Geometric Algebras of Compatible Null Vectors

A (Clifford) geometric algebra is usually defined in terms of a quadratic form. A null vector v is an algebraic quantity with the property that \(v^2=0\) . The universal algebra generated by taking the sums and products of null vectors over the real or complex numbers is denoted by \(\mathcal{N}\) . The rules of addition and multiplication are taken to be the familiar rules of addition and multiplication of real or complex square matrices. In a series of ten definitions, the concepts of a Grassmann algebra, its dual Grassmann algebra, the associated real and complex geometric algebras, and their isomorphic real or complex coordinate matrix algebras are set down. This is followed by a discussion of affine transformations, the horosphere and conformal transformations on pseudoeuclidean spaces.