# Symplectic forms are isomorphic

Let $V$ be a symplectic vector space, i.e. a vector space with a non-degenerate alternating bilinear form, a so-called symplectic form. There is a theorem:

All symplectic forms on $V$ are ismorphic.

1) Can somebody explain what it means precisely that two symplectic forms are isomorphic?

2) Could somebode give references on this statement with or without proof?

Thank you very much.

It can be proven that any finite-dimensional symplectic vector space of dimension $2n$ has a basis in which the symplectic form $\omega$ has the form:
$$\begin{pmatrix} 0 & I_n \\ -I_n & 0 \end{pmatrix}$$
• It means that for any $\omega_1,\omega_2$ two symplectic forms, you can find an invertible linear transformation $\varphi$ such that for any $x,y \in V$ $\omega_1(x,y) = \omega_2(\varphi(x),\varphi(y))$. – mathcounterexamples.net Aug 27 '18 at 16:24