Let $(E , \omega)$ be a symplectic vector space of dimension $n = 2m$ and let $u \in \operatorname { Symp } _ { \omega } ( E )$. Let $P = \operatorname { Vect } \left( e _ { 1 } , f _ { 1 } \right)$ where $\omega(e_1, f_1) = 1$. Let $\delta = \delta_2 \circ \delta_1$ be a composition of at most four symplectic transvections satisfying $\delta(u(e_1)) = e_1$ and $\delta(u(f_1)) = f_1$. Set $v = \delta \circ u$. Show that $P$ is stable under $v$ and determine $v _ { P }$, the endomorphism induced by $v$ on $P$.
Let $(E , \omega)$ be a symplectic vector space of dimension $n = 2m$ and let $u \in \operatorname { Symp } _ { \omega } ( E )$. Let $P = \operatorname { Vect } \left( e _ { 1 } , f _ { 1 } \right)$ where $\omega(e_1, f_1) = 1$. Let $\delta = \delta_2 \circ \delta_1$ be a composition of at most four symplectic transvections satisfying $\delta(u(e_1)) = e_1$ and $\delta(u(f_1)) = f_1$. Set $v = \delta \circ u$. Show that $P$ is stable under $v$ and determine $v _ { P }$, the endomorphism induced by $v$ on $P$.