grandes-ecoles 2022 Q9

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

Let $F$ be a vector subspace of a symplectic space $(E,\omega)$, and let $F^{\omega} = \{ x \in E \mid \forall y \in F , \omega(x,y) = 0 \}$. Show that the restriction $\omega _ { F }$ of $\omega$ to $F ^ { 2 }$ defines a symplectic form on $F$ if and only if $F \oplus F ^ { \omega } = E$.

Let $F$ be a vector subspace of a symplectic space $(E,\omega)$, and let $F^{\omega} = \{ x \in E \mid \forall y \in F , \omega(x,y) = 0 \}$. Show that the restriction $\omega _ { F }$ of $\omega$ to $F ^ { 2 }$ defines a symplectic form on $F$ if and only if $F \oplus F ^ { \omega } = E$.

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