Quadratic forms and their duals

There are many specific results, spread over the literature, regarding the dualisation of quadrics in projective spaces and quadratic forms on vector spaces. In the present work we aim at generalising and unifying some of these. We start with a quadratic form $Q$ that is defined on a subspace $S$ of a finite-dimensional vector space $V$ over a field $F$. Whenever $Q$ satisfies a certain condition, which comes into effect only when $F$ is of characteristic two, $Q$ gives rise to a dual quadratic form $\hat{Q}$. The domain of the latter is a particular subspace $\hat{S}$ of the dual vector space of $V$. The connection between $Q$ and $\hat{Q}$ is given by a binary relation between vectors of $S$ and linear forms belonging to $\hat{S}$.