Kunze Linear Algebra · Problem 10.2.17

Bilinear Forms

Mathematics — Methods · orthogonal-complement-subspace

Problem

Let VV be a finite-dimensional vector space over a field FF and ff a symmetric bilinear form on VV. For each subspace WW of VV, let W⊥W^\perp be the set of all vectors α\alpha in VV such that f(α,β)=0f(\alpha, \beta) = 0 for every β\beta in WW. Show that (a) W⊥W^\perp is a subspace. (b) V⊥={0}⊥V^\perp = \{0\}^\perp. (c) V⊥={0}V^\perp = \{0\} if and only if ff is non-degenerate. (d) rank f=dim⁡V−dim⁡V⊥\text{rank}\, f = \dim V - \dim V^\perp. (e) If dim⁡V=n\dim V = n and dim⁡W=m\dim W = m, then dim⁡W⊥≥n−m\dim W^\perp \geq n - m. (Hint: Let {β1,…,βm}\{\beta_1, \ldots, \beta_m\} be a basis of WW and consider the mapping α→(f(α,β1),…,f(α,βm))\alpha \to (f(\alpha, \beta_1), \ldots, f(\alpha, \beta_m)) of VV into FmF^m.) (f) The restriction of ff to WW is non-degenerate if and only if W∩W⊥={0}W \cap W^\perp = \{0\}. (g) V=W⊕W⊥V = W \oplus W^\perp if and only if the restriction of ff to WW is non-degenerate.

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