A symplectic vector space is a kind of vector space that is equipped with a bilinear, skew-symmetric form known as a symplectic form.
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A symplectic vector space is a vector space equipped with a nondegenerate antisymmetric bilinear form. Every finite-dimensional symplectic vector space has even dimension: splitting off one symplectic two-plane leaves a smaller nondegenerate antisymmetric space, and induction completes the decomposition.