grandes-ecoles 2015 Q15

grandes-ecoles · France · x-ens-maths1__mp Linear transformations
We denote by $E$ the vector space $\mathbb { R } ^ { 2 }$ equipped with the standard inner product and the canonical basis $\mathcal { B } = \left\{ e _ { 1 } , e _ { 2 } \right\}$. We define a basis $\mathcal { C } = \left\{ \omega _ { 1 } , \omega _ { 2 } \right\}$ of $E$ by $\omega _ { 1 } = e _ { 1 }$ and $\omega _ { 2 } = \frac { 1 } { 2 } \left( e _ { 1 } + \sqrt { 3 } e _ { 2 } \right)$. Let $s_1$ be the orthogonal reflection with respect to $\mathbb{R}\omega_1$ and $s_2$ the orthogonal reflection with respect to $\mathbb{R}\omega_2$. Let $H$ be the set of vectors $\left( m _ { 1 } , m _ { 2 } , m _ { 3 } \right) \in \mathbb { R } ^ { 3 }$ such that $m _ { 1 } + m _ { 2 } + m _ { 3 } = 0$. We denote by $H ^ { + }$ the subset of $\left( m _ { 1 } , m _ { 2 } , m _ { 3 } \right) \in H$ such that $m _ { 1 } \geqslant m _ { 2 } \geqslant m _ { 3 }$. We consider the application $$\begin{array} { c c c c } \varphi : & H & \longrightarrow & E \\ & \left( m _ { 1 } , m _ { 2 } , m _ { 3 } \right) & \longmapsto & \left( m _ { 1 } - m _ { 2 } \right) \omega _ { 1 } + \left( m _ { 2 } - m _ { 3 } \right) \omega _ { 2 } \end{array}$$ (a) Show that $\varphi$ is a linear isomorphism. Describe $\varphi \left( H ^ { + } \right)$.
(b) Show that, for all $\left( m _ { 1 } , m _ { 2 } , m _ { 3 } \right) \in H$, we have $$s _ { 1 } \circ \varphi \left( m _ { 1 } , m _ { 2 } , m _ { 3 } \right) = \varphi \left( m _ { 1 } , m _ { 3 } , m _ { 2 } \right) \quad \text { and } \quad s _ { 2 } \circ \varphi \left( m _ { 1 } , m _ { 2 } , m _ { 3 } \right) = \varphi \left( m _ { 2 } , m _ { 1 } , m _ { 3 } \right) .$$ (c) Let $\widehat { \lambda } = \left( \lambda _ { 1 } , \lambda _ { 2 } , \lambda _ { 3 } \right) \in H$ such that $\lambda _ { 1 } > \lambda _ { 2 } > \lambda _ { 3 }$. We denote by $\mathcal { Q } _ { \widehat { \lambda } }$ the set of $\left( m _ { 1 } , m _ { 2 } , m _ { 3 } \right) \in H ^ { + }$ such that $m _ { 1 } \leqslant \lambda _ { 1 }$ and $m _ { 1 } + m _ { 2 } \leqslant \lambda _ { 1 } + \lambda _ { 2 }$. Show that $\varphi \left( \mathcal { Q } _ { \widehat { \lambda } } \right)$ is a quadrilateral whose vertices will be described. 
We denote by $E$ the vector space $\mathbb { R } ^ { 2 }$ equipped with the standard inner product and the canonical basis $\mathcal { B } = \left\{ e _ { 1 } , e _ { 2 } \right\}$. We define a basis $\mathcal { C } = \left\{ \omega _ { 1 } , \omega _ { 2 } \right\}$ of $E$ by $\omega _ { 1 } = e _ { 1 }$ and $\omega _ { 2 } = \frac { 1 } { 2 } \left( e _ { 1 } + \sqrt { 3 } e _ { 2 } \right)$. Let $s_1$ be the orthogonal reflection with respect to $\mathbb{R}\omega_1$ and $s_2$ the orthogonal reflection with respect to $\mathbb{R}\omega_2$.\\
Let $H$ be the set of vectors $\left( m _ { 1 } , m _ { 2 } , m _ { 3 } \right) \in \mathbb { R } ^ { 3 }$ such that $m _ { 1 } + m _ { 2 } + m _ { 3 } = 0$. We denote by $H ^ { + }$ the subset of $\left( m _ { 1 } , m _ { 2 } , m _ { 3 } \right) \in H$ such that $m _ { 1 } \geqslant m _ { 2 } \geqslant m _ { 3 }$. We consider the application
$$\begin{array} { c c c c } \varphi : & H & \longrightarrow & E \\ & \left( m _ { 1 } , m _ { 2 } , m _ { 3 } \right) & \longmapsto & \left( m _ { 1 } - m _ { 2 } \right) \omega _ { 1 } + \left( m _ { 2 } - m _ { 3 } \right) \omega _ { 2 } \end{array}$$
(a) Show that $\varphi$ is a linear isomorphism. Describe $\varphi \left( H ^ { + } \right)$.\\
(b) Show that, for all $\left( m _ { 1 } , m _ { 2 } , m _ { 3 } \right) \in H$, we have
$$s _ { 1 } \circ \varphi \left( m _ { 1 } , m _ { 2 } , m _ { 3 } \right) = \varphi \left( m _ { 1 } , m _ { 3 } , m _ { 2 } \right) \quad \text { and } \quad s _ { 2 } \circ \varphi \left( m _ { 1 } , m _ { 2 } , m _ { 3 } \right) = \varphi \left( m _ { 2 } , m _ { 1 } , m _ { 3 } \right) .$$
(c) Let $\widehat { \lambda } = \left( \lambda _ { 1 } , \lambda _ { 2 } , \lambda _ { 3 } \right) \in H$ such that $\lambda _ { 1 } > \lambda _ { 2 } > \lambda _ { 3 }$. We denote by $\mathcal { Q } _ { \widehat { \lambda } }$ the set of $\left( m _ { 1 } , m _ { 2 } , m _ { 3 } \right) \in H ^ { + }$ such that $m _ { 1 } \leqslant \lambda _ { 1 }$ and $m _ { 1 } + m _ { 2 } \leqslant \lambda _ { 1 } + \lambda _ { 2 }$. Show that $\varphi \left( \mathcal { Q } _ { \widehat { \lambda } } \right)$ is a quadrilateral whose vertices will be described.
Paper Questions
Q1 Q2 Q3 Q4 Q5 Q6 Q7 Q8 Q9 Q10 Q11 Q12 Q13 Q14 Q15 Q16