Symplectic basis

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Definition: A symplectic vector space is a vector space V equipped with a nondegenerate alternating bilinear form ω.

A standard symplectic basis is a basis consisting of pairs e1, ..., en and f1, ..., fn such that
- ω(ei, ej) = 0 for all i, j
- ω(fi, fj) = 0 for all i, j
- ω(ei, fj) = δij (1 if i = j, 0 otherwise)

Such a basis always exists and can be constructed by a process similar to Gram–Schmidt. A consequence is that the dimension of a finite-dimensional symplectic space is always even: dim V = 2n.