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M(Lo), because, from M(Lo) = M'(Lo), it follows that M-IM' E GLo. In a similar vein, we can describe another family of interesting subspaces of a given symplectic space. Fix T (L), for a given Lagrangian subspace L C V, to be the collection of all Lagrangian subspaces L' transverse to L, so that T(L):= {L' EL(V), LPL'=V}. The proofs already given show that GL operates transitively on T (L). ) a basis of L' (so that both together form a symplectic basis of V).

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