Symplectic vector space

Summary

A symplectic vector space is a vector space with a nondegenerate alternating bilinear form, called the symplectic form. A finite-dimensional symplectic vector space has even dimension and a standard basis in which the form has one fixed shape. The linear maps that keep the form make up the symplectic group. Symplectic vector spaces are the linear model of phase space in classical mechanics and of symplectic manifolds.