A linear transformation $T : \mathbb{R}^8 \rightarrow \mathbb{R}^8$ is defined on the standard basis $e_1, \ldots, e_8$ by $$\begin{aligned} & T e_j = e_{j+1} \quad j = 1, \ldots, 5 \\ & T e_6 = e_7 \\ & T e_7 = e_6 \\ & T e_8 = e_2 + e_4 + e_6 + e_8. \end{aligned}$$ What is the nullity of $T$?

Let $a, b \in \mathbb{R}$, and consider the $\mathbb{R}$-linear map $f : \mathbb{C} \longrightarrow \mathbb{C},\ z \mapsto az + b\bar{z}$. Choose the correct statement(s) from below:
(A) $f$ is onto (i.e., surjective) if $ab \neq 0$;
(B) $f$ is one-one (i.e., injective) if $ab \neq 0$;
(C) $f$ is onto if $a^2 \neq b^2$;
(D) if $a^2 = b^2$, $f$ is not one-one.

Let $V$ be a subspace of the complex vector space $M_n(\mathbb{C})$. Suppose that every non-zero element of $V$ is an invertible matrix. Show that $\dim_{\mathbb{C}} V \leq 1$.

(A) (5 marks) Let $n \geq 2$ be an integer. Let $V$ be the $\mathbb { R }$-vector-space of homogeneous real polynomials in three variables $X , Y , Z$ of degree $n$. Let $p = ( 1,0,0 )$. Let
$$W = \left\{ f \in V \left\lvert \, f ( p ) = \frac { \partial f } { \partial X } ( p ) \right. \right\}$$
Determine the dimension of $V / W$.
(B) (5 marks) A linear transformation $T : \mathbb { R } ^ { 9 } \longrightarrow \mathbb { R } ^ { 9 }$ is defined on the standard basis $e _ { 1 } , \ldots , e _ { 9 }$ by
$$\begin{aligned} & T e _ { i } = e _ { i - 1 } , \quad i = 3 , \ldots , 9 \\ & T e _ { 2 } = e _ { 3 } \\ & T e _ { 1 } = e _ { 1 } + e _ { 3 } + e _ { 8 } . \end{aligned}$$
Determine the nullity of $T$.

Denote by $V$ the $\mathbb { Q }$-vector-space $\mathbb { Q } [ X ]$ (polynomial ring in one variable $X$ ). Show that $V ^ { * }$ is not isomorphic to $V$.

We consider the matrix with real coefficients $C \in \mathscr { M } _ { 7 } ( \mathbb { R } )$
$$C = \left( \begin{array} { l l l l l l l } 0 & 0 & \mathbf { 1 } & \mathbf { 1 } & 0 & 0 & 0 \\ 0 & \mathbf { 1 } & 0 & 0 & \mathbf { 1 } & 0 & 0 \\ \mathbf { 1 } & 0 & 0 & 0 & 0 & 0 & 0 \\ \mathbf { 1 } & 0 & 0 & 0 & 0 & 0 & 0 \\ \mathbf { 1 } & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & \mathbf { 1 } & 0 & 0 & \mathbf { 1 } & 0 & 0 \\ 0 & 0 & \mathbf { 1 } & \mathbf { 1 } & 0 & 0 & 0 \end{array} \right)$$
We denote $(e_1, e_2, e_3, e_4, e_5, e_6, e_7)$ the canonical basis of $\mathbb{R}^7$, and $c$ the endomorphism of $\mathbb{R}^7$ whose matrix in the canonical basis is $C$. We denote $f_1, f_2, f_3, f_4, f_5, f_6, f_7$ the column vectors of the matrix $C$.
Determine a basis of the kernel and a basis of the image of $c$, as well as the rank of $c$.

We consider the matrix with real coefficients $C \in \mathscr { M } _ { 7 } ( \mathbb { R } )$
$$C = \left( \begin{array} { l l l l l l l } 0 & 0 & \mathbf { 1 } & \mathbf { 1 } & 0 & 0 & 0 \\ 0 & \mathbf { 1 } & 0 & 0 & \mathbf { 1 } & 0 & 0 \\ \mathbf { 1 } & 0 & 0 & 0 & 0 & 0 & 0 \\ \mathbf { 1 } & 0 & 0 & 0 & 0 & 0 & 0 \\ \mathbf { 1 } & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & \mathbf { 1 } & 0 & 0 & \mathbf { 1 } & 0 & 0 \\ 0 & 0 & \mathbf { 1 } & \mathbf { 1 } & 0 & 0 & 0 \end{array} \right)$$
We denote $c$ the endomorphism of $\mathbb{R}^7$ whose matrix in the canonical basis is $C$, and $f_1, f_2, f_3, f_4, f_5, f_6, f_7$ the column vectors of the matrix $C$. We denote $F$ the vector subspace of $\mathbb{R}^7$ spanned by the first three column vectors $f_1, f_2$ and $f_3$ of $C$.
I.B.1) Show that $F$ is stable under $c$. I.B.2) Show that $(f_1, f_2, f_3)$ is a basis of $F$, and calculate the matrix $\Phi$ in this basis of the endomorphism $\varphi$ of $F$ induced by $c$.

We consider the matrix with real coefficients $C \in \mathscr { M } _ { 7 } ( \mathbb { R } )$
$$C = \left( \begin{array} { l l l l l l l } 0 & 0 & \mathbf { 1 } & \mathbf { 1 } & 0 & 0 & 0 \\ 0 & \mathbf { 1 } & 0 & 0 & \mathbf { 1 } & 0 & 0 \\ \mathbf { 1 } & 0 & 0 & 0 & 0 & 0 & 0 \\ \mathbf { 1 } & 0 & 0 & 0 & 0 & 0 & 0 \\ \mathbf { 1 } & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & \mathbf { 1 } & 0 & 0 & \mathbf { 1 } & 0 & 0 \\ 0 & 0 & \mathbf { 1 } & \mathbf { 1 } & 0 & 0 & 0 \end{array} \right)$$
and $c$ the endomorphism of $\mathbb{R}^7$ whose matrix in the canonical basis is $C$.
Notation: if $f$ is a function of class $C^1$ from an open set $\mathscr{U}$ of $\mathbb{R}^d$ ($d \geqslant 1$) to $\mathbb{R}$, we denote, for every integer $i$ such that $1 \leqslant i \leqslant d$, $\partial_i f$ the partial derivative of $f$ with respect to its $i$-th variable.
In this section, we propose to study functions $f$ of class $C^1$ from $\mathbb{R}^7$ to $\mathbb{R}$ that satisfy the condition $f \circ c = f$, that is, such that $$f\left(x_3 + x_4, x_2 + x_5, x_1, x_1, x_1, x_2 + x_5, x_3 + x_4\right) = f\left(x_1, x_2, x_3, x_4, x_5, x_6, x_7\right)$$ for all $(x_1, x_2, x_3, x_4, x_5, x_6, x_7) \in \mathbb{R}^7$.
I.E.1) What structure does the set $\mathscr{S}$ of functions $f$ of class $C^1$ from $\mathbb{R}^7$ to $\mathbb{R}$ such that $f \circ c = f$ possess? I.E.2) Show that such a function satisfies $f \circ c^n = f$ for every integer $n \geqslant 1$. I.E.3) Let $f \in \mathscr{S}$. Calculate the Jacobian matrix of $f \circ c$ at $X = (x_1, x_2, x_3, x_4, x_5, x_6, x_7)$. Deduce a system of equations relating the partial derivatives $\partial_1 f(X), \ldots, \partial_7 f(X)$ of $f$ at a point $X$ of $\mathbb{R}^7$. I.E.4) For $f \in \mathscr{S}$, calculate the Jacobian matrix of $f \circ c^2$ at $X = (x_1, x_2, x_3, x_4, x_5, x_6, x_7)$. Complete the system of equations relating the partial derivatives $\partial_1 f(X), \ldots, \partial_7 f(X)$ of $f$ at a point $X$ of $\mathbb{R}^7$ obtained in the previous question. I.E.5) Application: without further calculation, determine the linear forms $f$ on $\mathbb{R}^7$ that belong to $\mathscr{S}$.

Show that $\mathcal { M } _ { 0 } ( n , \mathbb { K } )$ is a $\mathbb { K }$-vector space; specify its dimension.

Show that the application $$\begin{aligned} j : & \mathbb { K } ^ { 3 } \longrightarrow \mathcal { M } _ { 0 } ( 2 , \mathbb { K } ) \\ \left( \begin{array} { l } x \\ y \\ z \end{array} \right) & \longmapsto \left( \begin{array} { c c } x & y + z \\ y - z & - x \end{array} \right) \end{aligned}$$ is an isomorphism of $\mathbb { K }$-vector spaces.

Let $A , B$ and $M$ be three elements of $\mathcal { M } _ { 0 } ( 2 , \mathbb { K } )$.
Prove that the matrix $M$ is nilpotent if and only if the trace of the matrix $M ^ { 2 }$ is zero.

Let $A , B$ and $M$ be three elements of $\mathcal { M } _ { 0 } ( 2 , \mathbb { K } )$. We assume that the matrices $A$ and $[ A , B ]$ commute.
Prove that the matrix $[ A , B ]$ is nilpotent.

Let $X, H, Y$ be three elements of $\mathcal { M } _ { 0 } ( 2 , \mathbb { K } )$ such that $(X, H, Y)$ forms an admissible triple. Set $P = QT$ as defined in question II.G.4. Let $Y ^ { \prime } \in \mathcal { M } _ { 0 } ( 2 , \mathbb { K } )$ such that $( X , H , Y ^ { \prime } )$ is an admissible triple.
a) Deduce from question II.G.1 the matrices of $\mathcal { M } _ { 0 } ( 2 , \mathbb { K } )$ that commute with $X$.
b) Compute the matrices $\Phi _ { X } \left( Y - Y ^ { \prime } \right)$ and $\Phi _ { H } \left( Y - Y ^ { \prime } \right)$.
c) Deduce that we have $Y ^ { \prime } = Y$.

$\mathcal{P}$ denotes the vector space of polynomial functions with complex coefficients, with inner product $\langle P, Q \rangle = \displaystyle\int_0^{+\infty} e^{-t}\bar{P}(t)Q(t)\,dt$.
We denote by $D$ the differentiation endomorphism and $U$ the endomorphism of $\mathcal{P}$ defined by $$U(P)(t) = e^t D\left(te^{-t}P^{\prime}(t)\right).$$
Verify that $U$ is an endomorphism of $\mathcal{P}$.

Let $x$ be a non-zero linear recurrent sequence, of order $m \geqslant 1$. Let $p = \operatorname{rang}(H_m(x))$. For any integer $n$ in $\mathbb{N}^*$, we denote by $H_n(x)$ the matrix in $\mathcal{M}_n(\mathbb{K})$ defined by $\forall (i,j) \in \{1,\ldots,n\}^2, [H_n(x)]_{i,j} = x_{i+j-2}$.
Show that $x$ is of minimal order $p$ and that the kernel of $H_{p+1}(x)$ is a one-dimensional vector space whose a direction vector can be written $(b_0, \ldots, b_{p-1}, 1)$, where $b_0, \ldots, b_{p-1}$ are in $\mathbb{K}$.

Let $B \in M_3(\mathbb{R})$ be antisymmetric.
a) Show that $\det B = 0$.
b) Show that $\left(\operatorname{Ker} u_B\right)^\perp$ is stable under $u_B$.
c) Deduce that $B$ has rank 0 or 2.

Let $A \in M_p(\mathbb{C})$ and $k \in \mathbb{N}^*$ such that $A^k = 0$ and $A^{k-1} \neq 0$ (we say that $A$ is nilpotent of order $k$).
Show that, for every integer $j$ such that $1 \leqslant j \leqslant k$, $\operatorname{Ker} A^{j-1}$ is strictly included in $\operatorname{Ker} A^j$.

Let $A \in M_p(\mathbb{C})$ and $k \in \mathbb{N}^*$ such that $A^k = 0$ and $A^{k-1} \neq 0$ (we say that $A$ is nilpotent of order $k$).
Deduce that $k \leqslant p$.

Let $A \in M_p(\mathbb{C})$ be nilpotent of order $k$.
Show that $E(A) - I_p$ is nilpotent.

Let $A \in M_p(\mathbb{C})$. Let $k \in \mathbb{N}^*$ and $\lambda_1, \lambda_2, \ldots, \lambda_k$ be the roots of $\chi_A$ pairwise distinct, whose respective multiplicities we denote by $n_1, n_2, \ldots, n_k$.
For every integer $q$ between 1 and $p$, we denote by $J_q$ the matrix of $M_q(\mathbb{C})$ whose coefficients are all zero except those located just above the diagonal which equal 1.
Show that, for every $x \in \mathbb{C}$, for every integer $q$ between 1 and $p$, the family $\left\{\left(xI_q + J_q\right)^i,\ 0 \leqslant i \leqslant q-1\right\}$ is free.

Let $A \in M_p(\mathbb{C})$ and $B = \operatorname{diag}\left\{\lambda_1 I_{n_1} + J_{n_1}, \ldots, \lambda_k I_{n_k} + J_{n_k}\right\}$ as defined in V.A.4. Let $P$ be a non-zero annihilating polynomial of the matrix $B$.
a) Show that the degree of $P$ is $\geqslant p$.
b) Deduce that the family $\left\{B^i,\ 0 \leqslant i \leqslant p-1\right\}$ is free.

We assume that the conditions of question 12 are satisfied and that $\lambda(0) \neq 0$. Show that $E \circ F = F \circ E + H - H^{-1}$ if and only if for all $i \in \mathbf{Z}$, $$\mu(i) = \mu(i-1) + \lambda(0) q^{-2i} - \lambda(0)^{-1} q^{2i}$$

We assume that the conditions of questions 12 and 14 are satisfied and that $\lambda(0) \neq 0$.
15a. Show that $\lambda$ and $\mu$ are periodic on $\mathbf{Z}$, with periods dividing $\ell$.
15b. Show that the period of $\lambda$ is equal to $\ell$.
15c. Show that the period of $\mu$ is also equal to $\ell$.

Let $\ell, W_{\ell}, a, P_a$ be as in Part II. We say that an element $\phi$ of $\mathcal{L}(V)$ is compatible with $P_a$ if $P_a \circ \phi \circ P_a = P_a \circ \phi$.
17a. Show that if $\phi \in \mathcal{L}(V)$ commutes with $P_a$, then $\phi$ is compatible with $P_a$.
17b. Show that $H$ and $H^{-1}$ are compatible with $P_a$.

Let $\mathcal{U}_q$ be the set of endomorphisms $\phi \in \mathcal{L}(V)$ that are compatible with $P_a$. Show that $E \in \mathcal{U}_q$ and $F \in \mathcal{U}_q$.