Symplectic Vector Space - Symplectic Map

Suppose that (V,ω) and (W,ρ) are symplectic vector spaces. Then a linear map ƒ : V → W is called a symplectic map if the pullback preserves the symplectic form, i.e. ƒ*ρ = ω, where the pullback form is defined by (ƒ*ρ)(u,v) = ρ(ƒ(u),ƒ(v)),. Note that symplectic maps are volume-preserving, orientation-preserving, and are vector space isomorphisms.