Let $G$ be a group $\mathbb { F }$ a field and $n$ a positive integer. A linear action of $G$ on $\mathbb { F } ^ { n }$ is a map $\alpha : G \times \mathbb { F } ^ { n } \rightarrow \mathbb { F } ^ { n }$ such that $\alpha ( g , v ) = \rho ( g ) v$ for some group homomorphism $\rho : G \rightarrow \mathrm { GL } _ { n } ( \mathbb { F } )$. Show that for every finite group $G$, there is an $n$ such that there is a linear action $\alpha$ of $G$ on $\mathbb { F } ^ { n }$ and such that there is a nonzero vector $v \in \mathbb { F } ^ { n }$ such that $\alpha ( g , v ) = v$ for all $g \in G$.

Let $$G = \left\{\left(\begin{array}{cc}a & b \\ 0 & a^{-1}\end{array}\right) : a, b \in \mathbb{R}, a > 0\right\}, \quad N = \left\{\left(\begin{array}{cc}1 & b \\ 0 & 1\end{array}\right) : b \in \mathbb{R}\right\}.$$ Which of the following are true?
(A) $G/N$ is isomorphic to $\mathbb{R}$ under addition.
(B) $G/N$ is isomorphic to $\{a \in \mathbb{R} : a > 0\}$ under multiplication.
(C) There is a proper normal subgroup $N'$ of $G$ which properly contains $N$.
(D) $N$ is isomorphic to $\mathbb{R}$ under addition.

Write $V$ for the space of $3 \times 3$ skew-symmetric real matrices.
(A) Show that for $A \in SO_3(\mathbb{R})$ and $M \in V$, $AMA^t \in V$. Write $A \cdot M$ for this action.
(B) Let $\Phi : \mathbb{R}^3 \longrightarrow V$ be the map $$\begin{bmatrix} u \\ v \\ w \end{bmatrix} \mapsto \begin{bmatrix} 0 & w & -v \\ -w & 0 & u \\ v & -u & 0 \end{bmatrix}$$ With the usual action of $SO_3(\mathbb{R})$ on $\mathbb{R}^3$ and the above action on $V$, show that $\Phi(Av) = A \cdot \Phi(v)$ for every $A \in SO_3(\mathbb{R})$ and $v \in \mathbb{R}^3$.
(C) Show that there does not exist $M \in V$, $M \neq 0$ such that for every $A \in SO_3(\mathbb{R})$, $A \cdot M$ belongs to the span of $M$.

Let $h \in \mathscr{L}(E)$ be an endomorphism of $E$. Show that the following properties are equivalent: i) $h$ is an element of $\operatorname{Sim}(E)$; ii) $h^{*}h$ is collinear to $\operatorname{Id}_{E}$; iii) the matrix of $h$ in an orthonormal basis of $E$ is collinear to an orthogonal matrix.

In this question, the space $E$ has dimension $n = 3$. Let $(e _ { 1 } , e _ { 2 } , e _ { 3 })$ be an orthonormal basis of $E$ and $\mathcal { R } _ { 0 } = \left\{ e _ { i } - e _ { j } \mid 1 \leq i , j \leq 3 , i \neq j \right\}$.
Draw graphically $\mathcal { R } _ { 0 }$ in the plane $\operatorname { Vect } \left( \mathcal { R } _ { 0 } \right)$. Recognize one of the root systems represented in question I.D.2.

Let $D \in M_p(\mathbb{K})$ be a diagonal matrix. Let $(\Delta, +)$ be the additive subgroup of $M_p(\mathbb{R})$ formed by diagonal matrices.
Show that $E$ defines a group morphism from $(\Delta, +)$ to $(GL_p(\mathbb{R}), \times)$.

We assume that the conditions of questions 4 and 5 are satisfied and that $\lambda(0) = 0, \mu(0) = 1$.
7a. Show that $F \in \mathrm{GL}(V)$.
7b. Show that $E$ and $F$ are not of finite order in the group $\mathrm{GL}(V)$.
7c. Calculate the kernel of $H$ and show that $H^r \neq \operatorname{Id}_V$ for $r \geq 1$.

Let $\lambda, \mu \in \mathbf{C}^{\mathbf{Z}}$ and $H \in \mathcal{L}(V)$ defined by $H(v_i) = \lambda(i) v_i$. Show that $H \circ E = q^2 E \circ H$ if and only if for all $i \in \mathbf{Z}, \lambda(i) = \lambda(0) q^{-2i}$.

Verify that the map which associates to every real $t$ the matrix $R_t$ is a surjective homomorphism from the group $(\mathbb{R},+)$ onto the group $(\mathrm{SO}(2),\times)$. Is this homomorphism bijective?

Show that, for all $t$ in $\mathbb{R}$ and all non-zero $u$ in $\mathbb{R}^2$, $t$ is a measure of the oriented angle $(u\widehat{\rho_t(u)})$, where $\rho_t$ is the endomorphism (the rotation of angle $t$) $f_{R_t}$ canonically associated with $R_t$.

Show that for all $A$ in $\mathcal{M}_n(\mathbb{R})$ we have $A$ dos $A$, that for all $(A,B)$ in $\mathcal{M}_n(\mathbb{R})^2$ if $A$ dos $B$ then $B$ dos $A$, and that for all $(A,B,C)$ in $\mathcal{M}_n(\mathbb{R})^3$ if $A$ dos $B$ and $B$ dos $C$ then $A$ dos $C$.

What are the matrices directly orthogonally similar to $\alpha I_n$ for $\alpha$ real?

What are the matrices directly orthogonally similar to $A$ if $A$ belongs to $\mathrm{SO}(2)$?

What are the matrices directly orthogonally similar to $K_2 = \left(\begin{array}{rr} 1 & 0 \\ 0 & -1 \end{array}\right)$?

Show that $\left(\begin{array}{ll} 0 & 0 \\ 0 & 2 \end{array}\right)$ and $\left(\begin{array}{ll} 1 & 1 \\ 1 & 1 \end{array}\right)$ are directly orthogonally similar.

Show that $\left(\begin{array}{ll} 1 & 0 \\ 0 & 2 \end{array}\right)$ and $\left(\begin{array}{rr} 3 & 2 \\ -1 & 0 \end{array}\right)$ are similar but are not orthogonally similar.

Show that $\left(\begin{array}{rr} 3 & 2 \\ -1 & 0 \end{array}\right)$ and its transpose are orthogonally similar but are not directly orthogonally similar.

We equip $\mathbb{C}[X]$ with the internal composition law given by composition, denoted $\circ$. We seek families $(F_n)_{n \in \mathbb{N}}$ of complex polynomials satisfying $$\forall n \in \mathbb{N}, \quad \deg F_n = n \quad \text{and} \quad \forall (m,n) \in \mathbb{N}^2, \quad F_n \circ F_m = F_m \circ F_n \tag{III.1}$$
Show that the family $(T_n)_{n \in \mathbb{N}}$ satisfies property (III.1). One may compare $T_n \circ T_m$ and $T_{mn}$.

We return to the case where $\mathbb { K }$ is an arbitrary field of characteristic zero. Let $\left( V _ { i } \right) _ { 1 \leq i \leq 3 }$ be three $\mathbb { K }$-vector spaces of finite dimension and $q _ { i } \in \mathcal { Q } \left( V _ { i } \right)$ for $1 \leq i \leq 3$ satisfying $q _ { 1 } \perp q _ { 3 } \cong q _ { 2 } \perp q _ { 3 }$. Show that $q _ { 1 } \cong q _ { 2 }$.
Hint: one may reason by induction and use questions 17 and 18.

We consider the set of square matrices of size 3 that are strictly upper triangular: $$\mathbf{L} = \left\{ M_{p,q,r} \mid (p,q,r) \in \mathbf{R}^3 \right\} \quad \text{where} \quad M_{p,q,r} = \begin{pmatrix} 0 & p & r \\ 0 & 0 & q \\ 0 & 0 & 0 \end{pmatrix},$$ and the group law $M * N = M + N + \frac{1}{2}[M,N]$ on $\mathbf{L}$.
Show that for all matrices $M, N \in \mathbf{L}$, we have: $$(\exp M) \times (\exp N) = \exp(M * N)$$

We consider the set of square matrices of size 3 that are strictly upper triangular: $$\mathbf{L} = \left\{ M_{p,q,r} \mid (p,q,r) \in \mathbf{R}^3 \right\} \quad \text{where} \quad M_{p,q,r} = \begin{pmatrix} 0 & p & r \\ 0 & 0 & q \\ 0 & 0 & 0 \end{pmatrix},$$ and the group law $M * N = M + N + \frac{1}{2}[M,N]$ on $\mathbf{L}$.
Let $M$ and $N$ be two elements of $\mathbf{L}$. Show that $$\exp([M,N]) = \exp(M)\exp(N)\exp(-M)\exp(-N)$$

Let $E$ be the set of continuous functions from $[0,1]$ to $\mathbb{R}$ equipped with the norm $\|.\|_{\infty}$: $$\|f\|_{\infty} = \max_{x \in [0,1]} |f(x)|$$ We denote by $T$ the application defined on $E$ such that: $$\forall f \in E, \quad \forall x \in [0,1], \quad T(f)(x) = x f\left(\frac{x}{2}\right)$$ Show that $T \in \mathcal{L}(E)$.

Let $E$ be the set of continuous functions from $[0,1]$ to $\mathbb{R}$ equipped with the norm $\|.\|_{\infty}$: $$\|f\|_{\infty} = \max_{x \in [0,1]} |f(x)|$$ We denote by $T$ the application defined on $E$ such that: $$\forall f \in E, \quad \forall x \in [0,1], \quad T(f)(x) = x f\left(\frac{x}{2}\right)$$ Calculate the minimal possible value for the constant $M$ in the relation $\exists M \geq 0, \forall f \in E, \|T(f)\|_E \leq M\|f\|_E$.

Let $E$ be the set of continuous functions from $[0,1]$ to $\mathbb{R}$ equipped with the norm $\|.\|_{\infty}$: $$\|f\|_{\infty} = \max_{x \in [0,1]} |f(x)|$$ We denote by $T$ the application defined on $E$ such that: $$\forall f \in E, \quad \forall x \in [0,1], \quad T(f)(x) = x f\left(\frac{x}{2}\right)$$ Determine $\operatorname{Ker}(T)$ and $\operatorname{Im}(T)$.

Let $E$ be the set of continuous functions from $[0,1]$ to $\mathbb{R}$ equipped with the norm $\|.\|_2$: $$\|f\|_2 = \sqrt{\int_0^1 |f(x)|^2 \, dx}$$ We denote by $T$ the application defined on $E$ such that: $$\forall f \in E, \quad \forall x \in [0,1], \quad T(f)(x) = x f\left(\frac{x}{2}\right)$$ Show that $T \in \mathcal{L}(E)$ with this norm.