grandes-ecoles 2019 Q26

grandes-ecoles · France · centrale-maths1__mp Matrices Linear Transformation and Endomorphism Properties

We denote by $d$ the degree of $\pi_f$, and $x_1$ is a vector of $E$ such that $\left(x_1, f(x_1), \ldots, f^{d-1}(x_1)\right)$ is free. We set $e_1 = x_1, e_2 = f(x_1), \ldots, e_d = f^{d-1}(x_1)$ and $E_1 = \operatorname{Vect}(e_1, e_2, \ldots, e_d)$.
Show that $E_1$ is stable under $f$ and that $E_1 = \{P(f)(x_1) \mid P \in \mathbb{K}[X]\}$.

We denote by $d$ the degree of $\pi_f$, and $x_1$ is a vector of $E$ such that $\left(x_1, f(x_1), \ldots, f^{d-1}(x_1)\right)$ is free. We set $e_1 = x_1, e_2 = f(x_1), \ldots, e_d = f^{d-1}(x_1)$ and $E_1 = \operatorname{Vect}(e_1, e_2, \ldots, e_d)$.

Show that $E_1$ is stable under $f$ and that $E_1 = \{P(f)(x_1) \mid P \in \mathbb{K}[X]\}$.

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