Editing Symplectic vector space (section)

==Subspaces==
Let ''W'' be a [[linear subspace]] of ''V''. Define the '''symplectic complement''' of ''W'' to be the subspace
:<math>W^\perp = \{v \in V \mid \omega(v,w) = 0 \mbox{ for all } w \in W\}.</math>

The symplectic complement satisfies:
:<math>\begin{align}
  \left(W^\perp\right)^\perp &= W \\
       \dim W + \dim W^\perp &= \dim V.
\end{align}</math>

However, unlike [[orthogonal complement]]s, ''W''<sup>⊥</sup> ∩ ''W'' need not be 0. We distinguish four cases:
* ''W'' is '''symplectic''' if {{nowrap|1=''W''<sup>⊥</sup> ∩ ''W'' = {0}}}. This is true [[if and only if]] ''ω'' restricts to a nondegenerate form on ''W''. A symplectic subspace with the restricted form is a symplectic vector space in its own right.
* ''W'' is '''isotropic''' if {{nowrap|''W'' ⊆ ''W''<sup>⊥</sup>}}. This is true if and only if ''ω'' restricts to 0 on ''W''. Any one-dimensional subspace is isotropic.
* ''W'' is '''coisotropic''' if {{nowrap|''W''<sup>⊥</sup> ⊆ ''W''}}. ''W'' is coisotropic if and only if ''ω'' descends to a nondegenerate form on the [[Quotient space (linear algebra)|quotient space]] ''W''/''W''<sup>⊥</sup>. Equivalently ''W'' is coisotropic if and only if ''W''<sup>⊥</sup> is isotropic. Any [[codimension]]-one subspace is coisotropic.
* ''W'' is '''Lagrangian''' if {{nowrap|1=''W'' = ''W''<sup>⊥</sup>}}. A subspace is Lagrangian if and only if it is both isotropic and coisotropic. In a finite-dimensional vector space, a Lagrangian subspace is an isotropic one whose dimension is half that of ''V''. Every isotropic subspace can be extended to a Lagrangian one.

Referring to the canonical vector space '''R'''<sup>2''n''</sup> above,
* the subspace spanned by {''x''<sub>1</sub>, ''y''<sub>1</sub>} is symplectic
* the subspace spanned by {''x''<sub>1</sub>, ''x''<sub>2</sub>} is isotropic
* the subspace spanned by {''x''<sub>1</sub>, ''x''<sub>2</sub>, ..., ''x''<sub>''n''</sub>, ''y''<sub>1</sub>} is coisotropic
* the subspace spanned by {''x''<sub>1</sub>, ''x''<sub>2</sub>, ..., ''x''<sub>''n''</sub>} is Lagrangian.