Let $(E , \omega)$ be a symplectic vector space of dimension $n = 2m$. Let $a \in E$ be a non-zero vector and $\lambda$ be a real number. The symplectic transvection is defined by $\tau _ { a } ^ { \lambda } ( x ) = x + \lambda \omega ( a , x ) a$. Show that $\operatorname { det } \left( \tau _ { a } ^ { \lambda } \right) > 0$.