# Which subspaces retain nondegeneracy of a bilinear form?

1. Oct 7, 2012

### [sauce]

Suppose I have a nondegenerate alternating bilinear form <,> on a vector space V. Under what conditions would a subspace U of V retain nondegeneracy? That is, if u ∈ U and u ≠ 0, then could I find a w ∈ U such that <u,w> ≠ 0?

So for example, it's clear that no one-dimensional subspace W of V could retain nondegeneracy since every vector in W could be written as a scalar multiple of any other. But would, say, a two-dimensional subspace retain nondegeneracy?

Last edited: Oct 7, 2012
2. Oct 9, 2012

### quasar987

A 2-dimensional space my or may not retain nondegeneracy: pick any nonzero v. Then by nondegeneracy, there exists w s.t. <v,w> doesn't vanish. Ok, well the form is nondegenerate on W:=span{v,w}. Such a subspace is called symplectic by the way.

More generally, by the "canonical form theorem" for such forms (see p.1 of the free book by anna canna silva on symplectic geometry), there exists symplectic subspaces of dimension d iff d is even. Moreover, if W is such a symplectic subspace, then A(W) is too for any symplectic linear transformation A. So there are a lot of them in each dimension too.