grandes-ecoles 2022 Q27

grandes-ecoles · France · centrale-maths1__official Matrices Bilinear and Symplectic Form Properties

Let $(E, \omega)$ be a symplectic vector space of dimension $n = 2m$. Let $a \in E$ be a non-zero vector and $\lambda \in \mathbb { R }$ be a real number. Show that the map $\tau _ { a } ^ { \lambda }$ defined by
$$\forall x \in E , \quad \tau _ { a } ^ { \lambda } ( x ) = x + \lambda \omega ( a , x ) a$$
is a transvection of $E$ and that it is a symplectic endomorphism of this same space.

Let $(E, \omega)$ be a symplectic vector space of dimension $n = 2m$. Let $a \in E$ be a non-zero vector and $\lambda \in \mathbb { R }$ be a real number. Show that the map $\tau _ { a } ^ { \lambda }$ defined by

$$\forall x \in E , \quad \tau _ { a } ^ { \lambda } ( x ) = x + \lambda \omega ( a , x ) a$$

is a transvection of $E$ and that it is a symplectic endomorphism of this same space.

Paper Questions

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