Let $f : \Omega \rightarrow \mathbb{R}$ be a harmonic application of class $C^2$ such that $\partial_1 f$ and $\partial_2 f$ are of class $C^2$ on $\Omega$. Show that the applications $\partial_1 f$ and $\partial_2 f$ are also harmonic on $\Omega$.

Justify that $\operatorname{Vect}\left(x, f(x), f^2(x), \ldots, f^{p-1}(x)\right)$ is stable under $f$.

Justify that $\operatorname{Vect}\left(x, f(x), f^2(x), \ldots, f^{p-1}(x)\right)$ is stable under $f$.

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.

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 $\mathcal{V}^{\bullet} \backslash \{0\}$. We denote by $p$ the generic nilindex of $\mathcal{V}$ (with $p \geq n-1$), 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 $L^{\perp} = \operatorname{Vect}(x) \oplus \mathcal{V} x$.
Suppose that there exists $v_{0}$ in $\mathcal{V}$ such that $v_{0}(x) \neq 0$. Let $v \in \mathcal{V}$. By considering $v + tv_{0}$ for $t$ real, show that $\operatorname{Im} v^{p-1} \subset \operatorname{Vect}(x) \oplus \mathcal{V} x$.

We identify $M_3(\mathbb{R})$ with the linear endomorphisms of $V$. Let $G$ be the set of endomorphisms $g$ such that $$B(gu,gv) = B(u,v)$$ for all $u,v\in V$.
Show that $G$ is a group under composition of linear maps.

Let $G$ be the group of endomorphisms $g$ of $V$ such that $B(gu,gv)=B(u,v)$ for all $u,v\in V$.
Show that for all $g\in G$, we have $g(\mathcal{H})=\mathcal{H}$ or $-g(\mathcal{H})=\mathcal{H}$.

For all $w\in V$ such that $B(w,w)>0$, define $$s_w : v \mapsto v - 2\frac{B(v,w)}{B(w,w)}w.$$ Show that $s_w \in G_0$.

For all $w\in V$ such that $B(w,w)>0$, define $$s_w : v \mapsto v - 2\frac{B(v,w)}{B(w,w)}w.$$ Show that for all $u,v\in\mathcal{H}$, there exists $w\in V$ such that $B(w,w)>0$ and $s_w(u)=v$.

If $\varphi : \mathbb{R} \rightarrow \mathbb{R}$ is a continuous function, the support of $\varphi$ is defined by: $$\operatorname{Supp}(\varphi) = \overline{\{x \in \mathbb{R} : \varphi(x) \neq 0\}}$$ We say that $\varphi$ has compact support if $\operatorname{Supp}(\varphi)$ is a bounded subset of $\mathbb{R}$. We denote by $\mathcal{C}_{c}(\mathbb{R})$ the set of continuous functions with compact support on $\mathbb{R}$.
Show that $\mathcal{C}_{c}(\mathbb{R})$ is a vector subspace of the space of continuous functions on $\mathbb{R}$.

a) Show that $N(ZZ') = N(Z)N(Z')$ for all $Z, Z' \in \mathbb{H}$. b) Show that $S$ is a subgroup of $\mathbb{H}^\times$ and that $\frac{1}{\sqrt{N(Z)}}Z \in S$ for all $Z \in \mathbb{H}^\times$.

a) Deduce that $\alpha(S \times S) = \mathrm{SO}(\mathbb{H})$. b) Show that $N := \alpha(S \times \{E\})$ is a subgroup of $\mathrm{SO}(\mathbb{H})$, then that $gng^{-1} \in N$ for all $n \in N$ and $g \in \mathrm{SO}(\mathbb{H})$ and that $\{\pm\mathrm{id}\} \subsetneq N \subsetneq \mathrm{SO}(\mathbb{H})$.

Let $A$ be a commutative ring. Let $S _ { 1 }$ and $S _ { 2 }$ be two subsets of $A$ such that $S _ { 1 } \subset \mathcal { A } \left( S _ { 2 } \right)$. Show that $\mathcal { A } \left( S _ { 1 } \right) \subset \mathcal { A } \left( S _ { 2 } \right)$.

Show that if an abelian group $M$ has property (F), then every subgroup of $M$ also has it.

Let $u = (u_k)_{k \geqslant 0}$ be a sequence of $\mathbb{C}$ such that $\mathbb{M}_n(u) \neq \emptyset$. Let $A \in \mathbb{M}_n(u)$. Show that $\mathscr{V}(A)$ is nonempty.

Let $A \in \mathscr{M}_n(\mathbb{C})$. We denote by $\mathbb{C}[A]$ the set of elements of $\mathscr{M}_n(\mathbb{C})$ of the form $P(A)$ where $P \in \mathbb{C}[X]$ is a polynomial. We denote $$(\mathbb{C}[A])^* = \left\{B \in \mathbb{C}[A] \cap \mathrm{GL}_n(\mathbb{C}) \mid B^{-1} \in \mathbb{C}[A]\right\}$$ Justify that $(\mathbb{C}[A])^*$ is an abelian subgroup of $\mathrm{GL}_n(\mathbb{C})$.

Let $A \in \mathscr{M}_n(\mathbb{C})$. We denote by $\mathbb{C}[A]$ the set of elements of $\mathscr{M}_n(\mathbb{C})$ of the form $P(A)$ where $P \in \mathbb{C}[X]$ is a polynomial. We denote $$(\mathbb{C}[A])^* = \left\{B \in \mathbb{C}[A] \cap \mathrm{GL}_n(\mathbb{C}) \mid B^{-1} \in \mathbb{C}[A]\right\}$$ Show that $(\mathbb{C}[A])^* = \mathbb{C}[A] \cap \mathrm{GL}_n(\mathbb{C})$.

Let $A \in \mathscr{M}_n(\mathbb{C})$. We denote by $\mathbb{C}[A]$ the set of elements of $\mathscr{M}_n(\mathbb{C})$ of the form $P(A)$ where $P \in \mathbb{C}[X]$ is a polynomial. We denote $$(\mathbb{C}[A])^* = \left\{B \in \mathbb{C}[A] \cap \mathrm{GL}_n(\mathbb{C}) \mid B^{-1} \in \mathbb{C}[A]\right\}$$ Show that $\exp(\mathbb{C}[A]) \subset (\mathbb{C}[A])^*$.

Let $A \in \mathscr{M}_n(\mathbb{C})$. We denote by $\mathbb{C}[A]$ the set of elements of $\mathscr{M}_n(\mathbb{C})$ of the form $P(A)$ where $P \in \mathbb{C}[X]$ is a polynomial. We denote $$(\mathbb{C}[A])^* = \left\{B \in \mathbb{C}[A] \cap \mathrm{GL}_n(\mathbb{C}) \mid B^{-1} \in \mathbb{C}[A]\right\}.$$ Justify that $\mathbb{C}[A]^*$ is an abelian subgroup of $\mathrm{GL}_n(\mathbb{C})$.

Let $A \in \mathscr{M}_n(\mathbb{C})$. We denote by $\mathbb{C}[A]$ the set of elements of $\mathscr{M}_n(\mathbb{C})$ of the form $P(A)$ where $P \in \mathbb{C}[X]$ is a polynomial. We denote $$(\mathbb{C}[A])^* = \left\{B \in \mathbb{C}[A] \cap \mathrm{GL}_n(\mathbb{C}) \mid B^{-1} \in \mathbb{C}[A]\right\}.$$ Show that $(\mathbb{C}[A])^* = \mathbb{C}[A] \cap \mathrm{GL}_n(\mathbb{C})$.

Let $A \in \mathscr{M}_n(\mathbb{C})$. We denote by $\mathbb{C}[A]$ the set of elements of $\mathscr{M}_n(\mathbb{C})$ of the form $P(A)$ where $P \in \mathbb{C}[X]$ is a polynomial. We denote $$(\mathbb{C}[A])^* = \left\{B \in \mathbb{C}[A] \cap \mathrm{GL}_n(\mathbb{C}) \mid B^{-1} \in \mathbb{C}[A]\right\}.$$ Show that $\exp(\mathbb{C}[A]) \subset (\mathbb{C}[A])^*$.