A null vector is an algebraic quantity with the property that its square is zero. I denote the universal algebra generated by taking all sums and products of null vectors over the real or complex numbers by \({{\mathcal {N}}}\). The rules of addition and multiplication in \({{\mathcal {N}}}\) are taken to be the same as those for real and complex square matrices. A distinct pair of null vectors is positively or negatively correlated if their inner product is positive or negative, respectively. A basis of \(n+1\) null vectors, with pairwise inner products equal to plus or minus one half, defines the Clifford geometric algebras \({\mathbb {G}}_{1,n}\), or \({\mathbb {G}}_{n,1}\), respectively, and provides a foundation for a new Cayley–Grassman linear algebra, a theory of complete graphs, and other applications in pure and applied areas of science.

零向量是一个代数量，其平方为零。我用 \({{\mathcal {N}}}\)表示通过对实数或复数求空向量的所有和与积而生成的通用代数。\({{\mathcal {N}}}\)中的加法和乘法规则与实数和复数方阵的规则相同。如果一对不同的空向量的内积分别为正或负，则它们呈正相关或负相关。以\(n+1\)个零向量为基础，两两内积等于正负二分之一，定义了 Clifford 几何代数\({\mathbb {G}}_{1,n}\)或\ ({\mathbb {G}}_{n,1}\)分别为新的凯莱-格拉斯曼线性代数、完全图理论以及纯科学和应用科学领域的其他应用提供了基础。