Projective correlation

Orthogonal complement polarity

Let be a -dimensional vector subspace. The orthogonal complement defined as

is a -dimensional vector subspace, and defines a projective polarity. #m/thm/geo Moreover, if for some matrix , then .

Proof

First we will show the column/kernel characterization always exists. Let be a basis for , and let so . Then iff for all . Since any is a finite linear combination of , it follows for any . Thus .

Since it follows from the Rank-nullity theorem that .

Note that clearly reverses inclusion of vector subspaces: If then certainly all vectors orthogonal to are orthogonal to , i.e. . With the observation above, this shows that is incidence-preserving and is therefore a projective polarity.

It follows that every projective correlation of can be written as a collineätion followed by .

Properties

  1. The orthogonal complement commutes with any field automorphism.
  2. Let . Then .
Proof of 1–2

Let and . Then

so

Similarly

so

Therefore , proving ^P1. #to/check

Let and Then

thus

But since , it follows , proving ^P2.

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