grandes-ecoles 2022 Q32

grandes-ecoles · France · centrale-maths1__mp Groups Symplectic and Orthogonal Group Properties

Let $(E , \omega)$ be a symplectic vector space of dimension $n = 2m$. We fix $x$ and $y$, non-zero, in $E$. Suppose that $\omega ( x , y ) = 0$. Show that there exists a vector $z \in E$ such that $\omega ( x , z ) \neq 0$ and $\omega ( y , z ) \neq 0$.

Let $(E , \omega)$ be a symplectic vector space of dimension $n = 2m$. We fix $x$ and $y$, non-zero, in $E$. Suppose that $\omega ( x , y ) = 0$. Show that there exists a vector $z \in E$ such that $\omega ( x , z ) \neq 0$ and $\omega ( y , z ) \neq 0$.

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