Let $T$ be a non-zero shift-invariant endomorphism of $\mathbb{K}[X]$ that is invertible. Show that $T^{-1}$ is still a shift-invariant endomorphism.

Let $T$ be a delta endomorphism of $\mathbb{K}[X]$.
Show that there exists a sequence of scalars $\left(\alpha_k\right)_{k \in \mathbb{N}}$ satisfying $\alpha_0 = 0$, $\alpha_1 \neq 0$ and $T = \sum_{k=1}^{+\infty} \alpha_k D^k$.

Let $W$ be a vector subspace of $V$ stable by $u$ and $h$. We assume that $W$ admits a complement $W'$ stable by $u$ and we seek a complement of $W$ stable by $u$ and $h$. Let $p$ be the projector onto $W$ parallel to $W'$. Verify that $u$ and $p$ commute.

We denote $$\bar{p} = \frac{1}{N} \sum_{k=0}^{N-1} h^k \circ p \circ h^{-k}.$$ Prove that the image of $\bar{p}$ is contained in $W$ and that for $w$ in $W$, we have $\bar{p}(w) = w$.

Deduce that $\bar{p}$ is a projector and that its image is $W$.

Prove carefully that $\bar{p}$ commutes with $u$ and $h$.

Deduce that the kernel of $\bar{p}$ is a complement of $W$ and that it is stable under $u$ and $h$.

131- The images of points $A(2,4)$ and $B(-6,2)$ under the transformation $D(x,y) = \left(-\dfrac{1}{2}y,\ \dfrac{1}{2}x+1\right)$ are called $A'$ and $B'$. What is the angle between lines $AB$ and $A'B'$?
(1) $30°$ (2) $60°$ (3) $90°$ (4) $180°$

131- The image of the line $2x + 3y = 6$ under the transformation $T(x,y) = (2y-1, x+3)$ passes through which point?
(1) $(-3, 2)$ (2) $(1,-1)$ (3) $(5, \circ)$ (4) $(7, \circ)$

131. Consider line $\Delta$ with equation $3x + 2y = 6$. Under rotation about the origin by $\dfrac{\pi}{2}$, in the direction of line $\Delta'$, the equation of line $\Delta'$ under the translation $T(x,y) = (x-3, y+1)$ is:
(1) $3y - 2x = 12$ (2) $3y - 2x = 15$ (3) $2y - 3x = 8$ (4) $2y + 2x = 9$

138- Matrix $A = \begin{bmatrix} 1 & -\sqrt{3} \\ \sqrt{3} & 1 \end{bmatrix}$ is given. If matrix $A^2$ acts on point $(1, 2, -2)$, what are the coordinates of the resulting point?
(1) $(-16, 8)$ (2) $(-8, 16)$ (3) $(8, -16)$ (4) $(16, -8)$

2. For ALL APPLICANTS.

Let $S$ and $T$ denote transformations of the $x y$-plane
$$S ( x , y ) = ( x + 1 , y ) , \quad T ( x , y ) = ( - y , x )$$
We will write, for example, $T S$ to denote the composition of applying $S$ then $T$, that is
$$T S ( x , y ) = T ( S ( x , y ) )$$
and write $T ^ { n }$ to denote $n$ applications of $T$ where $n$ is a positive integer.
(i) Show that $T S ( x , y ) \neq S T ( x , y )$.
(ii) For what values of $n$ is it the case that $T ^ { n } ( x , y ) = ( x , y )$ for all $x , y$ ?
(iii) Show that applications of $S$ and $T$ in some order can produce the transformation
$$U ( x , y ) = ( x - 1 , y )$$
What is the least number of applications (of $S$ and $T$ in total) that can produce $U$ ? Justify your answer.
(iv) Show that for any integers $a$ and $b$ there is some sequence of applications of $S$ and $T$ that maps $( 0,0 )$ to $( a , b )$.
(v) The parabola $C$ has equation $y = x ^ { 2 } + 2 x + 2$.
What is the equation of the curve obtained by applying $S$ to $C$ ? What is the equation of the curve obtained by applying $T$ to $C$ ?

Suppose a $2 \times 2$ matrix $\left[\begin{array}{ll} a & b \\ c & d \end{array}\right]$ representing a linear transformation maps three points $O(0,0), A(1,0), B(0,1)$ on the coordinate plane to $O(0,0), A'(3, \sqrt{3}), B'(-\sqrt{3}, 3)$ respectively, and maps a point $C(x, y)$ at distance 1 from the origin to point $C'(x', y')$. Select the correct options.
(1) The determinant $\left|\begin{array}{ll} a & b \\ c & d \end{array}\right| = 6$
(2) $\overline{OC'} = 2\sqrt{3}$
(3) The angle between $\overrightarrow{OC}$ and $\overrightarrow{OC'}$ is $60^\circ$
(4) It is possible that $y = y'$
(5) If $x < y$ then $x' < y'$

On the coordinate plane, let $\Gamma$ be an ellipse with center at the origin and major axis on the $y$-axis. It is known that a linear transformation of counterclockwise rotation by angle $\theta$ about the origin (where $0 < \theta < \pi$) transforms $\Gamma$ to a new ellipse $\Gamma ^ { \prime } : 40 x ^ { 2 } + 4 \sqrt { 5 } x y + 41 y ^ { 2 } = 180$. The point $\left( - \frac { 5 } { 3 } , \frac { 2 \sqrt { 5 } } { 3 } \right)$ is one of the two points on $\Gamma ^ { \prime }$ farthest from the origin.
The length of the major axis of ellipse $\Gamma ^ { \prime }$ is (15-1) $\sqrt{\underline{(15-2)}}$. (Express as a simplified radical)

On the coordinate plane, let $\Gamma$ be an ellipse with center at the origin and major axis on the $y$-axis. It is known that a linear transformation of counterclockwise rotation by angle $\theta$ about the origin (where $0 < \theta < \pi$) transforms $\Gamma$ to a new ellipse $\Gamma ^ { \prime } : 40 x ^ { 2 } + 4 \sqrt { 5 } x y + 41 y ^ { 2 } = 180$. The point $\left( - \frac { 5 } { 3 } , \frac { 2 \sqrt { 5 } } { 3 } \right)$ is one of the two points on $\Gamma ^ { \prime }$ farthest from the origin.
Find the equation of the line containing the minor axis of $\Gamma ^ { \prime }$ and the length of the minor axis.

On the coordinate plane, let $\Gamma$ be an ellipse with center at the origin and major axis on the $y$-axis. It is known that a linear transformation of counterclockwise rotation by angle $\theta$ about the origin (where $0 < \theta < \pi$) transforms $\Gamma$ to a new ellipse $\Gamma ^ { \prime } : 40 x ^ { 2 } + 4 \sqrt { 5 } x y + 41 y ^ { 2 } = 180$. The point $\left( - \frac { 5 } { 3 } , \frac { 2 \sqrt { 5 } } { 3 } \right)$ is one of the two points on $\Gamma ^ { \prime }$ farthest from the origin.
It is known that a point $P$ on $\Gamma$ is transformed by this rotation to a point $P ^ { \prime }$ that falls on the $x$-axis, and the $x$-coordinate of $P ^ { \prime }$ is positive. Find the coordinates of point $P$.

Three points $A(1,0), B(0,1), C(-1,0)$ are given on the coordinate plane. Let $\Gamma$ be the graph obtained by transforming $\triangle ABC$ by the matrix $T = \left[\begin{array}{ll} 3 & 0 \\ a & 1 \end{array}\right]$, where $a$ is a real number. Select the correct options.
(1) If $a = 0$, then $\Gamma$ is an isosceles right triangle
(2) At least two points on the sides of $\triangle ABC$ have unchanged coordinates after transformation by $T$
(3) $\Gamma$ must have part of it in the fourth quadrant
(4) There exists a figure $\Omega$ on the plane such that after transformation by $T$ it becomes $\triangle ABC$
(5) The area of $\Gamma$ is a constant value

Let $A = \left[ \begin{array} { l l } a _ { 1 } & a _ { 2 } \\ a _ { 3 } & a _ { 4 } \end{array} \right]$ and $B = \left[ \begin{array} { l l } b _ { 1 } & b _ { 2 } \\ b _ { 3 } & b _ { 4 } \end{array} \right]$ both be rotation matrices on the coordinate plane with center at the origin $O$, rotating counterclockwise by an acute angle, and satisfying $A ^ { 2 } = B ^ { 3 } = \left[ \begin{array} { l l } 0 & c \\ 1 & d \end{array} \right]$, where $c$ and $d$ are real numbers.
Let point $P ( 1,1 )$ be transformed by $A ^ { 3 }$ to point $Q$, and point $Q$ be transformed by $B ^ { 4 }$ to point $R$.
What is the value of $c$? (Single choice question, 3 points)
(1) 0
(2) $- 1$
(3) 1
(4) $- \frac { 1 } { 2 }$
(5) $\frac { 1 } { 2 }$

Let $A = \left[ \begin{array} { l l } a _ { 1 } & a _ { 2 } \\ a _ { 3 } & a _ { 4 } \end{array} \right]$ and $B = \left[ \begin{array} { l l } b _ { 1 } & b _ { 2 } \\ b _ { 3 } & b _ { 4 } \end{array} \right]$ both be rotation matrices on the coordinate plane with center at the origin $O$, rotating counterclockwise by an acute angle, and satisfying $A ^ { 2 } = B ^ { 3 } = \left[ \begin{array} { l l } 0 & c \\ 1 & d \end{array} \right]$, where $c$ and $d$ are real numbers.
Let point $P ( 1,1 )$ be transformed by $A ^ { 3 }$ to point $Q$, and point $Q$ be transformed by $B ^ { 4 }$ to point $R$.
Find the coordinates of point $Q$ and the angle between $\overrightarrow { O R }$ and the vector $(1, 0)$. (Non-multiple choice question, 6 points)

Let $A = \left[ \begin{array} { l l } a _ { 1 } & a _ { 2 } \\ a _ { 3 } & a _ { 4 } \end{array} \right]$ and $B = \left[ \begin{array} { l l } b _ { 1 } & b _ { 2 } \\ b _ { 3 } & b _ { 4 } \end{array} \right]$ both be rotation matrices on the coordinate plane with center at the origin $O$, rotating counterclockwise by an acute angle, and satisfying $A ^ { 2 } = B ^ { 3 } = \left[ \begin{array} { l l } 0 & c \\ 1 & d \end{array} \right]$, where $c$ and $d$ are real numbers.
Let point $P ( 1,1 )$ be transformed by $A ^ { 3 }$ to point $Q$, and point $Q$ be transformed by $B ^ { 4 }$ to point $R$.
Let $L$ be the line passing through point $P$ and parallel to line $OQ$. Let point $S$ be the intersection of $L$ and line $OR$. Find $\angle O S P$ and the coordinates of point $S$. (Non-multiple choice question, 6 points)

Problem 2

For a square matrix $\boldsymbol { A } , e ^ { \boldsymbol { A } }$ is defined as:
$$e ^ { A } = \boldsymbol { E } + \sum _ { k = 1 } ^ { \infty } \frac { 1 } { k ! } \boldsymbol { A } ^ { k } ,$$
where $\boldsymbol { E }$ is the identity matrix and $e$ is the base of natural logarithm.
I. Let $\boldsymbol { A }$ be a $3 \times 3$ square matrix which can be diagonalized by a regular matrix $\boldsymbol { P }$, i.e., $\boldsymbol { A } = \boldsymbol { P } \boldsymbol { D } \boldsymbol { P } ^ { - 1 }$, where $\boldsymbol { D }$ is a diagonal matrix:
$$\boldsymbol { D } = \left( \begin{array} { c c c } \lambda _ { 1 } & 0 & 0 \\ 0 & \lambda _ { 2 } & 0 \\ 0 & 0 & \lambda _ { 3 } \end{array} \right)$$
Here, $\lambda _ { 1 } , \lambda _ { 2 }$, and $\lambda _ { 3 }$ are complex numbers. Prove the following equation:
$$e ^ { \boldsymbol { A } } = \boldsymbol { P } \left( \begin{array} { c c c } e ^ { \lambda _ { 1 } } & 0 & 0 \\ 0 & e ^ { \lambda _ { 2 } } & 0 \\ 0 & 0 & e ^ { \lambda _ { 3 } } \end{array} \right) \boldsymbol { P } ^ { - 1 }$$
II. Let $\boldsymbol { A } = \left( \begin{array} { c c c } - 1 & 4 & 4 \\ - 5 & 8 & 10 \\ 3 & - 3 & - 5 \end{array} \right)$.

1. Find the regular matrix $\boldsymbol { P }$ and the diagonal matrix $\boldsymbol { D }$ such that $\boldsymbol { A } = \boldsymbol { P } \boldsymbol { D } \boldsymbol { P } ^ { - 1 }$.
2. Calculate $e ^ { \boldsymbol { A } }$.

III. Consider $\boldsymbol { A } = \left( \begin{array} { c c c } 0 & - x & 0 \\ x & 0 & 0 \\ 0 & 0 & 1 \end{array} \right) , \boldsymbol { B } = \left( \begin{array} { c c c } 1 & 0 & 0 \\ 0 & - 1 & 0 \\ 0 & 0 & 1 \end{array} \right)$ and $\boldsymbol { a } = \left( \begin{array} { l } 1 \\ 1 \\ e \end{array} \right)$, where $x$ is a real number. In the following, the transpose of a vector $\boldsymbol { v }$ is denoted by $\boldsymbol { v } ^ { T }$.

1. Express the sum of the eigenvalues of $e ^ { \boldsymbol { A } }$ using $e$ and $x$.
2. Let $\boldsymbol { C } = \boldsymbol { B } e ^ { \boldsymbol { A } }$. Find the minimum and maximum values of $\frac { \boldsymbol { y } ^ { T } \boldsymbol { C } \boldsymbol { y } } { \boldsymbol { y } ^ { T } \boldsymbol { y } }$ for a real three-dimensional vector $\boldsymbol { y } ( \boldsymbol { y } \neq \mathbf { 0 } )$.
3. Let $f ( \boldsymbol { z } ) = \frac { 1 } { 2 } \boldsymbol { z } ^ { T } \boldsymbol { C } \boldsymbol { z } - \boldsymbol { a } ^ { T } \boldsymbol { z }$ for a real three-dimensional vector $\boldsymbol { z } = \left( \begin{array} { c } z _ { 1 } \\ z _ { 2 } \\ z _ { 3 } \end{array} \right)$ and $\boldsymbol { C }$ in III.2. Find $\sqrt { z _ { 1 } ^ { 2 } + z _ { 2 } ^ { 2 } + z _ { 3 } ^ { 2 } }$ for $\boldsymbol { z }$ such that $\frac { \partial f ( \boldsymbol { z } ) } { \partial z _ { 1 } } = \frac { \partial f ( \boldsymbol { z } ) } { \partial z _ { 2 } } = \frac { \partial f ( \boldsymbol { z } ) } { \partial z _ { 3 } } = 0$.

In the Cartesian coordinate plane, the square $ABCD$ with vertex coordinates
$$\mathrm { A } ( - 1 , - 1 ) , \mathrm { B } ( 1 , - 1 ) , \mathrm { C } ( 1,1 ) , \mathrm { D } ( - 1,1 )$$
is given below.
To this square, the following transformations are applied in order:
- Rotation counterclockwise by $45 ^ { \circ }$ about the origin, - Reflection with respect to the y-axis, - Rotation clockwise by $45 ^ { \circ }$ about the origin.
In the final state, which of the following are the vertex points of this square whose coordinates remain unchanged?
A) A and B B) A and C C) A and D D) B and C E) C and D