OpenLLM-France/wikimedia huggingface dataset:
# Determinant/Transposed matrix/Universal property/Fact/Proof

If $M$ is not invertible, then, due to fact, the determinant is $0$ and the rank is smaller than $n$. This does also hold for the transposed matrix, so that its determinant is again $0$. So suppose that $M$ is invertible. We reduce the statement in this case to the corresponding statement for the elementary matrices, which can be verified directly, see exercise. Because of fact, there exist elementary matrices $E_{1},\ldots ,E_{s}$ such that
    $D=E_{s}\cdots E_{1}M\,$
is a diagonal matrix. Due to exercise, we have
    ${D^{\text{tr}}}={M^{\text{tr}}}{E_{1}^{\text{tr}}}\cdots {E_{s}^{\text{tr}}}\,$
and
    ${M^{\text{tr}}}={D^{\text{tr}}}({E_{s}^{\text{tr}}})^{-1}\cdots ({E_{1}^{\text{tr}}})^{-1}\,.$
The diagonal matrix $D$ is not changed under transposing it. Since the determinants of the elementary matrices are also not changed under transposition, we get, using fact,
    ${\begin{aligned}\det {M^{\text{tr}}}&=\det {\left({D^{\text{tr}}}({E_{s}^{\text{tr}}})^{-1}\cdots ({E_{1}^{\text{tr}}})^{-1}\right)}\\&=\det {D^{\text{tr}}}\cdot \det({E_{s}^{\text{tr}}})^{-1}\cdots \det({E_{1}^{\text{tr}}})^{-1}\\&=\det D\cdot \det {\left(E_{s}^{-1}\right)}\cdots \det {\left(E_{1}^{-1}\right)}\\&=\det {\left(E_{1}^{-1}\right)}\cdots \det {\left(E_{s}^{-1}\right)}\cdot \det D\cdot \\&=\det {\left(E_{1}^{-1}\cdots E_{s}^{-1}\cdot D\right)}\\&=\det M.\end{aligned}}$
# Linear algebra (Osnabrück 2024-2025)/Determinant/Transposed of a matrix/Elementary matrices/Exercise/Exercisereferencenumber

Exercise 16.13
# Linear algebra (Osnabrück 2024-2025)/Matrix/Elementary row operations/Elementary matrix from left/Fact/Factreferencenumber

Lemma 12.18
# Linear algebra (Osnabrück 2024-2025)/Determinant/Multiplication theorem/Fact/Factreferencenumber

Theorem 17.4
# Linear algebra (Osnabrück 2024-2025)/Determinant/Expansion for arbitrary row or column/Exercise/Exercisereferencenumber

Exercise 17.11
# Linear algebra (Osnabrück 2024-2025)/Permutation/Sign via inversions/Fact/Factreferencenumber

Lemma 18.10
# Sign/Transferring from ordered set to arbitrary set/Remark

Let $I$ be an arbitrary set with $n$ elements, but without an ordering, and let $\pi$ be a permutation on $I$. Then we can not talk about
inversions, and the definition of sign via products of differences is not directly applicable. However, we can look at fact in order to define the sign in this slightly more general situation. For this, we write $\pi$ as a product of $r$ transpositions and define
    $\operatorname {sgn} (\pi )={\begin{cases}1\,,{\text{ in case }}r{\text{ is even}}\,,\\-1\,,{\text{ in case }}r{\text{ is odd}}\,.\end{cases}}\,$
To see that this is well-defined, we consider a bijection
    $\varphi \colon I\longrightarrow \{1,\ldots ,n\}.$
The permutation $\pi$ on $I$ defines on $\{1,\ldots ,n\}$ the permutation $\pi '=\varphi \pi \varphi ^{-1}$. Let $\pi =\tau _{1}\cdots \tau _{r}$ be a representation as a product of $r$ transpositions on $I$. Then
    $\pi '=\varphi \pi \varphi ^{-1}=\varphi \tau _{1}\cdots \tau _{r}\varphi ^{-1}=\varphi \tau _{1}\varphi ^{-1}\varphi \tau _{2}\varphi ^{-1}\varphi \cdots \varphi ^{-1}\varphi \tau _{r}\varphi ^{-1}=\tau _{1}'\tau _{2}'\cdots \tau _{r}'\,,$
where $\tau _{j}'=\varphi \tau _{j}\varphi ^{-1}$. These are also transpositions, so that the parity of $r$ is determined by the sign of $\pi '$.
# Polynomial ring/Field/One variable/Explanations/Remark

A polynomial
    $P=\sum _{i=0}^{n}a_{i}X^{i}=a_{0}+a_{1}X+a_{2}X^{2}+\cdots +a_{n}X^{n}\,$
is, formally seen, nothing but the tuple $(a_{0},a_{1},\ldots ,a_{n})$, these numbers are called the coefficient of the polynomial. The polynomials are equal if and only if all their coefficients coincide. The letter $X$ is called the variable of the polynomial ring. In this context, the field $K$ is called the base field of the polynomial ring. Due to the componentwise definition of the addition, we have immediately a commutative group, with the zero polynomial (where all coefficients are $0$) as neutral element. The polynomials with $a_{i}=0$ for all $i\geq 1$ are called constant polynomials, they are simply written as $a_{0}$.
The way a polynomial is written suggests how the multiplication shall work, the product $X^{n}\cdot X^{m}$ is given by the addition of the exponents, thus $X^{n}\cdot X^{m}:=X^{n+m}$. For arbitrary polynomials, the multiplication arises from this simple multiplication rule by distributive continuation according to the law to multiply "everything with everything“. Explicitly, the multiplication is given by the following rule gegeben:
    ${\left(\sum _{i=0}^{n}a_{i}X^{i}\right)}\cdot {\left(\sum _{j=0}^{m}a_{j}X^{j}\right)}=\sum _{k=0}^{n+m}c_{k}X^{k}{\text{ where }}c_{k}=\sum _{r=0}^{k}a_{r}b_{k-r}.$
The multiplication is associative, commutative, distributive, and the constant polynomial $1$ is its neutral element, see exercise. Altogether, we have a commutative ring.
# Linear algebra (Osnabrück 2024-2025)/Polynomial ring/1/Field/Commutative ring/Fact/Proof/Exercise/Exercisereferencenumber

Exercise 19.4
# Linear algebra (Osnabrück 2024-2025)/Polynomial ring/Field/Replacement/Structure/Exercise/Exercisereferencenumber

Exercise 19.8
# Linear algebra (Osnabrück 2024-2025)/Polynomial ring/Field/One variable/Euclidean division/Fact/Factreferencenumber

Theorem 19.4
# Linear algebra (Osnabrück 2024-2025)/Polynomial ring/Field/Zero/Linear factor/Fact/Factreferencenumber

Lemma 19.8
# Linear algebra (Osnabrück 2024-2025)/Polynomial ring/Field/Zeroes/Number/Fact/Factreferencenumber

Corollary 19.9
# Linear algebra (Osnabrück 2024-2025)/Polynomial ring/Evaluation/Remark/Remarkreferencenumber

Remark 19.7
# Linear algebra (Osnabrück 2024-2025)/Polynomial/K/Interpolation/Fact/Factreferencenumber

Theorem 20.1
# MDLD/normed (polynomial)

Start=normed (polynomial) Target=Polynomial ring/Field/Polynomial function/Introduction/Section
# MDLD/polynomial

Start=polynomial Target=Polynomial ring/Field/Polynomial function/Introduction/Section
# Linear algebra (Osnabrück 2024-2025)/Endomorphism/Eigenspaces are linear subspaces/Zero/Fact/Factreferencenumber

Lemma 21.6
# Linear algebra (Osnabrück 2024-2025)/Linear mapping/Eigenvalue zero/Characterization/Fact/Proof/Exercise/Exercisereferencenumber

Exercise 21.9
# Linear algebra (Osnabrück 2024-2025)/Endomorphism/Finite-dimensional/Minimal polynomial/Principal ideal/Fact/Proof/Exercise/Exercisereferencenumber

Exercise 20.10
# Linear algebra (Osnabrück 2024-2025)/Endomorphism/Finite-dimensional/Minimal polynomial/Principal ideal/Fact/Factreferencenumber

Corollary 20.12
# Linear algebra (Osnabrück 2024-2025)/Eigenvalues/Endomorphism/Under Isomorphism/Fact/Factreferencenumber

Lemma 21.10
# Linear algebra (Osnabrück 2024-2025)/Linear mapping/Eigenvalue zero/Characterization/Fact/Factreferencenumber

Lemma 21.7
# Linear algebra (Osnabrück 2024-2025)/Linear mapping/Finite dimensional/Eigenvalues bounded by dimension/Fact/Proof/Exercise/Exercisereferencenumber

Exercise 22.3
# Linear algebra (Osnabrück 2024-2025)/Diagonal matrix/Eigenvalues/Example/Examplereferencenumber

Example 21.4
# Linear algebra (Osnabrück 2024-2025)/Endomorphism/Eigenvectors/Linearly independent/Fact/Factreferencenumber

Lemma 22.3
# Linear algebra (Osnabrück 2024-2025)/Matrix/Eigenvalues/0510/Q and R/Example/Examplereferencenumber

Example 21.5
# Linear algebra (Osnabrück 2024-2025)/Linear mapping/Different eigenvalues/Diagonalizable/Fact/Factreferencenumber

Corollary 22.10
# Linear algebra (Osnabrück 2024-2025)/Linear mapping/Eigenspace for different eigenvalues/Zero/Fact/Factreferencenumber

Lemma 22.2
# Linear algebra (Osnabrück 2024-2025)/Characteristic polynomial/Insert and direct determinant/Exercise/Exercisereferencenumber

Exercise 23.3
# Linear algebra (Osnabrück 2024-2025)/Endomorphism/Characteristic polynomial/Well-defined/Exercise/Exercisereferencenumber

Exercise 23.24
# Linear algebra (Osnabrück 2024-2025)/Linear mapping/Eigenspace as kernel/Fact/Factreferencenumber

Lemma 22.1
# Linear algebra (Osnabrück 2024-2025)/Determinant/Diagonal block matrix/Left lower block 0/Exercise/Exercisereferencenumber

Exercise 16.23
# Linear algebra (Osnabrück 2024-2025)/Endomorphism/Eigenvalue and characteristic polynomial/Fact/Factreferencenumber

Theorem 23.2
# Linear algebra (Osnabrück 2024-2025)/Eigenspaces/Direct sum/Dimension/Fact/Factreferencenumber

Lemma 22.6
# Linear algebra (Osnabrück 2024-2025)/Linear mapping/Diagonalizable/Direct sum of eigenspaces/Fact/Factreferencenumber

Lemma 22.11
# Linear algebra (Osnabrück 2024-2025)/Endomorphism/Characteristic polynomial/Different linear factors/Diagonalizable/Fact/Proof/Exercise/Exercisereferencenumber

Exercise 23.28
# Linear algebra (Osnabrück 2024-2025)/Polynomial ring/Endomorphism/Inserting/Ring homomorphism (without concept)/Fact/Factreferencenumber

Lemma 20.3
# Linear algebra (Osnabrück 2024-2025)/Cayley-Hamilton/Matrix version/Fact/Factreferencenumber

Theorem 24.1
# Linear algebra (Osnabrück 2024-2025)/Cayley-Hamilton/Fact/Factreferencenumber

Theorem 24.2
# Linear algebra (Osnabrück 2024-2025)/Permutation matrix/C/Diagonalizable/Fact/Proof/Exercise/Exercisereferencenumber

Exercise 24.23
# MDLD/fixed vector

Start=fixed vector Target=Mapping/Fixed point/Definition
# Linear algebra (Osnabrück 2024-2025)/Linear mapping/Trigonalizable/Characterizations/1/Fact/Factreferencenumber

Theorem 25.10
# Linear algebra (Osnabrück 2024-2025)/Matrix/2x2/Shearing matrices/Eigenvalues and diagonalizable/Example/Examplereferencenumber

Example 22.12
# Linear algebra (Osnabrück 2024-2025)/Trigonalizable/Direct sum/Fact/Proof/Exercise/Exercisereferencenumber

Exercise 25.5
# Linear algebra (Osnabrück 2024-2025)/Vector space/Endomorphism/Invariant linear subspace/Polynomial/Exercise/Exercisereferencenumber

Exercise 23.31
# Linear algebra (Osnabrück 2024-2025)/Endomorphism/Invariant linear subspace/Polynomial/Action/Fact/Factreferencenumber

Lemma 25.5
# Linear algebra (Osnabrück 2024-2025)/Endomorphism/Invariant linear subspace/Minimal polynomial/Divisibility/Fact/Factreferencenumber

Corollary 25.6
# Linear algebra (Osnabrück 2024-2025)/Cayley-Hamilton/Minimal polynomial and characteristic polynomial/Fact/Factreferencenumber

Corollary 24.3
# Linear algebra (Osnabrück 2024-2025)/Endomorphism/Eigenvalue/Invariant hyperplane/Fact/Factreferencenumber

Lemma 25.4
# Linear algebra (Osnabrück 2024-2025)/Linear mapping/Trigonalizable/Characterization with characteristic polynomial/Fact/Factreferencenumber

Theorem 25.10
# Linear algebra (Osnabrück 2024-2025)/Polynomial ring over a field/One variable/Principal ideal domain/2/Fact/Factreferencenumber

Theorem 20.10
# Linear algebra (Osnabrück 2024-2025)/Polynomial ring/Field/Lemma of Bezout/Fact/Factreferencenumber

Theorem 26.4
# Linear algebra (Osnabrück 2024-2025)/Divisibility theory (Z)/Lemma of Bezout/Fact/Proof/Exercise/Exercisereferencenumber

Exercise 26.10
# Linear algebra (Osnabrück 2024-2025)/Vector space/Endomorphism/Polynomial/Kernel/Invariant linear subspace/Exercise/Exercisereferencenumber

Exercise 26.28
# Linear algebra (Osnabrück 2024-2025)/Diagonalizable/Generalized eigenspace/Exercise/Exercisereferencenumber

Exercise 26.22
# Linear algebra (Osnabrück 2024-2025)/Polynomial/Bezout/X^2+X+1 and X-1/Example/Examplereferencenumber

Example 26.7
# Linear algebra (Osnabrück 2024-2025)/Characteristic polynomial/Direct sum decomposition/Fact/Factreferencenumber

Lemma 26.10
# Linear algebra (Osnabrück 2024-2025)/Endomorphism/Generalized eigenspace/Algebraic multiplicity/Fact/Factreferencenumber

Theorem 26.12
# Linear algebra (Osnabrück 2024-2025)/Matrix/Nilpotent/Typical form/Example/Examplereferencenumber

Example 27.3
# Linear algebra (Osnabrück 2024-2025)/Minimal polynomial and characteristic polynomial/Same zeroes/Fact/Factreferencenumber

Lemma 24.5
# Linear algebra (Osnabrück 2024-2025)/Nilpotent endomorphism/Minimal polynomial/Fact/Factreferencenumber

Lemma 27.7
# Linear algebra (Osnabrück 2024-2025)/Nilpotent endomorphism/Successive kernels/Fact/Factreferencenumber

Lemma 27.10
# Linear algebra (Osnabrück 2024-2025)/Endomorphism/Trigonalizable/Direct sum/Fact/Factreferencenumber

Lemma 26.14
# Linear algebra (Osnabrück 2024-2025)/Jordan matrix/Eigenvector/One-dimensional/Exercise/Exercisereferencenumber

Exercise 28.22
# Linear algebra (Osnabrück 2024-2025)/Trigonalizable endomorphism/Jordan normal form/Fact/Factreferencenumber

Theorem 28.5
# Linear algebra (Osnabrück 2024-2025)/Affine space/Affine subspace/Intersection property/Fact/Factreferencenumber

Lemma 30.1
# Linear algebra (Osnabrück 2024-2025)/Affine space/Point set/Barycentric combination/Point/Fact/Proof/Exercise/Exercisereferencenumber

Exercise 29.14
# MDLD/barycentric representation

Start=barycentric representation Target=Affine space/Point set/Barycentric combination/Definition
# MDLD/barycentric coordinates

Start=barycentric coordinates Target=Affine space/Point set/Barycentric combination/Definition
# Linear algebra (Osnabrück 2024-2025)/Affine space/Finite point set/Barycentric combinations/Barycentric combination/Exercise/Exercisereferencenumber

Exercise 29.20
# Linear algebra (Osnabrück 2024-2025)/Affine space/Finite point family/Affinely independent/Characterization/Fact/Proof/Exercise/Exercisereferencenumber

Exercise 30.3
# Linear algebra (Osnabrück 2024-2025)/Affine space/Affine subspace/Characterization/Fact/Factreferencenumber

Lemma 29.14
# Linear algebra (Osnabrück 2024-2025)/Affine mapping/One point suffices/Exercise/Exercisereferencenumber

Exercise 30.7
# Linear algebra (Osnabrück 2024-2025)/Barycentric coordinates/Point representation/Exercise/Exercisereferencenumber

Exercise 29.15
# MDLD/affine-linear mapping

Start=affine-linear mapping Target=Affine space/Affine-linear mapping/Definition
# MDLD/affine isomorphism

Start=affine isomorphism Target=Affine-linear mapping/Affine isomorphism/Definition
# Linear algebra (Osnabrück 2024-2025)/Set/Disjoint union/Bijection of the power sets/2/Exercise/Exercisereferencenumber

Exercise 2.16
# Linear algebra (Osnabrück 2024-2025)/Four tuple/Modulus of differences/R/Infinite dilation/Exercise/Exercisereferencenumber

Exercise 23.16
# MDLD/operation

Start=operation See=binary operation MDLD/binary operation Target=Binary operation/Definition
# Linear mapping/Field/Isomorphism/Definition

Let $K$ denote a field and let $V$ and $W$ denote $K$-vector spaces. A bijective linear mapping
    $\varphi \colon V\longrightarrow W$
is called isomorphism.
# MDLD/automorphism (vs)

Start=automorphism (vs) Target=Isomorphic vector spaces/Introduction/Section
# MDLD/translation

Start=translation Target=Vector space/Translation/Definition
# MDLD/automorphisms (vs)

Start=automorphisms (vs) See=automorphism (vs) MDLD/automorphism (vs) Target=Isomorphic vector spaces/Introduction/Section
# MDLD/coordinate function (vs)

Start=coordinate function (vs) Target=Vector space/Basis and isomorphic to K^n/Remark
# Linear algebra (Osnabrück 2024-2025)/Hyperplane/Kernel of a linear form/Fact/Proof/Exercise/Exercisereferencenumber

Exercise 14.5
# MDLD/trace (linear)

Start=trace (linear) Target=Linear mapping/V over K/Trace/Definition
# MDLD/orthogonal space (dual)

Start=orthogonal space (dual) Target=Linear subspace/Orthogonal space in dual space/Definition
# MDLD/orthogonal spaces (dual space)

Start=orthogonal spaces (dual space) See=orthogonal space (dual space) MDLD/orthogonal space (dual space) Target=Linear subspace/Orthogonal space in dual space/Definition
# MDLD/bidual

Start=bidual Target=Vector space/Bidual space/Definition
# MDLD/generating systems (vs)

Start=generating systems (vs) See=generating system (vs) MDLD/generating system (vs) Target=Vector space/Generating system/Definition
# Multilinear mapping/Zero/No linear subspace/Exercise

Let $K$ be a field, and let $V_{1},\ldots ,V_{n}$ and $W$ denote vector spaces over $K$. Let
    $\Phi \colon V_{1}\times \cdots \times V_{n}\longrightarrow W$
be a multilinear mapping. Show that the set
    ${\left\{\left(v_{1},\,\ldots ,\,v_{n}\right)\in V_{1}\times \cdots \times V_{n}\mid \Phi \left(v_{1},\,\ldots ,\,v_{n}\right)=0\right\}}$
is, in general, not a linear subspace of $V_{1}\times \cdots \times V_{n}$.
# MDLD/injectivity

Start=injectivity See=injective MDLD/injective Target=Mapping/Injective/Definition
# MDLD/surjectivity

Start=surjectivity See=surjective MDLD/surjective Target=Mapping/Surjective/Definition
# Permutation/10/Explicit example/Exercise

Compute, for the permutation $\sigma$ given by
the powers $\sigma ^{2}$ and $\sigma ^{3}$, and determine the cycle representation of these three permutations.
# MDLD/finite permutation

Start=finite permutation Target=Permutation group/Definition
# Permutation/8/25731486/Inversions/Sign/Exercise

Compute, for the permutation
the number of inversions and the sign.
# MDLD/cycle representation

Start=cycle representation Target=Permutation_group/Number/2/Introduction/Section
# Permutation/8/25371486/45286713/Product/Inversions/Sign/Exercise

We consider the two permutations given by
and
Compute $\sigma \tau$ and $\tau \sigma$. Determine the number of inversions and the sign of $\tau$. Describe the cycle representation of $\sigma$ and of $\sigma ^{3}$. What is the order of $\sigma$?
# MDLD/subgroup

Start=subgroup Target=Group/Subgroup/Definition
# MDLD/even permutation

Start=even permutation Target=Permutation/Sign/Introduction/Section
# MDLD/polynomial ring (1 K)

Start=polynomial ring (1 K) Target=Polynomial ring/Field/One variable/Definition
# Polynomial ring/Field/Degree/Simple rules/Fact/Proof/Exercise

Let $K$ be a field and let $K[X]$ be the polynomial ring over $K$. Prove the following properties concerning the degree of a polynomial:
    $\operatorname {deg} \,(P+Q)\leq \max\{\operatorname {deg} \,(P),\,\operatorname {deg} \,(Q)\}\,,$
    $\operatorname {deg} \,(P\cdot Q)=\operatorname {deg} \,(P)+\operatorname {deg} \,(Q)\,.$
# MDLD/normed polynomials

Start=normed polynomials See=normed polynomial MDLD/normed polynomial Target=[[MDLD/normed polynomial]]
# MDLD/normed polynomial

Start=normed polynomial Target=Polynomial ring/Field/Polynomial function/Introduction/Section
# MDLD/divisor (polynomial)

Start=divisor (polynomial) Target=Polynomial ring/K/Division/Definition
# MDLD/rational functions (K)

Start=rational functions (K) See=rational function (K) MDLD/rational function (K) Target=[[MDLD/rational function (K)]]
# MDLD/rational function (K)

Start=rational function (K) Target=Rational function/Field/Definition
# MDLD/compositions

Start=compositions See=composition MDLD/composition Target=Mapping/Composition/Definition
# MDLD/minimal polynomial (matrix)

Start=minimal polynomial (matrix) Target=Endomorphism/Finite-dimensional/Minimal polynomial/Definition
# MDLD/minimal polynomials (matrix)

Start=minimal polynomials (matrix) See=minimal polynomial (matrix) MDLD/minimal polynomial (matrix) Target=Endomorphism/Finite-dimensional/Minimal polynomial/Definition
# MDLD/complex (number)

Start=complex (number) Target=Complex numbers/Pairs/Definition
# Diagonalizable/Minimal polynomial/Exercise

Let $M$ be a diagonalizable matrix. Show that the minimal polynomial of $M$ has the form
    $(X-\lambda _{1})\cdots (X-\lambda _{k})$
with different $\lambda _{i}$.
# Matrix/n different eigenvalues/Trace is sum/Exercise

Let
    $M\in \operatorname {Mat} _{n}(K)\,$
be a matrix with $n$ different eigenvalues. Show that the trace of $M$ is the sum of these eigenvalues.
# Linear algebra (Osnabrück 2024-2025)/Minimal polynomial/Splits and reduced/Diagonalizable/Exercise/Exercisereferencenumber

Exercise 24.7
# MDLD/diagonalizable matrix (ev)

Start=diagonalizable matrix (ev) Target=Linear mapping/Diagonalizable/Eigenvectors/Definition
# MDLD/diagonalizable endomorphism

Start=diagonalizable endomorphism Target=Linear mapping/Diagonalizable/Eigenvectors/Definition
# MDLD/geometric multiplicities

Start=geometric multiplicities See=geometric multiplicity MDLD/geometric multiplicity Target=Linear mapping/Multiplicities/Introduction/Section
# MDLD/geometric multiplicity

Start=geometric multiplicity Target=Endomorphism/Eigenvalue/Geometric multiplicity/Definition
# MDLD/base change

Start=base change Target=Base change/Transformation matrix/Definition
# MDLD/field of rational functions (1)

Start=field of rational functions (1) Target=Rational function/Field/Definition
# MDLD/endomorphism (vs)

Start=endomorphism (vs) Target=Linear mapping/Matrix/Relation/2/Section
# MDLD/invariant linear subspace (endomorphism)

Start=invariant linear subspace (endomorphism) Target=Endomorphism/Invariant subspace/Definition
# Endomorphism/Invariant linear subspace/Decomposition/Matrix/Exercise

Let $\varphi \colon V\rightarrow V$ be a linear mapping on the finite-dimensional $K$-vector space $V$. Let $k\leq n$. Show that there exists a direct sum decomposition $V=U\oplus W$ into invariant linear subspaces $U,W\subseteq V$ of dimension $k$ and $n-k$, 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
    ${\begin{pmatrix}a_{11}&\ldots &a_{1k}&0&\ldots &0\\\vdots &\vdots &\vdots &\vdots &\vdots &\vdots \\a_{k1}&\ldots &a_{kk}&0&\ldots &0\\0&\ldots &0&a_{k+1k+1}&\ldots &a_{k+1n}\\\vdots &\vdots &\vdots &\vdots &\vdots &\vdots \\0&\ldots &0&a_{nk+1}&\ldots &a_{nn}\end{pmatrix}}.$
# Linear algebra (Osnabrück 2024-2025)/4-number-problem/Ordered by size/Matrix/Exercise/Exercisereferencenumber

Exercise 10.16
# MDLD/characteristic polynomials

Start=characteristic polynomials See=characteristic polynomial MDLD/characteristic polynomial Target=Matrix/Field/Characteristic polynomial/Definition
# MDLD/invariant linear subspaces (endomorphism)

Start=invariant linear subspaces (endomorphism) See=invariant linear subspace (endomorphism) MDLD/invariant linear subspace (endomorphism) Target=Endomorphism/Invariant subspace/Definition
# MDLD/roots of unity

Start=roots of unity See=root of unity MDLD/root of unity Target=Roots of unity/Field/Definition
# MDLD/primitive root of unity

Start=primitive root of unity Target=Roots of unity/Primitive root/Definition
# MDLD/order (matrix)

Start=order (matrix) Target=Group theory/Order of an element/Definition
# MDLD/minimal polynomials (endomorphism)

Start=minimal polynomials (endomorphism) See=minimal polynomial (endomorphism) MDLD/minimal polynomial (endomorphism) Target=Endomorphism/Finite-dimensional/Minimal polynomial/Definition
# Endomorphism/Trigonalizable/Lower triangular matrix/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. Show that $\varphi$ is trigonalizable if and only if $\varphi$ is described by a lower triangular matrix
    ${\begin{pmatrix}a_{1}&0&\cdots &\cdots &0\\\ast &a_{2}&0&\cdots &0\\\vdots &\ddots &\ddots &\ddots &\vdots \\\ast &\cdots &\ast &a_{n-1}&0\\\ast &\cdots &\cdots &\ast &a_{n}\end{pmatrix}}$
with respect to a suitable basis.
# Vector space/Flag/Bijection/Exercise

Let $V$ and $W$ be vector spaces over $K$ of the same dimension $n$, and let
    $0=V_{0}\subset V_{1}\subset \ldots \subset V_{n-1}\subset V_{n}=V\,$
and
    $0=W_{0}\subset W_{1}\subset \ldots \subset W_{n-1}\subset W_{n}=W\,$
be flags in $V$ and in $W$, respectively. Show that there exists a bijective linear mapping
    $\varphi \colon V\longrightarrow W$
such that
    $\varphi (V_{i})=W_{i}\,$
for all $i=0,1,\ldots ,n$.
# MDLD/flags (vs)

Start=flags (vs) See=flag (vs) MDLD/flag (vs) Target=[[MDLD/flag (vs)]]
# MDLD/flag (vs)

Start=flag (vs) Target=Vector space/Flag/Definition
# Matrix/Nilpotent/Jordan/Successive kernels/1/Exercise

Determine the kernels of the powers $M^{i}$ of the matrix
    $M={\begin{pmatrix}0&1&0&0&0\\0&0&1&0&0\\0&0&0&1&0\\0&0&0&0&1\\0&0&0&0&0\end{pmatrix}}\,.$
# MDLD/kernels (linear)

Start=kernels (linear) See=kernel (linear) MDLD/kernel (linear) Target=[[MDLD/kernel (linear)]]
# MDLD/kernel (linear)

Start=kernel (linear) Target=Linear mapping/Kernel/Definition
# MDLD/diagonalizable mapping

Start=diagonalizable mapping Target=Linear mapping/Diagonalizable/Eigenvectors/Definition
# Matrix/-24-12/Nilpotent/Exercise

Show that the matrix
    ${\begin{pmatrix}-2&4\\-1&2\end{pmatrix}}$
is nilpotent.