Tangential and Anisotropic Bases in Quadratic Spaces

This paper presents an explicit method for constructing tangential and anisotropic bases in quadratic spaces. We will also provide three theorems that allow us to explicitly calculate vectors that are orthogonal or tangent to given vectors or that provide necessary and sufficient conditions for the existence of such vectors. A classical theorem on quadratic space states that every finite-dimensional quadratic space has an orthogonal basis. A result that seems to be unanswered is under what conditions a finite-dimensional quadratic space has a basis consisting of isotropic vectors. A second problem is whether or not every finite-dimensional quadratic space has a basis consisting of mutually tangent vectors, where two vectors v,w are tangent if (uv)2=(uu)(vv). In this paper, we attack these two problems.