Let $\xi_1, \ldots, \xi_k$, $k$ elements of $\mathbb{R}^d$ and $$F := \left\{\sum_{i=1}^k \lambda_i \xi_i, (\lambda_1, \ldots, \lambda_k) \in \mathbb{R}_+^k\right\}$$ show that $F$ is a closed convex cone. Let $\xi \in \mathbb{R}^d$, show the equivalence between:

• $\xi \in F$,
• $\xi \cdot x \geqslant 0$ for all $x \in \mathbb{R}^d$ such that $\xi_i \cdot x \geqslant 0, i = 1, \ldots, k$.

Let $\xi_1, \ldots, \xi_k$, $k$ elements of $\mathbb{R}^d$ and $$F := \left\{\sum_{i=1}^k \lambda_i \xi_i, (\lambda_1, \ldots, \lambda_k) \in \mathbb{R}_+^k\right\}$$ show that $F$ is a closed convex cone. Let $\xi \in \mathbb{R}^d$, show the equivalence between:

• $\xi \in F$,
• $\xi \cdot x \geq 0$ for all $x \in \mathbb{R}^d$ such that $\xi_i \cdot x \geq 0, i = 1, \ldots, k$.

Let $n$ be a nonzero natural integer, $I = [a,b]$ with $a < b$, and $a_1 < \cdots < a_n$ distinct real numbers in $I$. Let $f$ be a real-valued function of class $\mathcal{C}^n$ on $I$ and $P = \Pi(f)$ its Lagrange interpolation polynomial. Deduce that $$\sup _ { x \in [ a , b ] } | f ( x ) - P ( x ) | \leqslant \frac { M _ { n } ( b - a ) ^ { n } } { n ! }$$ where $M _ { n } = \sup _ { x \in [ a , b ] } \left| f ^ { ( n ) } ( x ) \right|$.

Let $M \in \mathcal{M}_{k \times d}(\mathbb{R}), b = (b_1, \ldots, b_k) \in \mathbb{R}^k$ and $p \in \mathbb{R}^d$. Set $$\alpha := \inf\left\{p \cdot x : x \in \mathbb{R}^d, x \geqslant 0, Mx \leqslant b\right\}$$ and $$\beta := \sup\left\{b \cdot q : q \in \mathbb{R}^k, q \leqslant 0, M^T q \leqslant p\right\}$$ (adopting the convention: $\inf \emptyset = +\infty$ and $\sup \emptyset = -\infty$). Show that $\alpha \geqslant \beta$.

Let $M \in \mathcal{M}_{k \times d}(\mathbb{R}), b = (b_1, \ldots, b_k) \in \mathbb{R}^k$ and $p \in \mathbb{R}^d$. Set $$\alpha := \inf\left\{p \cdot x : x \in \mathbb{R}^d, x \geq 0, Mx \leqslant b\right\}$$ and $$\beta := \sup\left\{b \cdot q : q \in \mathbb{R}^k, q \leqslant 0, M^T q \leqslant p\right\}$$ (adopting the convention: $\inf \emptyset = +\infty$ and $\sup \emptyset = -\infty$).
Show that $\alpha \geqslant \beta$.

Let $I = [ a , b ]$ where $a < b$, and let $f(x) = \exp(x)$ for all $x \in I$. For all $n \in \mathbb { N } ^ { * }$, let $P _ { n } = \Pi _ { n } ( f )$ be the Lagrange interpolation polynomial of $f$ at $n$ distinct points $a_{1,n} < \cdots < a_{n,n}$ of $I$. Show that the sequence $\left( P _ { n } \right) _ { n \in \mathbb { N } ^ { * } }$ converges uniformly towards $f$ on $I$.

Let $M \in \mathcal{M}_{k \times d}(\mathbb{R}), b = (b_1, \ldots, b_k) \in \mathbb{R}^k$ and $p \in \mathbb{R}^d$. Set $$\alpha := \inf\left\{p \cdot x : x \in \mathbb{R}^d, x \geqslant 0, Mx \leqslant b\right\}$$ and $$\beta := \sup\left\{b \cdot q : q \in \mathbb{R}^k, q \leqslant 0, M^T q \leqslant p\right\}.$$ Suppose that there exists $\bar{x} = (\bar{x}_1, \ldots, \bar{x}_d) \in \mathbb{R}^d$ such that $$\bar{x} \geqslant 0, M\bar{x} \leqslant b \text{ and } p \cdot \bar{x} = \alpha.$$ Denoting by $M_i$ the vector of $\mathbb{R}^d$ whose coordinates are the coefficients of the $i$-th row of $M$, set: $$I := \left\{i \in \{1, \ldots, k\} : M_i \cdot \bar{x} = b_i\right\}$$ and $$J := \left\{j \in \{1, \ldots, d\} : \bar{x}_j = 0\right\}$$

• a) Show that $p \cdot z \geqslant 0$ for all $z \in \mathbb{R}^d$ such that $$z_j \geqslant 0 \text{ for all } j \in J \text{ and } M_i \cdot z \leqslant 0 \text{ for all } i \in I.$$
• b) Show that there exists $\bar{q} \in \mathbb{R}^k$ such that: $$\bar{q} \leqslant 0, M^T \bar{q} \leqslant p, \bar{q} \cdot (M\bar{x} - b) = 0 \text{ and } (p - M^T \bar{q}) \cdot \bar{x} = 0.$$
• c) Show that $b \cdot \bar{q} = \alpha = \beta$.

Let $M \in \mathcal{M}_{k \times d}(\mathbb{R}), b = (b_1, \ldots, b_k) \in \mathbb{R}^k$ and $p \in \mathbb{R}^d$. Set $$\alpha := \inf\left\{p \cdot x : x \in \mathbb{R}^d, x \geq 0, Mx \leqslant b\right\}$$ and $$\beta := \sup\left\{b \cdot q : q \in \mathbb{R}^k, q \leqslant 0, M^T q \leqslant p\right\}.$$ Suppose that there exists $\bar{x} = (\bar{x}_1, \ldots, \bar{x}_d) \in \mathbb{R}^d$ such that $$\bar{x} \geq 0, M\bar{x} \leqslant b \text{ and } p \cdot \bar{x} = \alpha.$$ Denoting by $M_i$ the vector of $\mathbb{R}^d$ whose coordinates are the coefficients of the $i$-th row of $M$, set: $$I := \left\{i \in \{1, \ldots, k\} : M_i \cdot \bar{x} = b_i\right\}$$ and $$J := \left\{j \in \{1, \ldots, d\} : \bar{x}_j = 0\right\}$$
a) Show that $p \cdot z \geq 0$ for all $z \in \mathbb{R}^d$ such that $$z_j \geq 0 \text{ for all } j \in J \text{ and } M_i \cdot z \leqslant 0 \text{ for all } i \in I.$$
b) Show that there exists $\bar{q} \in \mathbb{R}^k$ such that: $$\bar{q} \leqslant 0, M^T \bar{q} \leqslant p, \bar{q} \cdot (M\bar{x} - b) = 0 \text{ and } (p - M^T \bar{q}) \cdot \bar{x} = 0.$$
c) Show that $b \cdot \bar{q} = \alpha = \beta$.

Let $I = [ a , b ]$ where $a < b$, and let $f(x) = \exp(x)$ for all $x \in I$. Show that there exists a sequence of polynomials $\left( Q _ { n } \right) _ { n \in \mathbb { N } ^ { * } }$ that converges uniformly towards $f$ on $I$ and such that, for all $n \in \mathbb { N } ^ { * }$, the function $Q _ { n }$ does not coincide with $f$ at any point of $I$, except possibly at zero: $$\forall n \in \mathbb { N } ^ { * } , \quad \forall x \in I \backslash \{ 0 \} , \quad Q _ { n } ( x ) \neq \exp ( x ).$$

For all $x = (x_1, \ldots, x_d) \in \mathbb{R}^d$, we set $$\|x\|_1 := \sum_{i=1}^d |x_i|, \quad \|x\|_\infty := \max\{|x_i|, i = 1, \ldots, d\}.$$ Show that for all $x \in \mathbb{R}^d$, we have $$\|x\|_1 = \max\left\{x \cdot y, y \in \mathbb{R}^d, \|y\|_\infty \leqslant 1\right\}$$ and $$\|x\|_\infty = \max\left\{x \cdot y, y \in \mathbb{R}^d, \|y\|_1 \leqslant 1\right\}.$$

For all $x = (x_1, \ldots, x_d) \in \mathbb{R}^d$, we set $$\|x\|_1 := \sum_{i=1}^d |x_i|, \quad \|x\|_\infty := \max\{|x_i|, i = 1, \ldots, d\}.$$ Show that for all $x \in \mathbb{R}^d$, we have $$\|x\|_1 = \max\left\{x \cdot y, y \in \mathbb{R}^d, \|y\|_\infty \leqslant 1\right\},$$ and $$\|x\|_\infty = \max\left\{x \cdot y, y \in \mathbb{R}^d, \|y\|_1 \leqslant 1\right\}.$$

Let $M \in \mathcal{M}_{k \times d}(\mathbb{R})$ with $\operatorname{rank}(M) = k$, $b \in \mathbb{R}^k \backslash \{0\}$, and $$r := \inf\left\{\|x\|_1, x \in \mathbb{R}^d, Mx = b\right\}.$$ Denote by $C$ the set: $$C := \left\{x \in \mathbb{R}^d : Mx = b, \|x\|_1 = r\right\}.$$ Show that $C$ is non-empty, convex, closed and bounded.

Let $M \in \mathcal{M}_{k \times d}(\mathbb{R})$ with $\operatorname{rank}(M) = k$, $b \in \mathbb{R}^k \setminus \{0\}$, and $$r := \inf\left\{\|x\|_1, x \in \mathbb{R}^d, Mx = b\right\}.$$ Let $C$ be the set: $$C := \left\{x \in \mathbb{R}^d : Mx = b, \|x\|_1 = r\right\}$$ Show that $C$ is non-empty, convex, closed and bounded.

Let $M \in \mathcal{M}_{k \times d}(\mathbb{R})$ with $\operatorname{rank}(M) = k$, $b \in \mathbb{R}^k \backslash \{0\}$, and $$r := \inf\left\{\|x\|_1, x \in \mathbb{R}^d, Mx = b\right\}, \quad C := \left\{x \in \mathbb{R}^d : Mx = b, \|x\|_1 = r\right\}.$$ Fix $\bar{x} \in C$. Show that there exists $q \in \operatorname{Ker}(M)^\perp \backslash \{0\}$ such that for all $i \in \{1, \ldots, d\}$, we have $$q_i \bar{x}_i = \|q\|_\infty |\bar{x}_i|$$

Let $M \in \mathcal{M}_{k \times d}(\mathbb{R})$ with $\operatorname{rank}(M) = k$, $b \in \mathbb{R}^k \setminus \{0\}$, and $$r := \inf\left\{\|x\|_1, x \in \mathbb{R}^d, Mx = b\right\}, \quad C := \left\{x \in \mathbb{R}^d : Mx = b, \|x\|_1 = r\right\}.$$ Fix $\bar{x} \in C$. Show that there exists $q \in \operatorname{Ker}(M)^\perp \setminus \{0\}$ such that for all $i \in \{1, \ldots, d\}$, we have $$q_i \bar{x}_i = \|q\|_\infty |\bar{x}_i|.$$

Let $M \in \mathcal{M}_{k \times d}(\mathbb{R})$ with $\operatorname{rank}(M) = k$, $b \in \mathbb{R}^k \backslash \{0\}$, and fix $\bar{x} \in C$ where $C := \{x \in \mathbb{R}^d : Mx = b, \|x\|_1 = r\}$. Let $q \in \operatorname{Ker}(M)^\perp \backslash \{0\}$ be as in question 23. Let $K$ be the set of $y \in \mathbb{R}^d$ such that $$My = b, \quad y_i = 0 \ \forall i \in I_0(\bar{x}), \quad q_i y_i \geqslant 0 \ \forall i \in \{1, \ldots, d\}$$ Show that $K$ is non-empty and included in $C$.

Let $M \in \mathcal{M}_{k \times d}(\mathbb{R})$ with $\operatorname{rank}(M) = k$, $b \in \mathbb{R}^k \setminus \{0\}$, $\bar{x} \in C$ (where $C := \{x \in \mathbb{R}^d : Mx = b, \|x\|_1 = r\}$), and $q \in \operatorname{Ker}(M)^\perp \setminus \{0\}$ as in question 23. Let $K$ be the set of $y \in \mathbb{R}^d$ such that $$My = b, \quad y_i = 0 \quad \forall i \in I_0(\bar{x}), \quad q_i y_i \geq 0 \quad \forall i \in \{1, \ldots, d\}.$$ Show that $K$ is non-empty and included in $C$.

Let $M \in \mathcal{M}_{k \times d}(\mathbb{R})$ with $\operatorname{rank}(M) = k$, $b \in \mathbb{R}^k \backslash \{0\}$, and fix $\bar{x} \in C$ where $C := \{x \in \mathbb{R}^d : Mx = b, \|x\|_1 = r\}$. Let $K$ be as defined in question 24. Show that if $y \in \operatorname{Ext}(K)$ then $$h \in \operatorname{Ker}(M) \text{ and } I_0(y) \subset I_0(h) \Rightarrow h = 0.$$

With the notation of questions 23 and 24, show that if $y \in \operatorname{Ext}(K)$ then $$h \in \operatorname{Ker}(M) \text{ and } I_0(y) \subset I_0(h) \Rightarrow h = 0.$$

Let $M \in \mathcal{M}_{k \times d}(\mathbb{R})$ with $\operatorname{rank}(M) = k$, $b \in \mathbb{R}^k \backslash \{0\}$, and fix $\bar{x} \in C$ where $C := \{x \in \mathbb{R}^d : Mx = b, \|x\|_1 = r\}$. Let $K$ be as defined in question 24. Deduce that if $y \in \operatorname{Ext}(K)$ then the cardinality of $I_+(y) \cup I_-(y)$ is at most $k$.

With the notation of questions 23, 24 and 25, deduce that if $y \in \operatorname{Ext}(K)$ then the cardinality of $I_+(y) \cup I_-(y)$ is less than or equal to $k$.

We denote by $I$ a subset of $\mathbb { N }$ having at least two elements and by $u = \left( u _ { i } \right) _ { i \in I }$ a sequence of unit vectors of $\mathcal { M } _ { n , 1 } ( \mathbb { R } )$.
Prove that the real number $$C ( u ) = \sup \left\{ \left| \left\langle u _ { i } \mid u _ { j } \right\rangle \right| , ( i , j ) \in I ^ { 2 } , i \neq j \right\}$$ exists and belongs to the interval $[ 0,1 ]$.

We denote by $I$ a subset of $\mathbb { N }$ having at least two elements and by $u = \left( u _ { i } \right) _ { i \in I }$ a sequence of unit vectors of $\mathcal { M } _ { n , 1 } ( \mathbb { R } )$. $C(u)$ denotes the coherence parameter $C ( u ) = \sup \left\{ \left| \left\langle u _ { i } \mid u _ { j } \right\rangle \right| , ( i , j ) \in I ^ { 2 } , i \neq j \right\}$.
Show that if $C ( u ) = 0$, then the set $\left\{ u _ { i } , i \in I \right\}$ is finite and give an upper bound for its cardinality.

Let $a \in \mathbb{K}$. For all $p \in \mathbb{K}[X]$, we set $E_a(p) = E_a p = p(X+a)$.
Show that $E_a$ is an automorphism of $\mathbb{K}[X]$.

Let $n$ be a non-zero natural number. We denote $E _ { n } = \mathbb { R } _ { n } [ X ]$ and for all $k \in \llbracket 0 , n \rrbracket , P _ { k } = X ^ { k }$.
Let $\alpha$ be a real number.

1. Justify that the family $\mathcal { E } = \left( 1 , X - \alpha , \ldots , ( X - \alpha ) ^ { n } \right)$ is a basis of $E _ { n }$.
2. Let $P$ be a polynomial in $E _ { n }$. Give without proof the decomposition of $P$ in the basis $\mathcal { E }$ using the successive derivatives of the polynomial $P$.
3. Suppose that $\alpha$ is a root of order $r \in \llbracket 1 , n \rrbracket$ of $P$. Determine the quotient and remainder of the Euclidean division of $P$ by $( X - \alpha ) ^ { r }$.

To every polynomial $P$ of $E _ { n }$, we associate the polynomial $Q$ defined by: $$Q ( X ) = X P ( X ) - \frac { 1 } { n } \left( X ^ { 2 } - 1 \right) P ^ { \prime } ( X )$$ and we denote by $T$ the application that associates $Q$ to $P$.

1. Let $k \in \llbracket 0 , n \rrbracket$. Determine $T \left( P _ { k } \right)$.
2. Show that $T$ is an endomorphism of $E _ { n }$.
3. Write the matrix $M$ of $T$ in the basis $\mathscr { B } = \left( P _ { 0 } , P _ { 1 } , \ldots , P _ { n } \right)$ of $E _ { n }$.
4. Suppose that $\lambda$ is a real eigenvalue of the endomorphism $T$ and let $P$ be a monic polynomial, eigenvector associated with the eigenvalue $\lambda$.
   1. [7.1.] Show that $P$ has degree $n$.
   2. [7.2.] Let $z _ { 0 }$ be a complex root of $P$ with multiplicity order $r \in \mathbb { N } ^ { * }$. Prove that $z _ { 0 } ^ { 2 } - 1 = 0$.
   3. [7.3.] Deduce an expression for $P$.
5. Determine the eigenvectors of the endomorphism $T$. Is the endomorphism $T$ diagonalisable?