grandes-ecoles

Papers (176)
2025
centrale-maths1__official 40 centrale-maths2__official 36 mines-ponts-maths1__mp 17 mines-ponts-maths1__pc 21 mines-ponts-maths1__psi 21 mines-ponts-maths2__mp 28 mines-ponts-maths2__pc 23 mines-ponts-maths2__psi 25 polytechnique-maths-a__mp 35 polytechnique-maths__fui 9 polytechnique-maths__pc 27 x-ens-maths-a__fui 10 x-ens-maths-a__mp 18 x-ens-maths-b__mp 6 x-ens-maths-c__mp 6 x-ens-maths-d__mp 31 x-ens-maths__pc 27 x-ens-maths__psi 30
2024
centrale-maths1__official 21 centrale-maths2__official 28 geipi-polytech__maths 9 mines-ponts-maths1__mp 23 mines-ponts-maths1__psi 9 mines-ponts-maths2__mp 14 mines-ponts-maths2__pc 19 mines-ponts-maths2__psi 20 polytechnique-maths-a__mp 42 polytechnique-maths-b__mp 27 x-ens-maths-a__mp 43 x-ens-maths-b__mp 29 x-ens-maths-c__mp 22 x-ens-maths-d__mp 41 x-ens-maths__pc 20 x-ens-maths__psi 23
2023
centrale-maths1__official 37 centrale-maths2__official 32 e3a-polytech-maths__mp 4 mines-ponts-maths1__mp 14 mines-ponts-maths1__pc 21 mines-ponts-maths1__psi 21 mines-ponts-maths2__mp 21 mines-ponts-maths2__pc 13 mines-ponts-maths2__psi 22 polytechnique-maths__fui 3 x-ens-maths-a__mp 24 x-ens-maths-b__mp 10 x-ens-maths-c__mp 10 x-ens-maths-d__mp 10 x-ens-maths__pc 22
2022
centrale-maths1__mp 22 centrale-maths1__pc 33 centrale-maths1__psi 42 centrale-maths2__mp 26 centrale-maths2__pc 37 centrale-maths2__psi 40 mines-ponts-maths1__mp 26 mines-ponts-maths1__pc 20 mines-ponts-maths1__psi 23 mines-ponts-maths2__mp 22 mines-ponts-maths2__pc 9 mines-ponts-maths2__psi 18 x-ens-maths-a__mp 8 x-ens-maths-b__mp 19 x-ens-maths-c__mp 17 x-ens-maths-d__mp 47 x-ens-maths1__mp 13 x-ens-maths2__mp 26 x-ens-maths__pc 7 x-ens-maths__pc_cpge 14 x-ens-maths__psi 22 x-ens-maths__psi_cpge 26
2021
centrale-maths1__mp 34 centrale-maths1__pc 36 centrale-maths1__psi 28 centrale-maths2__mp 21 centrale-maths2__pc 38 centrale-maths2__psi 28 x-ens-maths2__mp 35 x-ens-maths__pc 29
2020
centrale-maths1__mp 42 centrale-maths1__pc 36 centrale-maths1__psi 38 centrale-maths2__mp 2 centrale-maths2__pc 35 centrale-maths2__psi 39 mines-ponts-maths1__mp_cpge 22 mines-ponts-maths2__mp_cpge 19 x-ens-maths-a__mp_cpge 10 x-ens-maths-b__mp_cpge 19 x-ens-maths-c__mp 10 x-ens-maths-d__mp 13 x-ens-maths1__mp 13 x-ens-maths2__mp 20 x-ens-maths__pc 6
2019
centrale-maths1__mp 37 centrale-maths1__pc 40 centrale-maths1__psi 38 centrale-maths2__mp 37 centrale-maths2__pc 39 centrale-maths2__psi 46 x-ens-maths1__mp 24 x-ens-maths__pc 18 x-ens-maths__psi 9
2018
centrale-maths1__mp 21 centrale-maths1__pc 31 centrale-maths1__psi 39 centrale-maths2__mp 23 centrale-maths2__pc 35 centrale-maths2__psi 30 x-ens-maths1__mp 18 x-ens-maths2__mp 13 x-ens-maths__pc 17 x-ens-maths__psi 20
2017
centrale-maths1__mp 45 centrale-maths1__pc 22 centrale-maths1__psi 17 centrale-maths2__mp 30 centrale-maths2__pc 28 centrale-maths2__psi 44 x-ens-maths1__mp 24 x-ens-maths2__mp 7 x-ens-maths__pc 17 x-ens-maths__psi 19
2016
centrale-maths1__mp 41 centrale-maths1__pc 31 centrale-maths1__psi 33 centrale-maths2__mp 25 centrale-maths2__pc 42 centrale-maths2__psi 17 x-ens-maths1__mp 10 x-ens-maths2__mp 32 x-ens-maths__pc 1 x-ens-maths__psi 20
2015
centrale-maths1__mp 18 centrale-maths1__pc 11 centrale-maths1__psi 42 centrale-maths2__mp 44 centrale-maths2__pc 1 centrale-maths2__psi 14 x-ens-maths1__mp 16 x-ens-maths2__mp 19 x-ens-maths__pc 30 x-ens-maths__psi 20
2014
centrale-maths1__mp 28 centrale-maths1__pc 26 centrale-maths1__psi 36 centrale-maths2__mp 24 centrale-maths2__pc 23 centrale-maths2__psi 29 x-ens-maths2__mp 13
2013
centrale-maths1__mp 3 centrale-maths1__pc 45 centrale-maths1__psi 20 centrale-maths2__mp 32 centrale-maths2__pc 50 centrale-maths2__psi 32 x-ens-maths1__mp 14 x-ens-maths2__mp 10 x-ens-maths__pc 22 x-ens-maths__psi 9
2012
centrale-maths1__pc 23 centrale-maths1__psi 20 centrale-maths2__mp 27 centrale-maths2__psi 20
2011
centrale-maths1__mp 27 centrale-maths1__pc 15 centrale-maths1__psi 21 centrale-maths2__mp 29 centrale-maths2__pc 8 centrale-maths2__psi 28
2010
centrale-maths1__mp 7 centrale-maths1__pc 23 centrale-maths1__psi 9 centrale-maths2__mp 10 centrale-maths2__pc 36 centrale-maths2__psi 27
2020 mines-ponts-maths1__mp_cpge

22 maths questions

Q1 3x3 Matrices Linear Transformation and Endomorphism Properties View
Let $u \in \mathcal{N}(E)$. Show that $\operatorname{tr} u^{k} = 0$ for every $k \in \mathbf{N}^{*}$.
Q2 3x3 Matrices Linear Transformation and Endomorphism Properties View
We fix a basis $\mathbf{B}$ of $E$. We denote by $\mathcal{N}_{\mathbf{B}}$ the set of endomorphisms of $E$ whose matrix in $\mathbf{B}$ is strictly upper triangular. Justify that $\mathcal{N}_{\mathbf{B}}$ is a nilpotent vector subspace of $\mathcal{L}(E)$ and that its dimension equals $\frac{n(n-1)}{2}$.
Q3 Proof Linear Transformation and Endomorphism Properties View
Let $\mathbf{B}$ be a basis of $E$. Show that
$$\left\{\nu(u) \mid u \in \mathcal{N}_{\mathbf{B}}\right\} = \{\nu(u) \mid u \in \mathcal{N}(E)\} = \llbracket 1, n \rrbracket$$
Q4 Proof Linear Transformation and Endomorphism Properties View
Let $u \in \mathcal{L}(E)$. We are given two vectors $x$ and $y$ of $E$, as well as two integers $p \geq q \geq 1$ such that $u^{p}(x) = u^{q}(y) = 0$ and $u^{p-1}(x) \neq 0$. Show that the family $(x, u(x), \ldots, u^{p-1}(x))$ is free, and that if $(u^{p-1}(x), u^{q-1}(y))$ is free then $(x, u(x), \ldots, u^{p-1}(x), y, u(y), \ldots, u^{q-1}(y))$ is free.
Q5 Proof Linear Transformation and Endomorphism Properties View
Let $u \in \mathcal{N}(E)$, with nilindex $p$. Deduce from the previous question that if $p \geq n-1$ and $p \geq 2$ then $\operatorname{Im} u^{p-1} = \operatorname{Im} u \cap \operatorname{Ker} u$ and $\operatorname{Im} u^{p-1}$ has dimension 1.
Q6 Proof Linear Transformation and Endomorphism Properties View
We consider a Euclidean vector space $(E, (-\mid-))$. Given $a \in E$ and $x \in E$, we denote by $a \otimes x$ the map from $E$ to itself defined by:
$$\forall z \in E, (a \otimes x)(z) = (a \mid z) \cdot x$$
We fix $x \in E \backslash \{0\}$. Show that the map $a \in E \mapsto a \otimes x$ is linear and constitutes a bijection from $E$ onto $\{u \in \mathcal{L}(E) : \operatorname{Im} u \subset \operatorname{Vect}(x)\}$.
Q7 Proof Linear Transformation and Endomorphism Properties View
We consider a Euclidean vector space $(E, (-\mid-))$. Given $a \in E$ and $x \in E$, we denote by $a \otimes x$ the map from $E$ to itself defined by:
$$\forall z \in E, (a \otimes x)(z) = (a \mid z) \cdot x$$
Let $a \in E$ and $x \in E \backslash \{0\}$. Show that $\operatorname{tr}(a \otimes x) = (a \mid x)$.
Q8 Proof Matrix Algebra and Product Properties View
We consider an $\mathbf{R}$-vector space $E$ of dimension $n > 0$. Let $\mathcal{V}$ be a nilpotent vector subspace of $\mathcal{L}(E)$ containing a non-zero element, with generic nilindex $p := \max_{u \in \mathcal{V}} \nu(u)$. In questions 8 to 11, we are given two arbitrary elements $u$ and $v$ of $\mathcal{V}$.
Let $k \in \mathbf{N}^{*}$. Show that there exists a unique family $(f_{0}^{(k)}, \ldots, f_{k}^{(k)})$ of endomorphisms of $E$ such that
$$\forall t \in \mathbf{R}, (u + tv)^{k} = \sum_{i=0}^{k} t^{i} f_{i}^{(k)}$$
Show in particular that $f_{0}^{(k)} = u^{k}$ and $f_{1}^{(k)} = \sum_{i=0}^{k-1} u^{i} v u^{k-1-i}$.
Q9 Proof Matrix Algebra and Product Properties View
We consider an $\mathbf{R}$-vector space $E$ of dimension $n > 0$. Let $\mathcal{V}$ be a nilpotent vector subspace of $\mathcal{L}(E)$ containing a non-zero element, with generic nilindex $p := \max_{u \in \mathcal{V}} \nu(u)$. In questions 8 to 11, we are given two arbitrary elements $u$ and $v$ of $\mathcal{V}$.
Show that $\sum_{i=0}^{p-1} u^{i} v u^{p-1-i} = 0$.
Q10 Proof Deduction or Consequence from Prior Results View
We consider an $\mathbf{R}$-vector space $E$ of dimension $n > 0$. Let $\mathcal{V}$ be a nilpotent vector subspace of $\mathcal{L}(E)$ containing a non-zero element, with generic nilindex $p := \max_{u \in \mathcal{V}} \nu(u)$. In questions 8 to 11, we are given two arbitrary elements $u$ and $v$ of $\mathcal{V}$.
Given $k \in \mathbf{N}$, give a simplified expression for $\operatorname{tr}(f_{1}^{(k+1)})$, and deduce from this the validity of Lemma A: for $u, v \in \mathcal{V}$, $\operatorname{tr}(u^{k} v) = 0$ for every natural integer $k$.
Q11 Proof Deduction or Consequence from Prior Results View
We consider an $\mathbf{R}$-vector space $E$ of dimension $n > 0$. Let $\mathcal{V}$ be a nilpotent vector subspace of $\mathcal{L}(E)$ containing a non-zero element, with generic nilindex $p := \max_{u \in \mathcal{V}} \nu(u)$. We introduce the subset $\mathcal{V}^{\bullet}$ of $E$ formed by vectors belonging to at least one of the sets $\operatorname{Im} u^{p-1}$ for $u$ in $\mathcal{V}$, and the vector subspace $K(\mathcal{V}) := \operatorname{Vect}(\mathcal{V}^{\bullet})$. In questions 8 to 11, we are given two arbitrary elements $u$ and $v$ of $\mathcal{V}$.
Let $y \in E$. Prove that $f_{1}^{(p-1)}(y) \in K(\mathcal{V})$. Using a relation between $u(f_{1}^{(p-1)}(y))$ and $v(u^{p-1}(y))$, deduce that $v(x) \in u(K(\mathcal{V}))$ for every $x \in \operatorname{Im} u^{p-1}$.
Q12 Proof Linear Transformation and Endomorphism Properties View
We consider an $\mathbf{R}$-vector space $E$ of dimension $n > 0$. Let $\mathcal{V}$ be a nilpotent vector subspace of $\mathcal{L}(E)$ containing a non-zero element, with generic nilindex $p := \max_{u \in \mathcal{V}} \nu(u)$. We introduce the subset $\mathcal{V}^{\bullet}$ of $E$ formed by vectors belonging to at least one of the sets $\operatorname{Im} u^{p-1}$ for $u$ in $\mathcal{V}$, the vector subspace $K(\mathcal{V}) := \operatorname{Vect}(\mathcal{V}^{\bullet})$, and given $x \in E$, $\mathcal{V} x := \{v(x) \mid v \in \mathcal{V}\}$.
Lemma B states: Let $x$ be in $\mathcal{V}^{\bullet} \backslash \{0\}$. If $K(\mathcal{V}) \subset \operatorname{Vect}(x) + \mathcal{V} x$, then $v(x) = 0$ for every $v$ in $\mathcal{V}$.
Let $x \in \mathcal{V}^{\bullet} \backslash \{0\}$ such that $K(\mathcal{V}) \subset \operatorname{Vect}(x) + \mathcal{V} x$. We choose $u \in \mathcal{V}$ such that $x \in \operatorname{Im} u^{p-1}$.
Given $y \in K(\mathcal{V})$, show that for every $k \in \mathbf{N}$ there exist $y_{k} \in K(\mathcal{V})$ and $\lambda_{k} \in \mathbf{R}$ such that $y = \lambda_{k} x + u^{k}(y_{k})$. Deduce that $K(\mathcal{V}) \subset \operatorname{Vect}(x)$ and then that $v(x) = 0$ for every $v \in \mathcal{V}$.
Q13 Matrices Linear Transformation and Endomorphism Properties View
We fix a real vector space $E$ of dimension $n$, as well as a nilpotent vector subspace $\mathcal{V}$ of $\mathcal{L}(E)$, equipped with an inner product $(-\mid-)$. We consider an arbitrary vector $x$ of $E \backslash \{0\}$, and set
$$H := \operatorname{Vect}(x)^{\perp}, \quad \mathcal{V} x := \{v(x) \mid v \in \mathcal{V}\} \text{ and } \mathcal{W} := \{v \in \mathcal{V} : v(x) = 0\}$$
We denote by $\pi$ the orthogonal projection of $E$ onto $H$. For $u \in \mathcal{W}$, we denote by $\bar{u}$ the endomorphism of $H$ defined by $\forall z \in H, \bar{u}(z) = \pi(u(z))$. We consider the sets $\overline{\mathcal{V}} := \{\bar{u} \mid u \in \mathcal{W}\}$ and $\mathcal{Z} := \{u \in \mathcal{W} : \bar{u} = 0\}$.
Show that $\mathcal{V} x$, $\mathcal{W}$, $\overline{\mathcal{V}}$ and $\mathcal{Z}$ are vector subspaces of $E$, $\mathcal{V}$, $\mathcal{L}(H)$ and $\mathcal{V}$ respectively.
Q14 Matrices Determinant and Rank Computation View
We fix a real vector space $E$ of dimension $n$, as well as a nilpotent vector subspace $\mathcal{V}$ of $\mathcal{L}(E)$, equipped with an inner product $(-\mid-)$. We consider an arbitrary vector $x$ of $E \backslash \{0\}$, and set
$$H := \operatorname{Vect}(x)^{\perp}, \quad \mathcal{V} x := \{v(x) \mid v \in \mathcal{V}\} \text{ and } \mathcal{W} := \{v \in \mathcal{V} : v(x) = 0\}$$
We denote by $\pi$ the orthogonal projection of $E$ onto $H$. For $u \in \mathcal{W}$, we denote by $\bar{u}$ the endomorphism of $H$ defined by $\forall z \in H, \bar{u}(z) = \pi(u(z))$. We consider the sets $\overline{\mathcal{V}} := \{\bar{u} \mid u \in \mathcal{W}\}$ and $\mathcal{Z} := \{u \in \mathcal{W} : \bar{u} = 0\}$.
Show that
$$\operatorname{dim} \mathcal{V} = \operatorname{dim}(\mathcal{V} x) + \operatorname{dim} \mathcal{Z} + \operatorname{dim} \overline{\mathcal{V}}.$$
Q15 Matrices Linear Transformation and Endomorphism Properties View
We fix a real vector space $E$ of dimension $n$, as well as a nilpotent vector subspace $\mathcal{V}$ of $\mathcal{L}(E)$, equipped with an inner product $(-\mid-)$. We consider an arbitrary vector $x$ of $E \backslash \{0\}$, and set
$$H := \operatorname{Vect}(x)^{\perp}, \quad \mathcal{V} x := \{v(x) \mid v \in \mathcal{V}\} \text{ and } \mathcal{W} := \{v \in \mathcal{V} : v(x) = 0\}$$
We denote by $\pi$ the orthogonal projection of $E$ onto $H$. For $u \in \mathcal{W}$, we denote by $\bar{u}$ the endomorphism of $H$ defined by $\forall z \in H, \bar{u}(z) = \pi(u(z))$. We consider the sets $\overline{\mathcal{V}} := \{\bar{u} \mid u \in \mathcal{W}\}$ and $\mathcal{Z} := \{u \in \mathcal{W} : \bar{u} = 0\}$. Given $a \in E$ and $x \in E$, $(a \otimes x)(z) = (a \mid z) \cdot x$ for all $z \in E$.
Show that there exists a vector subspace $L$ of $E$ such that
$$\mathcal{Z} = \{a \otimes x \mid a \in L\} \quad \text{and} \quad \operatorname{dim} L = \operatorname{dim} \mathcal{Z},$$
and show that then $x \in L^{\perp}$.
Q16 Matrices Linear Transformation and Endomorphism Properties View
We fix a real vector space $E$ of dimension $n$, as well as a nilpotent vector subspace $\mathcal{V}$ of $\mathcal{L}(E)$, equipped with an inner product $(-\mid-)$. We consider an arbitrary vector $x$ of $E \backslash \{0\}$, and set
$$H := \operatorname{Vect}(x)^{\perp}, \quad \mathcal{V} x := \{v(x) \mid v \in \mathcal{V}\} \text{ and } \mathcal{W} := \{v \in \mathcal{V} : v(x) = 0\}$$
We consider the sets $\overline{\mathcal{V}} := \{\bar{u} \mid u \in \mathcal{W}\}$ and $\mathcal{Z} := \{u \in \mathcal{W} : \bar{u} = 0\}$. Given $a \in E$ and $x \in E$, $(a \otimes x)(z) = (a \mid z) \cdot x$ for all $z \in E$. There exists a vector subspace $L$ of $E$ such that $\mathcal{Z} = \{a \otimes x \mid a \in L\}$ and $x \in L^{\perp}$.
By considering $u$ and $a \otimes x$ for $u \in \mathcal{V}$ and $a \in L$, deduce from Lemma A that $\mathcal{V} x \subset L^{\perp}$, and that more generally $u^{k}(x) \in L^{\perp}$ for every $k \in \mathbf{N}$ and every $u \in \mathcal{V}$.
Q17 Matrices Linear Transformation and Endomorphism Properties View
We fix a real vector space $E$ of dimension $n$, as well as a nilpotent vector subspace $\mathcal{V}$ of $\mathcal{L}(E)$, equipped with an inner product $(-\mid-)$. We consider an arbitrary vector $x$ of $E \backslash \{0\}$, and set
$$H := \operatorname{Vect}(x)^{\perp}, \quad \mathcal{V} x := \{v(x) \mid v \in \mathcal{V}\}$$
There exists a vector subspace $L$ of $E$ such that $\mathcal{Z} = \{a \otimes x \mid a \in L\}$, $x \in L^{\perp}$, and $\mathcal{V} x \subset L^{\perp}$.
Justify that $\lambda x \notin \mathcal{V} x$ for every $\lambda \in \mathbf{R}^{*}$, and deduce from the two previous questions that
$$\operatorname{dim} \mathcal{V} x + \operatorname{dim} L \leq n-1$$
Q18 Matrices Linear Transformation and Endomorphism Properties View
We fix a real vector space $E$ of dimension $n$, as well as a nilpotent vector subspace $\mathcal{V}$ of $\mathcal{L}(E)$, equipped with an inner product $(-\mid-)$. We consider an arbitrary vector $x$ of $E \backslash \{0\}$, and set
$$H := \operatorname{Vect}(x)^{\perp}, \quad \mathcal{W} := \{v \in \mathcal{V} : v(x) = 0\}$$
We denote by $\pi$ the orthogonal projection of $E$ onto $H$. For $u \in \mathcal{W}$, we denote by $\bar{u}$ the endomorphism of $H$ defined by $\forall z \in H, \bar{u}(z) = \pi(u(z))$. We consider the set $\overline{\mathcal{V}} := \{\bar{u} \mid u \in \mathcal{W}\}$.
Let $u \in \mathcal{W}$. Show that $(\bar{u})^{k}(z) = \pi(u^{k}(z))$ for every $k \in \mathbf{N}$ and every $z \in H$. Deduce that $\overline{\mathcal{V}}$ is a nilpotent vector subspace of $\mathcal{L}(H)$.
Q19 Matrices Linear Transformation and Endomorphism Properties View
We fix a real vector space $E$ of dimension $n \geq 2$, as well as a nilpotent vector subspace $\mathcal{V}$ of $\mathcal{L}(E)$, equipped with an inner product $(-\mid-)$. We consider an arbitrary vector $x$ of $E \backslash \{0\}$, and set $H := \operatorname{Vect}(x)^{\perp}$, $\mathcal{V} x := \{v(x) \mid v \in \mathcal{V}\}$, $\mathcal{W} := \{v \in \mathcal{V} : v(x) = 0\}$, $\overline{\mathcal{V}} := \{\bar{u} \mid u \in \mathcal{W}\}$, $\mathcal{Z} := \{u \in \mathcal{W} : \bar{u} = 0\}$, and $L$ the vector subspace such that $\mathcal{Z} = \{a \otimes x \mid a \in L\}$.
We have $\operatorname{dim} \mathcal{V} = \operatorname{dim}(\mathcal{V} x) + \operatorname{dim} \mathcal{Z} + \operatorname{dim} \overline{\mathcal{V}}$, $\operatorname{dim} \mathcal{V} x + \operatorname{dim} L \leq n-1$, $\overline{\mathcal{V}}$ is a nilpotent subspace of $\mathcal{L}(H)$ with $\dim H = n-1$, and by induction hypothesis $\operatorname{dim} \overline{\mathcal{V}} \leq \frac{(n-1)(n-2)}{2}$.
Prove that
$$\operatorname{dim} \mathcal{V} \leq \frac{n(n-1)}{2}$$
Q20 Matrices Decomposition and Basis Construction View
We fix a real vector space $E$ of dimension $n \geq 2$, as well as a nilpotent vector subspace $\mathcal{V}$ of $\mathcal{L}(E)$ with $\operatorname{dim} \mathcal{V} = \frac{n(n-1)}{2}$, equipped with an inner product $(-\mid-)$. We consider an arbitrary vector $x$ of $E \backslash \{0\}$, and set $H := \operatorname{Vect}(x)^{\perp}$, $\mathcal{V} x := \{v(x) \mid v \in \mathcal{V}\}$, $\mathcal{W} := \{v \in \mathcal{V} : v(x) = 0\}$, $\overline{\mathcal{V}} := \{\bar{u} \mid u \in \mathcal{W}\}$, $\mathcal{Z} := \{u \in \mathcal{W} : \bar{u} = 0\}$, and $L$ the vector subspace such that $\mathcal{Z} = \{a \otimes x \mid a \in L\}$.
Prove that
$$\operatorname{dim} \overline{\mathcal{V}} = \frac{(n-1)(n-2)}{2}, \quad \operatorname{dim}(\operatorname{Vect}(x) \oplus \mathcal{V} x) + \operatorname{dim} L = n$$
and
$$L^{\perp} = \operatorname{Vect}(x) \oplus \mathcal{V} x$$
Deduce that $\operatorname{Vect}(x) \oplus \mathcal{V} x$ contains $v^{k}(x)$ for every $v \in \mathcal{V}$ and every $k \in \mathbf{N}$.
Q21 Matrices Decomposition and Basis Construction View
We fix a real vector space $E$ of dimension $n \geq 2$, as well as a nilpotent vector subspace $\mathcal{V}$ of $\mathcal{L}(E)$ with $\operatorname{dim} \mathcal{V} = \frac{n(n-1)}{2}$, equipped with an inner product $(-\mid-)$. We consider an arbitrary vector $x$ of $E \backslash \{0\}$, and set $H := \operatorname{Vect}(x)^{\perp}$, $\mathcal{V} x := \{v(x) \mid v \in \mathcal{V}\}$, $\mathcal{W} := \{v \in \mathcal{V} : v(x) = 0\}$, $\overline{\mathcal{V}} := \{\bar{u} \mid u \in \mathcal{W}\}$. We have established that $\operatorname{dim} \overline{\mathcal{V}} = \frac{(n-1)(n-2)}{2}$ and $L^{\perp} = \operatorname{Vect}(x) \oplus \mathcal{V} x$.
By applying the induction hypothesis, show that the generic nilindex of $\mathcal{V}$ is greater than or equal to $n-1$, and that if moreover $\mathcal{V} x = \{0\}$ then there exists a basis of $E$ in which every element of $\mathcal{V}$ is represented by a strictly upper triangular matrix.
Q22 Matrices Subgroup and Normal Subgroup Properties View
We fix a real vector space $E$ of dimension $n \geq 2$, as well as a nilpotent vector subspace $\mathcal{V}$ of $\mathcal{L}(E)$ with $\operatorname{dim} \mathcal{V} = \frac{n(n-1)}{2}$, equipped with an inner product $(-\mid-)$. We choose $x$ in $\mathcal{V}^{\bullet} \backslash \{0\}$ (where $\mathcal{V}^{\bullet}$ is the subset of $E$ formed by vectors belonging to at least one of the sets $\operatorname{Im} u^{p-1}$ for $u$ in $\mathcal{V}$). We denote by $p$ the generic nilindex of $\mathcal{V}$ (with $p \geq n-1$ by question 21), and we fix $u \in \mathcal{V}$ such that $x \in \operatorname{Im} u^{p-1}$. We have $\mathcal{V} x := \{v(x) \mid v \in \mathcal{V}\}$ and $L^{\perp} = \operatorname{Vect}(x) \oplus \mathcal{V} x$.
Let $v \in \mathcal{V}$ such that $v(x) \neq 0$. Show that $\operatorname{Im} v^{p-1} \subset \operatorname{Vect}(x) \oplus \mathcal{V} x$. One may use the results of questions 5 and 20.