We consider an endomorphism $f$ of a $\mathbb{K}$-vector space $E$ of dimension $n \geqslant 2$ such that $f^n = 0$ and $f^{n-1} \neq 0$. III.B.1) Determine the set of vectors $u$ of $E$ such that the family $\mathcal{B}_{f,u} = (f^{n-i}(u))_{1 \leqslant i \leqslant n}$ is a basis of $E$. III.B.2) In the case where $\mathcal{B}_{f,u}$ is a basis of $E$, what is the matrix of $f$ in $\mathcal{B}_{f,u}$? III.B.3) Determine a basis of $E$ such that the matrix of $f$ in this basis is $A_{n-1}$. III.B.4) Give all subspaces of $E$ stable by $f$. How many are there? Give a simple relation between these stable subspaces and the kernels $\ker(f^i)$ for $i$ in $\llbracket 0, n \rrbracket$.
In this part, $n$ is a non-zero natural integer, $M$ is in $\mathcal{M}_n(\mathbb{R})$ and $f$ is the endomorphism of $E = \mathcal{M}_{n,1}(\mathbb{R})$ defined by $f(X) = MX$ for all $X$ in $E$. If we denote $X_i = \begin{pmatrix} \delta_{1,i} \\ \vdots \\ \delta_{n,i} \end{pmatrix}$ where $\delta_{k,l} = \begin{cases} 1 & \text{if } k = l \\ 0 & \text{if } k \neq l \end{cases}$ and $\mathcal{B}_n = (X_i)_{1 \leqslant i \leqslant n}$ the canonical basis of $E$, what is the matrix of $f$ in $\mathcal{B}_n$?
In this part, $n$ is a non-zero natural integer, $M$ is in $\mathcal{M}_n(\mathbb{R})$ and $f$ is the endomorphism of $E = \mathcal{M}_{n,1}(\mathbb{R})$ defined by $f(X) = MX$ for all $X$ in $E$. Show that if $n$ is odd, then $f$ admits at least one real eigenvalue.
In this part, $n$ is a non-zero natural integer, $M$ is in $\mathcal{M}_n(\mathbb{R})$ and $f$ is the endomorphism of $E = \mathcal{M}_{n,1}(\mathbb{R})$ defined by $f(X) = MX$ for all $X$ in $E$. In this question, $\lambda = \alpha + \mathrm{i}\beta$, with $(\alpha, \beta)$ in $\mathbb{R}^2$, is a non-real eigenvalue of $M$ and $Z$ in $\mathcal{M}_{n,1}(\mathbb{C})$, non-zero, is such that $MZ = \lambda Z$. If $M = (m_{i,j})_{\substack{1 \leqslant i \leqslant n \\ 1 \leqslant j \leqslant n}}$ we denote $\bar{M} = (m_{i,j}')_{\substack{1 \leqslant i \leqslant n \\ 1 \leqslant j \leqslant n}}$ with $m_{i,j}' = \bar{m}_{i,j}$ (conjugate of the complex number $m_{i,j}$) for all $(i,j)$ in $\llbracket 1, n \rrbracket^2$ and if $Z = \begin{pmatrix} z_1 \\ \vdots \\ z_n \end{pmatrix}$ we denote $\bar{Z} = \begin{pmatrix} z_1' \\ \vdots \\ z_n' \end{pmatrix}$ with $z_i' = \bar{z}_i$ for all $i$ in $\llbracket 1, n \rrbracket$. We set $X = \frac{1}{2}(Z + \bar{Z})$ and $Y = \frac{1}{2\mathrm{i}}(Z - \bar{Z})$. IV.C.1) Verify that $X$ and $Y$ are in $E$ and show that the family $(X, Y)$ is free in $E$. IV.C.2) Show that the vector plane $F$ generated by $X$ and $Y$ is stable by $f$ and give the matrix of $f_F$ in the basis $(X, Y)$.
In this part, $n$ is a non-zero natural integer, $M$ is in $\mathcal{M}_n(\mathbb{R})$ and $f$ is the endomorphism of $E = \mathcal{M}_{n,1}(\mathbb{R})$ defined by $f(X) = MX$ for all $X$ in $E$. What do you think of the statement: ``every endomorphism of a finite-dimensional real vector space admits at least one line or one plane stable''?
In this question we consider the linear differential system $\mathcal{S} : X' = AX$ associated with the matrix $A = \begin{pmatrix} 1 & -4 & 0 \\ 1 & -2 & -1 \\ 1 & 1 & 0 \end{pmatrix}$. We call trajectories of $\mathcal{S}$ the arcs of the space $\mathbb{R}^3$ parametrized by the solutions of $\mathcal{S}$. We want to determine the rectilinear trajectories and the planar trajectories of $\mathcal{S}$. IV.F.1) Construct an invertible matrix $P$ and a matrix $T = \begin{pmatrix} \alpha & \beta & 0 \\ -\beta & \alpha & 0 \\ 0 & 0 & \gamma \end{pmatrix}$ with $(\alpha, \beta, \gamma)$ in $(\mathbb{R}^*)^3$ such that $P^{-1}AP = T$ and determine a plane $F$ and a line $G$ stable by the endomorphism of $\mathbb{R}^3$ canonically associated with $A$ and supplementary in $\mathbb{R}^3$. IV.F.2) Determine the unique solution of the Cauchy problem $\mathcal{P}_U : \begin{cases} X' = AX \\ X(0) = U \end{cases}$ when $U$ belongs to $G$. IV.F.3) For all $\sigma = (a, b)$ in $\mathbb{R}^2$, we consider the Cauchy problem $\mathcal{C}_{\sigma} : \begin{cases} x' = -x + 2y \\ y' = -2x - y \\ x(0) = a,\ y(0) = b \end{cases}$ and $\varphi = (x, y)$ in $\mathcal{C}^1(\mathbb{R}, \mathbb{R}^2)$ the unique solution of $\mathcal{C}_{\sigma}$. Specify $x'(0)$ and $y'(0)$; show that $x$ and $y$ are solutions of the same linear homogeneous differential equation of second order with constant coefficients and thus deduce $\varphi$ as a function of $a$ and $b$. IV.F.4) Determine the rectilinear trajectories and the planar trajectories of the differential system $X' = AX$.
In this part $E$ is a real vector space of dimension $n$ equipped with a basis $\mathcal{B} = (\varepsilon_i)_{1 \leqslant i \leqslant n}$. We consider an endomorphism $f$ of $E$ and we denote by $A$ its matrix in the basis $\mathcal{B}$. V.A.1) Show that there exists a unique inner product on $E$ for which $\mathcal{B}$ is orthonormal. This inner product is denoted in the usual way by $\langle u, v \rangle$ or more simply $u \cdot v$ for all $(u, v)$ in $E^2$. V.A.2) If $u$ and $v$ are represented by the respective column matrices $U$ and $V$ in the basis $\mathcal{B}$, what simple relation exists between $u \cdot v$ and the matrix product ${}^t U V$ (where ${}^t U$ is the transpose of $U$)?
In this part $E$ is a real vector space of dimension $n$ equipped with a basis $\mathcal{B} = (\varepsilon_i)_{1 \leqslant i \leqslant n}$. We consider an endomorphism $f$ of $E$ and we denote by $A$ its matrix in the basis $\mathcal{B}$. There exists a unique inner product on $E$ for which $\mathcal{B}$ is orthonormal, denoted $\langle u, v \rangle$ or $u \cdot v$. Let $H$ be a hyperplane of $E$ and $D$ its orthogonal complement. If $(u)$ is a basis of $D$ and if $U$ is the column matrix of $u$ in $\mathcal{B}$, show that $H$ is stable by $f$ if and only if $U$ is an eigenvector of the transpose of $A$.
In this part $E$ is a real vector space of dimension $n$ equipped with a basis $\mathcal{B} = (\varepsilon_i)_{1 \leqslant i \leqslant n}$. We consider an endomorphism $f$ of $E$ and we denote by $A$ its matrix in the basis $\mathcal{B}$. Determine thus the stable plane(s) of $f$ when $n = 3$ and $A$ is the matrix $A = \begin{pmatrix} 1 & -4 & 0 \\ 1 & -2 & -1 \\ 1 & 1 & 0 \end{pmatrix}$ considered in IV.F.
In this question, $E$ is a real vector space of dimension $n$ and $f$ is an endomorphism of $E$. V.D.1) Show that if $f$ is diagonalisable then there exist $n$ hyperplanes of $E$, $(H_i)_{1 \leqslant i \leqslant n}$, all stable by $f$ such that $\bigcap_{i=1}^{n} H_i = \{0\}$. V.D.2) Is an endomorphism $f$ of $E$ for which there exist $n$ hyperplanes of $E$ stable by $f$ and with intersection reduced to the zero vector necessarily diagonalisable?