Splitting of a maximal cyclic subspace Let $V$ be a finite-dimensional vector space over $\mathbb{C}$ and let $u$ be an endomorphism of $V$. We assume that $u$ is nilpotent of index $n$, that is $u^n = 0$ and $u^{n-1} \neq 0$. We choose a vector $v_0$ such that $u^{n-1}(v_0)$ is nonzero.
a) Verify that the family $(v_0, u(v_0), \ldots, u^{n-1}(v_0))$ is free and that the subspace $W$ it spans contains $v_0$ and is stable by $u$. Write the matrix of the induced endomorphism $u_W$ in this basis.
b) Prove that there exists an injective linear application $\varphi : W \rightarrow \mathcal{D}$ such that $\xi \circ \varphi = \varphi \circ u_W$. According to Part III, this linear application $\varphi$ admits an extension $\psi : V \rightarrow \mathcal{D}$ compatible with $u$.
c) Prove that the kernel of $\psi$ is a complement of $W$ stable by $u$.

Let $u$ be a nilpotent endomorphism of a finite-dimensional vector space $V$. Prove that there exists a basis of $V$, a natural integer $s$ and nonzero natural integers $r_1 \geqslant \cdots \geqslant r_s$ in which the matrix of $u$ is block diagonal and whose diagonal blocks are Jordan blocks $J_{r_1}, \ldots, J_{r_s}$ of respective sizes $r_1, \ldots, r_s$.

Decomposition theorem: uniqueness of block sizes Prove that the number $s$ and the sizes of the blocks $r_1, \ldots, r_s$ that appear in question $9^\circ$ depend only on $u$ and not on the choice of basis. You may use question $2^\circ$.

Conversely, let $K \in \mathcal{M}_n(\mathbb{R})$ be a square matrix of rank 1. Show that there exist $\mathbf{u}, \mathbf{v} \in \mathbb{R}^n \backslash \{\mathbf{0}\}$ such that $K = \mathbf{u v}^T$.

Let $n \geq 1$ be an integer and $P \subset \mathbb{R}^n$ a polytope. Show that $P$ has a finite number of faces and at least one vertex.

Let $n \geq 1$ be an integer and $P \subset \mathbb{R}^n$ a polytope. Let $V$ be the set of vertices of $P$. Show that $P = \operatorname{Conv}(V)$.

Let $V \subset \mathbb{R}^n$ be a non-empty finite set. Justify that to prove that $\operatorname{Conv}(V)$ is a polytope it suffices to treat the case where $\operatorname{Conv}(V)$ is not contained in a hyperplane of $\mathbb{R}^n$ and contains 0 in its interior.

We choose a vector $v_0$ such that $u^{n-1}(v_0)$ is nonzero. Verify that the family $(v_0, u(v_0), \ldots, u^{n-1}(v_0))$ is free and that the subspace $W$ it spans contains $v_0$ and is stable by $u$. Write the matrix of the induced endomorphism $u_W$ in this basis.

Let $u$ be a nilpotent endomorphism of a finite-dimensional vector space $V$. Prove that there exists a basis of $V$, a natural integer $s$ and nonzero natural integers $r_1 \geqslant \cdots \geqslant r_s$ in which the matrix of $u$ is block diagonal and whose diagonal blocks are Jordan blocks $J_{r_1}, \ldots, J_{r_s}$ of respective sizes $r_1, \ldots, r_s$.

Prove that the number $s$ and the sizes of the blocks $r_1, \ldots, r_s$ that appear in question $9^\circ$ depend only on $u$ and not on the choice of basis. One may use question $2^\circ$.

Let $j$ be a natural integer strictly less than $N$. By denoting $V_j = \ker(h - \zeta^j \mathrm{id}_V)$ and $V_N = V_0$, verify that $u(V_j) \subset V_{j+1}$.

In this question, we further assume that $N = 4$ and $\ker(h - \mathrm{id}_V) = \{0\}$. Verify that $u^3 = 0$.

Let $\chi_M$ be the characteristic polynomial of $M$. We write it as $\chi_M = X^r Q$ where $r$ is an integer and $Q$ is a polynomial whose constant coefficient is nonzero. Briefly justify that $$\mathbb{C}^{m+n} = \ker M^r \oplus \ker Q(M)$$ and verify that these subspaces are stable under $H$.

Let $A$ and $B$ be two $3 \times 3$ non-zero real matrices such that $A B$ is a zero matrix. Then
(1) The system of linear equations $A X = 0$ has a unique solution
(2) The system of linear equations $A X = 0$ has infinitely many solutions
(3) $B$ is an invertible matrix
(4) $\operatorname { adj } ( A )$ is an invertible matrix

A line on a two-dimensional plane can be expressed as $\alpha x + \beta y + \gamma = 0$, where $( x , y )$ is a point on the line in the Cartesian coordinate system. We call the column vector $( \alpha , \beta , \gamma ) ^ { \mathrm { T } }$ a coefficient vector of the line. Answer the following questions. Note that the coefficient vector in your answer must satisfy $\alpha ^ { 2 } + \beta ^ { 2 } = 1$.
(1) Find a coefficient vector of a line that passes through a point $\vec { a }$ and is perpendicular to a unit vector $\vec { v }$ on a two-dimensional plane.
(2) Let a line B pass through a point $\vec { b }$ and be perpendicular to a unit vector $\vec { n }$. Given a line $A$, let the line $A ^ { \prime }$ be the mirror transformation of the line $A$ over the line $B$. Using $\vec { b }$ and $\vec { n }$, write a three-dimensional square matrix that transforms a coefficient vector of the line A to a coefficient vector of the line $\mathrm { A } ^ { \prime }$.
(3) Find the determinant of the matrix derived in Question (2).
(4) Consider the movement of the line $\mathrm { D } _ { t }$ whose coefficient vector changes with the real variable $t$ as $\left( 4 t , 4 t ^ { 2 } - 1 , t \right) ^ { \mathrm { T } }$. This line passes through a point regardless of $t$. Find the coordinate of that point.
(5) Suppose that, with the mirror transformation over a line $M _ { t }$, which also changes with $t$, the line $\mathrm { D } _ { t }$ in Question (4) is transformed to the line with a coefficient vector $( 0,1 , - t ) ^ { \mathrm { T } }$. Find the coefficient vector $\left( \alpha _ { t } , \beta _ { t } , \gamma _ { t } \right) ^ { \mathrm { T } }$ of the line $\mathrm { M } _ { t }$, where $\alpha _ { t } > 0$ and $\beta _ { t } > 0$ for $t > 0$.
(6) When $t$ changes from 0 to $+ \infty$, consider the region where the line $\mathrm { M } _ { t }$ in Question (5) can exist. Describe the region using a simple mathematical expression and draw a diagram of the region.