Let $W_{\ell} = \bigoplus_{0 \leq i < \ell} \mathbf{C} v_i$ and $a \in \mathbf{C}^*$. We define a linear map $P_a : V \rightarrow V$ by $P_a(v_i) = a^p v_r$ where for $i \in \mathbf{Z}$, we define $r$ and $p$ respectively as the remainder and quotient of the Euclidean division of $i$ by $\ell$; in other words, $i = p\ell + r$ where $0 \leq r < \ell$ and $p \in \mathbf{Z}$. Show that $P_a$ is a projector with image $W_{\ell}$.

Is the application $\Phi : \left\{ \begin{array}{cll} G & \rightarrow & \mathbb{R}^2 \\ M(A, \vec{b}) & \mapsto & \vec{b} \end{array} \right.$ surjective? Is it injective?

If $f$ is a function defined on $\mathbb{R}^2$, we denote by $f^*$ the function $f \circ \Phi$, defined on $G$ by $f^*(g) = f(\Phi(g))$ where $\Phi : G \rightarrow \mathbb{R}^2$ is the function introduced in question I.A.5.
Prove that for all $g$ in $G$ and $r$ such that $\Phi(r) = \overrightarrow{0}$ we have $f^*(gr) = f^*(g)$.

We define $\hat{f}^\star$ on $G$ by composing $\hat{f}$ with $\Psi$: we set, for all $g \in G$, $\hat{f}^\star(g) = \hat{f}(\Psi(g))$.
Demonstrate that $\hat{f}^\star$ is $H$-invariant, that is, for all $g \in G$ and $h \in H$, $\hat{f}^\star(gh) = \hat{f}^\star(g)$.

After justifying the existence of the suprema, show that: $$\sup _ { \substack { x \in E \\ x \neq 0 } } \frac { \| u ( x ) \| } { \| x \| } = \sup _ { \substack { x \in E \\ \| x \| = 1 } } \| u ( x ) \| .$$

Let $A \in \mathscr{M}_{N}(\mathbf{R})$. Show that $A$ satisfies $(M_2)$ if and only if $AU = U$. Deduce that if $A$ and $B$ are two Markov kernels then $AB$ is also a Markov kernel.

Let $r$ and $s$ be strictly positive integers. Let $M \in M _ { s , r } ( A )$. We consider the application $u : A ^ { r } \rightarrow A ^ { s }$ defined by $u ( X ) = M X$, where we identify elements of $A ^ { r }$ and $A ^ { s }$ with column vectors. We assume that $u$ is surjective and that the ring $A$ is not reduced to $\{ 0 \}$. The purpose of this question is to prove that we then have $r \geq s$. For this, we reason by contradiction by assuming $r < s$. a) Show that there exists a matrix $N \in M _ { r , s } ( A )$ such that $M N = I _ { s }$. b) We define matrices of $M _ { s } ( A )$ by blocks: $$\begin{aligned} M _ { 1 } & = \left( \begin{array} { l l } M & 0 \end{array} \right) \\ N _ { 1 } & = \binom { N } { 0 } \end{aligned}$$ In other words, $M _ { 1 }$ is the matrix obtained by adding $s - r$ zero columns to $M$ and $N _ { 1 }$ is the matrix obtained by adding $s - r$ zero rows to $N$. Calculate $M _ { 1 } N _ { 1 }$. c) Reach a contradiction and conclude. d) We assume that $r = s$. Show the equivalence of the following properties: i) The application $u$ is surjective; ii) The determinant $\operatorname { det } M$ belongs to $A ^ { * }$; iii) There exists $N \in M _ { r } ( A )$ such that $M N = N M = I _ { r }$. iv) The application $u$ is bijective.

Let $C \subset E$ be a convex set. Let $f$ and $g$ be two convex functions from $C$ to $\mathbb{R}$.
(a) Show that $f + g$ is convex, and strictly convex if one of the two functions $f$ or $g$ is strictly convex.
(b) Assume $f$ is strictly convex. Verify that the minimum of $f$ is attained on $C$ at most at one point of $C$.

We consider $n$ a strictly positive integer and $\mathscr{E}_{d}^{n}(\mathbb{R}) = \{ \boldsymbol{z} = (\boldsymbol{z}_{i})_{1 \leqslant i \leqslant n} \mid \boldsymbol{z}_{i} \in \mathbb{R}^{d}, 1 \leqslant i \leqslant n \}$ the vector space of families of $n$ points in $\mathbb{R}^{d}$ equipped with the norm $\|\boldsymbol{z}\| = \sqrt{\sum_{i=1}^{n} |\boldsymbol{z}_{i}|^{2}}$. For all $g \in \operatorname{Dep}(\mathbb{R}^{d})$ and $\boldsymbol{z} \in \mathscr{E}_{d}^{n}(\mathbb{R})$ we denote $g \cdot \boldsymbol{z} = (\phi_{g}(\boldsymbol{z}_{i}))_{1 \leqslant i \leqslant n}$.

• [(a)] Show that for all $g, g^{\prime} \in \operatorname{Dep}(\mathbb{R}^{d})$ and $\boldsymbol{z} \in \mathscr{E}_{d}^{n}(\mathbb{R})$, we have $g \cdot (g^{\prime} \cdot \boldsymbol{z}) = (gg^{\prime}) \cdot \boldsymbol{z}$.
• [(b)] Show that for all $\boldsymbol{x}, \boldsymbol{y} \in \mathscr{E}_{d}^{n}(\mathbb{R})$ and all $g \in \operatorname{Dep}(\mathbb{R}^{d})$, if $\boldsymbol{x} = g \cdot \boldsymbol{y}$ then $\boldsymbol{y} = g^{-1} \cdot \boldsymbol{x}$.

We consider $n$ a strictly positive integer and $\mathscr{E}_{d}^{n}(\mathbb{R}) = \{ \boldsymbol{z} = (\boldsymbol{z}_{i})_{1 \leqslant i \leqslant n} \mid \boldsymbol{z}_{i} \in \mathbb{R}^{d}, 1 \leqslant i \leqslant n \}$. For all $\boldsymbol{x} \in \mathscr{E}_{d}^{n}(\mathbb{R})$, we denote $c(\boldsymbol{x}) = \{ \boldsymbol{y} \in \mathscr{E}_{d}^{n}(\mathbb{R}) \mid \exists g \in \operatorname{Dep}(\mathbb{R}^{d}), g \cdot \boldsymbol{x} = \boldsymbol{y} \}$.

• [(a)] Show that if $c(\boldsymbol{x}) \cap c(\boldsymbol{y}) \neq \emptyset$ then $c(\boldsymbol{x}) = c(\boldsymbol{y})$.
• [(b)] Show that if $c(\boldsymbol{x}) = c(\boldsymbol{y})$ then $\delta(\boldsymbol{x}, \boldsymbol{y}) = 0$.