Exercise 4 (5 points) -- Candidates who have chosen the specialization option
A software allows transforming a rectangular element of a photograph. Thus, the initial rectangle OEFG is transformed into a rectangle $OE'F'G'$, called the image of OEFG.
The purpose of this exercise is to study the rectangle obtained after several successive transformations.
Part A
The plane is referred to an orthonormal coordinate system $(O, \vec{\imath}, \vec{\jmath})$. The points E, F and G have coordinates respectively $(2; 2)$, $(-1; 5)$ and $(-3; 3)$. The software transformation associates with any point $M(x; y)$ of the plane the point $M'(x'; y')$, image of point $M$ such that: $$\left\{\begin{aligned} x' &= \frac{5}{4}x + \frac{3}{4}y \\ y' &= \frac{3}{4}x + \frac{5}{4}y \end{aligned}\right.$$

1. a. Calculate the coordinates of points $E'$, $F'$ and $G'$, images of points E, F and G by this transformation. b. Compare the lengths OE and $OE'$ on one hand, OG and $OG'$ on the other hand.

Give the square matrix of order 2, denoted $A$, such that: $\binom{x'}{y'} = A\binom{x}{y}$.
Part B
In this part, we study the coordinates of the successive images of vertex F of rectangle OEFG when the software transformation is applied multiple times.

1. Consider the algorithm intended to display the coordinates of these successive images.
2. a. Prove that, for every natural integer $n$, the point $E_{n}$ is located on the line with equation $y = x$. One may use the fact that, for every natural integer $n$, the coordinates $(x_{n}; y_{n})$ of point $E_{n}$ satisfy: $$\binom{x_{n}}{y_{n}} = A^{n}\binom{2}{2}$$ b. Prove that the length $\mathrm{O}E_{n}$ tends towards $+\infty$ when $n$ tends towards $+\infty$.

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 $A$ be a $2 \times 2$ orthogonal matrix such that $\operatorname{det}(A) = -1$. Show that $A$ represents reflection about a line in $\mathbb{R}^2$.

On the coordinate plane, three points $\mathrm { A } ( 3,0 ) , \mathrm { B } ( 3,3 ) , \mathrm { C } ( 0,3 )$ are mapped to $\mathrm { A } ^ { \prime } , \mathrm { B } ^ { \prime } , \mathrm { C } ^ { \prime }$ respectively by the linear transformation represented by the matrix $\left( \begin{array} { l l } k & 0 \\ 0 & k \end{array} \right) ( k > 1 )$. When the area of the common part of the interior of triangle ABC and the interior of triangle $\mathrm { A } ^ { \prime } \mathrm { B } ^ { \prime } \mathrm { C } ^ { \prime }$ is $\frac { 1 } { 2 }$, what is the value of $k$? [3 points]
(1) $\frac { 3 } { 2 }$
(2) $\frac { 5 } { 3 }$
(3) $\frac { 7 } { 4 }$
(4) $\frac { 9 } { 5 }$
(5) $\frac { 11 } { 6 }$

On the coordinate plane, a rotation transformation $f$ centered at the origin maps the point $( 1,0 )$ to a point $\left( \frac { \sqrt { 3 } } { 2 } , a \right)$ in the first quadrant. When the sum of all components of the matrix representing the rotation transformation $f$ is $b$, what is the value of $a ^ { 2 } + b ^ { 2 }$? [3 points]
(1) $\frac { 31 } { 12 }$
(2) $\frac { 11 } { 4 }$
(3) $\frac { 35 } { 12 }$
(4) $\frac { 37 } { 12 }$
(5) $\frac { 13 } { 4 }$

In the coordinate plane, let $f$ be the rotation transformation that rotates by $\frac { \pi } { 3 }$ about the origin, and let $g$ be the reflection transformation about the line $y = x$. When the line $x + 2 y + 5 = 0$ is mapped to the line $a x + b y + 5 = 0$ by the composite transformation $g ^ { - 1 } \circ f \circ g$, what is the value of $a + 2 b$? (Given that $a , b$ are constants.) [3 points]
(1) $\frac { 1 } { 2 }$
(2) 1
(3) $\frac { 3 } { 2 }$
(4) 2
(5) $\frac { 5 } { 2 }$

Let $A$ be the matrix representing the linear transformation $f : ( x , y ) \rightarrow ( 2 x - y , x - 2 y )$. When the point $( 5 , - 1 )$ is mapped to the point $( a , b )$ by the linear transformation represented by the matrix $A ^ { 4 }$, find the value of $a + b$. [3 points]

Let the matrices representing two linear transformations $f , g$ be $\left( \begin{array} { l l } 2 & 1 \\ 4 & 2 \end{array} \right) , \left( \begin{array} { r r } 2 & 0 \\ 1 & - 1 \end{array} \right)$ respectively. When the point $( 1,2 )$ is mapped to the point $( a , 6 )$ by the composite transformation $f \circ g$, what is the value of $a$? [3 points]
(1) 1
(2) 2
(3) 3
(4) 4
(5) 5

Let $\pi$ be the endomorphism of $\mathbb{R}^n$ whose representation in the canonical basis is the matrix $P = I_n - \frac{1}{n}J$, where $J = Z^t Z$ and $Z = \left(\begin{array}{c}1\\ \vdots \\ 1\end{array}\right) \in \mathcal{M}_{n,1}(\mathbb{R})$.
Show that $\pi$ is an orthogonal projector and specify its characteristic elements.

We consider the endomorphism $\Phi$ of $\mathcal{M}_n(\mathbb{R})$ defined by: $$\forall M \in \mathcal{M}_n(\mathbb{R}), \quad \Phi(M) = PMP$$ where $P = I_n - \frac{1}{n}J$.
1) Show that $\Phi$ is an orthogonal projector in the Euclidean space $(\mathcal{M}_n(\mathbb{R}), (\cdot \mid \cdot))$.
2) Show that $\operatorname{Im}\Phi = \left\{M \in \mathcal{M}_n(\mathbb{R}) \mid MZ = 0 \text{ and } {}^t MZ = 0\right\}$.

Let $M = (m_{ij}) \in \mathcal{S}_n(\mathbb{R})$. We set $$S(M) = MZ = \left(\begin{array}{c}\sum_{i=1}^n m_{1i}\\ \vdots \\ \sum_{i=1}^n m_{ni}\end{array}\right) = \left(\begin{array}{c}S(M)_1\\ \vdots \\ S(M)_n\end{array}\right) \quad \text{and} \quad \sigma(M) = \langle Z, S(M)\rangle$$
Show that $$\Phi(M) = M - \frac{1}{n}\left(S(M)^t Z + Z^t S(M)\right) + \frac{\sigma(M)}{n^2}J$$

Let $M = (m_{ij})_{(i,j) \in \llbracket 1,n\rrbracket^2} \in \mathcal{S}_n(\mathbb{R})$ such that for every pair $(i,j) \in \llbracket 1,n\rrbracket^2$, $m_{ij} \geqslant 0$ and $m_{ii} = 0$. We assume in this question that there exist $U_1, U_2, \cdots, U_n$ elements of $\mathbb{R}^p$ such that for every $(i,j) \in \llbracket 1,n\rrbracket^2$, $m_{ij} = \|U_i - U_j\|^2$.
1) Show that the eigenvalues of $\Phi(M)$ are all real and non-positive.
2) We further assume (if necessary by performing a translation) that the $(U_i)_{i \in \llbracket 1,n\rrbracket}$ are centered, that is, $\sum_{i=1}^n U_i = 0$. Show that $\operatorname{rg}(U) = \operatorname{rg}(U_1 | U_2 | \cdots | U_n) = \operatorname{rg}(\Phi(M))$ and that $p \geqslant \operatorname{rg}(\Phi(M))$.

Let $M = (m_{ij})_{(i,j) \in \llbracket 1,n\rrbracket^2} \in \mathcal{S}_n(\mathbb{R})$ such that for every pair $(i,j) \in \llbracket 1,n\rrbracket^2$, $m_{ij} \geqslant 0$ and $m_{ii} = 0$. Conversely, we assume that the eigenvalues of $\Phi(M)$ are all non-positive and we set $\Psi(M) = -\frac{1}{2}\Phi(M)$ and $r = \operatorname{rg}(\Psi(M))$.
1) Show that there exists a matrix $U \in \mathcal{M}_{r,n}(\mathbb{R})$ such that ${}^t UU = \Psi(M)$.
2) We denote $U_1, U_2, \cdots, U_n$ the columns of the matrix $U$. We seek to show that for every $(i,j) \in \llbracket 1,n\rrbracket^2$, $m_{ij} = \|U_i - U_j\|^2$.
a) Show that the $(U_i)$ are centered, that is, $\sum_{i=1}^n U_i = 0$.
b) Show that the matrix $N = (n_{ij})$ defined by: $$\forall (i,j) \in \llbracket 1,n\rrbracket^2, \quad n_{ij} = \|U_i - U_j\|^2$$ satisfies $\Psi(N) = \Psi(M)$.
c) Show that $M = N$ and conclude.

Show that $A \in \mathrm{SO}(2)$ if and only if there exists a real $t$ such that $A = R_t$ with $R_t = \left(\begin{array}{rr} \cos t & -\sin t \\ \sin t & \cos t \end{array}\right)$.

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 $R_t$.

We denote $K_2 = \left(\begin{array}{rr} 1 & 0 \\ 0 & -1 \end{array}\right)$. Verify that the endomorphism $\sigma_0 = f_{K_2}$ is a reflection (orthogonal symmetry with respect to a line in the plane) and specify its eigenelements.

For all real $t$, specify the endomorphism $\sigma_t$ canonically associated with $R_t^{-1} K_2 R_t$ and in particular its eigenelements.

Show that for every matrix $A$ of $\mathrm{O}(2)$ such that $\det(A) = -1$, there exists a real $t$ such that $$A = \left(\begin{array}{cr} \cos(2t) & \sin(2t) \\ \sin(2t) & -\cos(2t) \end{array}\right)$$

Suppose in this question that $n = 3$. What geometric interpretation can be given to $R_{p,q}(\theta)$?

We denote by $A$ a real square matrix of size 2 and we set, for all $x$ in $\mathbb{R}^2$, $f(x) = Ax$.
For $x$ in $\mathbb{R}^2$, express $\operatorname{div}_f(x)$ using only $A$.

We denote $\alpha > -1/2$, $E$ the $\mathbb{R}$-vector space of functions of class $\mathcal{C}^\infty$ on $[-1,1]$ with real values, and $$\varphi_\alpha(y) : t \mapsto \left(1-t^2\right)y''(t) - (2\alpha+1)t\,y'(t)$$ Justify that $\varphi_\alpha$ is an endomorphism of $E$. Is it injective?

We denote $\alpha > -1/2$, $F_n$ the vector subspace of $E$ of polynomial functions of degree less than or equal to $n$ (where $n \in \mathbb{N}$), and $$\varphi_\alpha(y) : t \mapsto \left(1-t^2\right)y''(t) - (2\alpha+1)t\,y'(t)$$ Justify that $\varphi_\alpha$ induces on $F_n$ an endomorphism and that this induced endomorphism (still denoted $\varphi_\alpha$) is diagonalizable.

We denote $\alpha > -1/2$, $F_n$ the vector subspace of $E$ of polynomial functions of degree less than or equal to $n$ (where $n \in \mathbb{N}$), and $$\varphi_\alpha(y) : t \mapsto \left(1-t^2\right)y''(t) - (2\alpha+1)t\,y'(t)$$ Show that there exists a basis of $F_n$ consisting of eigenvectors of $\varphi_\alpha$ of pairwise distinct degrees.

We denote $\alpha > -1/2$, $F_n$ the vector subspace of $E$ of polynomial functions of degree less than or equal to $n$ (where $n \in \mathbb{N}$), and $$\varphi_\alpha(y) : t \mapsto \left(1-t^2\right)y''(t) - (2\alpha+1)t\,y'(t)$$ Verify that two eigenvectors of $\varphi_\alpha$ of distinct degrees are associated with distinct eigenvalues. One may be interested in the leading coefficient of a judiciously chosen polynomial.