Projective geometry's sphere space has five-dimensional homogeneous coordinates within which linear algebra may be performed, but that is only for convenience in computation. Since all equations in homogeneous space require the vanishing of inner products, there is a matrix of coefficients whose determinant must vanish. Given such a restriction, the coefficient matrix will always have a maximum of four linearly independent rows or columns; consequently the projective dimension will be four.
For example, four points identify a unique sphere, just as it requires four spheres to intersect at a single point. There are many other situations in which four spheres define a unique environment.