Orthogonal Polarities
Let B be a symmetric bilinear form with a trivial radical on the space V over the field K with characteristic different from 2. One can now define a map from D(V), the set of all subspaces of V, to itself :
This map is an orthogonal polarity on the projective space PG(W). Conversely, one can prove all orthogonal polarities are induced in this way, and that two symmetric bilinear forms with trivial radical induce the same polarity if and only if they are equal up to scalar multiplication.
Read more about this topic: Symmetric Bilinear Form
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