ABSTRACT
A quadric consists of all points x in an affine space A n satisfying a quadratic equation which can be written as
where C = Cx is a symmetric non-zero n x n matrix. The quadric described by the equation Q(x) = 0 will also be denoted by Q . The equation can be visualized by blocks:
The intersection of Q with an affine subspace B of dimension r > 1 is a quadric again or a hyperplane in B, i.e., a subspace of dimension r — 1. To prove this let B be represented by
then substitution yields
which is abbreviated by
If C = B tCB # O, this equation represents a quadric in B. Note that C is symmetric. On the other hand, if C = O but c ^ o, the equation is linear in y and represents a hyperplane in B. If G — O and c = o, but c ^ 0 , the intersection is empty; if c = 0 the subspace B is completely contained in Q.
