Editing Orthogonal complement (section)

== General bilinear forms ==
Let <math>V</math> be a vector space over a [[Field (mathematics)|field]] <math>\mathbb{F}</math> equipped with a [[bilinear form]] <math>B.</math>  We define <math>\mathbf{u}</math> to be left-orthogonal to <math>\mathbf{v}</math>, and <math>\mathbf{v}</math> to be right-orthogonal to <math>\mathbf{u}</math>, when <math>B(\mathbf{u},\mathbf{v}) = 0.</math>  For a subset <math>W</math> of <math>V,</math> define the left-orthogonal complement <math>W^\perp</math> to be
<math display="block">W^\perp = \left\{ \mathbf{x} \in V : B(\mathbf{x}, \mathbf{y}) = 0 \ \ \forall \ \mathbf{y} \in W \right\}.</math>

There is a corresponding definition of the right-orthogonal complement. For a [[reflexive bilinear form]], where <math>B(\mathbf{u},\mathbf{v}) = 0 \implies B(\mathbf{v},\mathbf{u}) = 0 \ \ \forall \ \mathbf{u} , \mathbf{v} \in V</math>, the left and right complements coincide. This will be the case if <math>B</math> is a [[Symmetric bilinear form|symmetric]] or an [[Bilinear form#Symmetric, skew-symmetric and alternating forms|alternating form]].

The definition extends to a bilinear form on a [[free module]] over a [[commutative ring]], and to a [[sesquilinear form]] extended to include any free module over a commutative ring with [[Conjugate element (field theory)|conjugation]].<ref>Adkins & Weintraub (1992) p.359</ref>

=== Properties ===
* An orthogonal complement is a subspace of <math>V</math>;
* If <math>X \subseteq Y</math> then <math>X^\perp \supseteq Y^\perp</math>;
* The [[Radical of a quadratic space|radical]] <math>V^\perp</math> of <math>V</math> is a subspace of every orthogonal complement;
* <math>W \subseteq (W^\perp)^\perp</math>;
* If <math>B</math> is [[non-degenerate]] and <math>V</math> is finite-dimensional, then <math>\dim(W)+\dim (W^\perp)=\dim (V)</math>.
* If <math>L_1, \ldots, L_r</math> are subspaces of a finite-dimensional space <math>V</math> and <math>L_* = L_1 \cap \cdots \cap L_r,</math> then <math>L_*^\perp = L_1^\perp + \cdots + L_r^\perp</math>.