Linear symplectomorphism

A linear symplectomorhpism is a morphism of symplectic vector spaces. Also can be called a symplectic vector space isomorphism. It is a linear map that preserves the symplectic form $\omega$ on the symplectic vector space.

Let $(V,{\omega }_1)$ and $(W,{\omega }_2)$ be symplectic vector spaces and $\Phi :V⟶W$ be a linear map. $\Phi$ is a linear symplectomorphism if ${\Phi }^{\ast }{\omega }_2={\omega }_1$ As seen in this commutative diagram:

This can also be stated as ${\omega }_1(v_1,v_2)={\omega }_2(\Phi (v_1),\Phi (v_2))$