds.
# Polynomial ring/Substitution/R to Mat2R/2x^3-5x^2+7x-4/41/53/Exercise

Compute the result, when we replace in the polynomial
    $2X^{3}-5X^{2}+7X-4$
the variable $X$ by the $2\times 2$-matrix
    ${\begin{pmatrix}4&1\\5&3\end{pmatrix}}.$
# Endomorphism/Finite-dimensional/Minimal polynomial/Principal ideal/Fact/Proof/Exercise

Let $V$ be a finite-dimensional vector space over a field $K$, and let
    $f\colon V\longrightarrow V$
denote a linear mapping. Show that the set
    ${\left\{P\in K[X]\mid P(f)=0\right\}}$
is a principal ideal in the polynomial ring $K[X]$, which is generated by the minimal polynomial $\mu _{f}$.
# Polynomial ring/Substitution/Matrix product/Exercise

Let
    $f,g\colon V\longrightarrow V$
be endomorphisms on a $K$-vector space $V$, and let $P\in K[X]$ be a polynomial. Show that the equality
    $P(f\circ g)=P(f)\circ P(g)\,$
does not hold in general.
# Minimal polynomial/Linear form/Homothety/Exercise

Let $M$ be a matrix with minimal polynomial $X-a$. Show that $M$ is a homothety with factor $a$.
# Endomorphism/Conjugation/Polynomial substitution/Exercise

Let
    $f\colon V\longrightarrow V$
be an endomorphism on a $K$-vector space $V$, and let
    $g\colon V\longrightarrow V$
denote an isomorphism. Show that, for every polynomial $P\in K[X]$, the equality
    $gP(f)g^{-1}=P(gfg^{-1})\,$
holds.
# Polynomial ring/Substitution/R to Mat3R/-x^3+6x^2-6x+27/532/541/730/Exercise

Compute the result, when we replace in the polynomial
    $-X^{3}+6X^{2}-6X+27,$
the variable $X$ by the $3\times 3$-matrix
    ${\begin{pmatrix}5&3&2\\5&4&1\\7&3&0\end{pmatrix}}.$
# K^(N)/Index shift/No annihilation/Exercise

We consider the linear mapping
    $\varphi \colon K^{(\mathbb {N} )}\longrightarrow K^{(\mathbb {N} )}$
given by $e_{n}\mapsto e_{n+1}$. Show that $\varphi$ is only annihilated by the zero polynomial.
# Minimal polynomial/Factorization/Substitutions not bijective/Exercise

Let $M$ be an $n\times n$-matrix over $K$ with the minimal polynomial $P\in K[X]$. Let
    $P=F_{1}\cdots F_{k}\,$
be a factorization into polynomials $F_{i}$ of positive degree. Show that $F_{i}(M)$ is not bijective.
# Matrix/Minimal 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 minimal polynomial $P\in K[X]$ coincides with the minimal polynomial of $M$, considered as a matrix over $L$.
# Projection/Minimal polynomial/Exercise

Let $V\neq 0$ be a finite-dimensional $K$-vector space, and let
    $\varphi \colon V\longrightarrow V$
denote a projection. Show that the minimal polynomial of $\varphi$ is either $X$, or $X-1$, or $X(X-1)$.
# Finite field/Function/Interpolation theorem/Exercise

Let $K$ be a finite field with $q$ elements. Show that every function $\varphi \colon K\rightarrow K$ can be expressed in a unique way as a polynomial $P\in K[X]$ of degree $<q$.
# Elementary matrices/Minimal polynomial/Exercise

We discuss the minimal polynomials of the elementary matrices.
a) Show that the minimal polynomial of an exchange matrix $V_{ij}$ is $X^{2}-1$.
b) Show that the minimal polynomial of a scaling elementary matrix $S_{k}(s)$ with $s\neq 1$ is
    $X^{2}-(s+1)X+s.$
c) Show that the minimal polynomial of an addition matrix $A_{ij}(a)$ is of the form
    $(X-1)^{k}.$
What is $k$?
# 2x2-matrix/Characteristic polynomial directly/Annihilation/Exercise

For a $2\times 2$-matrix $M={\begin{pmatrix}a&b\\c&d\end{pmatrix}}$, let
    $P_{M}:=X^{2}-\operatorname {trace} {\left(M\right)}X+\det M\,.$
Show that $P_{M}(M)=0$ holds.
# Finite field/Exponential function/Linearly independent/Exercise

Let $K$ be a finite field with $q$ elements.
* Show that the polynomial functions with $0\leq d<q$ are linearly independent.
    $\varphi _{d}\colon K\longrightarrow K,x\longmapsto x^{d},$
* Show that the exponential functions with $0\leq b<q$ are linearly independent.
    $\psi _{b}\colon K\longrightarrow K,x\longmapsto b^{x},$
# One-dimensional/Eigenvector/Exercise

Let a linear mapping
    $\varphi \colon K\longrightarrow K$
be given. What are the eigenvalues and the eigenvectors of $\varphi$?
# Direct product/Eigenvectors/Exercise

Let $V_{1},\ldots ,V_{n}$ denote vector spaces over a field $K$, and let
    $\varphi _{i}\colon V_{i}\longrightarrow V_{i}$
denote linear mappings. Let $a\in K$ be an eigenvalue of $\varphi _{k}$ for a determined $k$. Show that $a$ is also an eigenvalue of the product mapping
    $\varphi _{1}\times \cdots \times \varphi _{n}\colon V_{1}\times \cdots \times V_{n}\longrightarrow V_{1}\times \cdots \times V_{n}.$
# Linear endomorphism/R^2/Characteristic polynomial/Eigenvalue/Exercise

Show that $\lambda \in K$ is an eigenvalue for the linear mapping
    $\varphi \colon K^{2}\longrightarrow K^{2}$
given by a matrix of the form ${\begin{pmatrix}a&b\\c&d\end{pmatrix}}$ if and only if $\lambda$ is a zero of the polynomial
    $X^{2}-(a+d)X+ad-cb.$
# Endomorphism/Finite/Eigenvalue not 0/Complete basis/Exercise

Let $K$ be a field, and let $V$ denote an $n$-dimensional vector space. Let
    $\varphi \colon V\longrightarrow V$
be a linear mapping. Let $\lambda \neq 0$ be an eigenvalue of $\varphi$, and $v$ a corresponding eigenvector. Show that, for a given basis $v,u_{2},\ldots ,u_{n}$ of $V$, there exists a basis $v,w_{2},\ldots ,w_{n}$ such that $\langle v,u_{j}\rangle =\langle v,w_{j}\rangle$ and such that
    $\varphi (w_{j})\in \langle u_{i},\,i=2,\ldots ,n\rangle \,$
for all $j=2,\ldots ,n$ holds.
Show also that this is not possible for $\lambda =0$.
# Matrix/R and C/Real number/Eigenvalue/Exercise

Let $M$ be a real $n\times n$-matrix. Let $a\in \mathbb {R}$ be a real number, and suppose that it is an eigenvalue of $M$, considered as a complex matrix. Show that $a$ is already over $\mathbb {R}$ an eigenvalue of $M$.
# Vector/Endomorphism with this eigenvector/Ring/Dimension/Exercise

Let $V$ be a finite-dimensional $K$-vector space, and $v\in V$, $v\neq 0$, a fixed vector. Show that
    $R(v)={\left\{\varphi \in \operatorname {End} _{\left(V\right)}\mid v{\text{ is eigenvector of }}\varphi \right\}}\,,$
with the natural addition and multiplication of endomorphisms, is a ring and a linear subspace of $\operatorname {End} _{\left(V\right)}$. Determine the dimension of this space.
# Eigenvector/Dual space/Dual basis/Exercise

Let
    $\varphi \colon V\longrightarrow V$
be an endomorphism on a finite-dimensional $K$-vector space $V$, and let $v\in V$ be an eigenvector for $\varphi$ with eigenvalue $\lambda \in K$. Let
    ${\varphi }^{*}\colon {V}^{*}\longrightarrow {V}^{*}$
be the dual mapping of $\varphi$. We consider bases of $V$ of the form $v,u_{1},\ldots ,u_{r}$ with the dual basis $v^{*},u_{1}^{*},\ldots ,u_{r}^{*}$. Give examples of the following behavior.
a) $v^{*}$ is an eigenvector of ${\varphi }^{*}$ with the eigenvalue $\lambda$ independent of $u_{1},\ldots ,u_{r}$.
b) $v^{*}$ is an eigenvector of ${\varphi }^{*}$ with the eigenvalue $\lambda$ with respect to some basis $v,u_{1},\ldots ,u_{r}$, but not with respect to another basis $v,w_{1},\ldots ,w_{r}$.
c) $v^{*}$ is for no basis $v,u_{1},\ldots ,u_{r}$ an eigenvector of ${\varphi }^{*}$.
# Eigenvector/Two mappings/Composition/Exercise

Let
    $\varphi ,\psi \colon V\longrightarrow V$
be endomorphisms on a $K$-vector space $V$, and let $v\in V$ be an eigenvector of $\varphi$ and of $\psi$. Show that $v$ is also an eigenvector of $\varphi \circ \psi$. What is its eigenvalue?
# Diagonal matrix/Eigenspaces/Geometric multiplicity/Exercise

Determine the eigenvalues, the eigenspaces, and the geometric multiplicities of the matrix
    ${\begin{pmatrix}2&0&\cdots &\cdots &\cdots &0\\0&1&\ddots &\ddots &\vdots &\vdots \\\vdots &\ddots &3&\ddots &\ddots &\vdots \\\vdots &\ddots &\ddots &1&\ddots &\vdots \\\vdots &\vdots &\ddots &\ddots &3&0\\0&\cdots &\cdots &\cdots &0&1\end{pmatrix}}.$
# Eigenvalue/Geometric multiplicity/Exercise

For a given $a\in K$ and $m$, $1\leq m\leq n$, describe an $n\times n$-matrix over $K$ such that $a$ is its only eigenvalue with geometric multiplicity $m$.
# Matrix/0110/F2/Not diagonalizable/Exercise

Show that the matrix
    ${\begin{pmatrix}0&1\\1&0\end{pmatrix}}$
over the field with two elements ${\mathbb {F} }_{2}$ is not diagonalizable.
# Matrix/01-10/Diagonalizable/R and C/Exercise

Show that the matrix
    ${\begin{pmatrix}0&1\\-1&0\end{pmatrix}}$
is diagonalizable over $\mathbb {C}$, but not over $\mathbb {R}$. Perform the diagonalization over $\mathbb {C}$.
# Endomorphism/Polynomial/Diagonalizable/Exercise

Let $\varphi \colon V\rightarrow V$ be a diagonalizable endomorphism on the finite-dimensional $K$-vector space $V$, and let $P\in K[X]$ be a polynomial. Show that $P(\varphi )$ is also diagonalizable.
# Projection/Diagonalizable/Exercise

Show that a projection is diagonalizable.
# Upper triangular form/Constant diagonal/Diagonalizable/Exercise

Let $M$ be an upper triangular matrix, where all diagonal entries equal $c$. Show that $M$ is diagonalizable if and only if it is already a diagonal matrix.
# Elementary matrices/Diagonalizable/Exercise

Determine which elementary matrices are diagonalizable.
# Invertible matrix/Inverse matrix/Diagonalizable/Exercise

Let $M$ be an invertible matrix over $K$. Show that $M$ is diagonalizable if and only if the inverse matrix $M^{-1}$ is diagonalizable.
# Matrix/610024007/R/Diagonalization/Base change/Exercise

Show that the matrix
    ${\begin{pmatrix}6&1&0\\0&2&4\\0&0&7\end{pmatrix}}$
is diagonalizable over $\mathbb {R}$, and determine a basis consisting of eigenvectors. Perform the base change, yielding the describing diagonal matrix.
# Matrix/2305/R/Diagonalization/Exercise

Show that the matrix
    ${\begin{pmatrix}2&3\\0&5\end{pmatrix}}$
is diagonalizable over $\mathbb {R}$, and determine a basis consisting of eigenvectors. Perform the base change, yielding the describing diagonal matrix.
# Endomorphism/Exponent in minimal polynomial/Geometric multiplicity/Compare/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 $\lambda \in K$. Show that the exponent of $X-\lambda$ in the minimal polynomial of $\varphi$ might be smaller, but also larger, than the geometric multiplicity of $\lambda$.
# Direct sum/Diagonalizable/Exercise

Let $V_{1},\ldots ,V_{n}$ be finite-dimensional vector spaces over the field $K$, and let
    $\varphi _{i}\colon V_{i}\longrightarrow V_{i}$
be linear mappings. Let
    $\varphi =\varphi _{1}\times \cdots \times \varphi _{n}\colon V_{1}\times \cdots \times V_{n}\longrightarrow V_{1}\times \cdots \times V_{n}$
be the product mapping. Show that $\varphi$ is diagonalizable if and only if all $\varphi _{i}$ are diagonalizable.
# Fibonacci-numbers/Matrix/Iterations/Eigenvectors/Exercise

We consider the real matrix
    $M={\begin{pmatrix}1&1\\1&0\end{pmatrix}}\,.$
a) Determine
    $M^{n}{\begin{pmatrix}1\\0\end{pmatrix}}$
for $n=1,2,3,4$.
b) Let
    ${\begin{pmatrix}x_{n+1}\\y_{n+1}\end{pmatrix}}:=M^{n}{\begin{pmatrix}1\\0\end{pmatrix}}\,.$
Establish a relation between the sequences $x_{n}$ and $y_{n}$, and determine a recursive formula for these sequences.
c) Determine the eigenvalues and the eigenvectors of $M$.
# Matrix/Characteristic polynomial/Eigenvalues/-2 -4 2 0 2 3 0 6 1/Exercise

Determine the characteristic polynomial and the eigenvalues of the linear mapping
    $\varphi \colon \mathbb {R} ^{3}\longrightarrow \mathbb {R} ^{3},$
given by the matrix
    $A={\begin{pmatrix}-2&-4&2\\0&2&3\\0&6&1\end{pmatrix}}\,$
with respect to the standard basis.
# Square matrix/Characteristic polynomial/Trace/Exercise

Let $K$ be a field, and let $M$ denote an $n\times n$-matrix over $K$. How can we find the $\operatorname {trace} {\left(M\right)}$ in the characteristic polynomial $\chi _{M}$?
# MDLD/disjoint sets

Start=disjoint sets Target=Sets/Operations/Introduction/Section
# MDLD/graph (map)

Start=graph (map) Target=Mapping/Graph/Definition
# MDLD/compositions (mapping)

Start=compositions (mapping) See=composition (mapping) MDLD/composition (mapping) Target=Mapping/Composition/Definition
# MDLD/surjective mappings

Start=surjective mappings Target=Mapping/Surjective/Definition
# MDLD/injective mappings

Start=injective mappings Target=Mapping/Injective/Definition
# MDLD/disjoint union

Start=disjoint union Target=Sets/Operations/Introduction/Section
# MDLD/power set

Start=power set Target=Set/Power set/Definition
# MDLD/taking preimages

Start=taking preimages See=taking preimage MDLD/taking preimage Target=[[MDLD/taking preimage]]
# MDLD/taking preimage

Start=taking preimage Target=Mapping/Preimage under a mapping/Definition