Exercice
Exercice 91 *
12 février 2021 11:03
— Par Emmanuel Vieillard-Baron
Alain Soyeur
François Capaces
Vérifier si les applications entre \(\mathbb{R}\)-espace vectoriels suivantes sont linéaires ou pas.
\(f: \left\{ \begin{array}{ccl} \mathbb{R}^2 & \longrightarrow & \mathbb{R} \\ \left(x,y\right) & \longmapsto & x-y \end{array} \right.\)
\(f: \left\{ \begin{array}{ccl} \mathbb{R}^2 & \longrightarrow & \mathbb{R} \\ \left(x,y\right) & \longmapsto & x+y+1 \end{array} \right.\)
\(f: \left\{ \begin{array}{ccl} \mathbb{R}^2 & \longrightarrow & \mathbb{R}^2 \\ \left(x,y\right) & \longmapsto & \left(x+y,x-y\right) \end{array} \right.\)
\(f: \left\{ \begin{array}{ccl} \mathbb{R}^3 & \longrightarrow & \mathbb{R}^2 \\ \left(x,y,z\right) & \longmapsto & \left(x-y,x+z\right) \end{array} \right.\)
\(f: \left\{ \begin{array}{ccl} \mathbb{R}^3 & \longrightarrow & \mathbb{R} \\ \left(x,y,z\right) & \longmapsto & xyz \end{array} \right.\)
\(\theta: \left\{ \begin{array}{ccl} \mathscr F\left(\mathbb{R},\mathbb{R}\right) & \longrightarrow & \mathbb{R} \\ f & \longmapsto & f\left(1\right) \end{array} \right.\)
\(\theta: \left\{ \begin{array}{ccl} \mathcal{C}^{1}\left(\mathbb{R}\right) & \longrightarrow & \mathcal{C}^{0}\left(\mathbb{R}\right) \\ f & \longmapsto & 2f+f' \end{array} \right.\)
\(\theta: \left\{ \begin{array}{ccl} \mathcal{C}^{0}\left(\mathbb{R}\right) & \longrightarrow & \mathbb{R} \\ f & \longmapsto & \int_{0}^{1} f\left(t\right)\,\textrm{d}t \end{array} \right.\)
\(\theta: \left\{ \begin{array}{ccl} \mathscr S\left(\mathbb{R}\right) & \longrightarrow & \mathbb{R}^2 \\ \left(u_n\right) & \longmapsto & \left(u_0,u_1\right) \end{array} \right.\)
\(\theta: \left\{ \begin{array}{ccl} \mathbb{R}\left[X\right] & \longrightarrow & \mathbb{R}\left[X\right] \\ P & \longmapsto & \left(X^2+1\right)P \end{array} \right.\)