ABSTRACT
We have given the name hyperplane to the solution set of any single non-trivial linear equation in Rn. Geometrically, hyperplanes in R2
and R3 correspond to lines in the plane and planes in space, respectively. We visualize hyperplanes as “copies” of Rn−1 situated in Rn.
More generally, we can “linearly situate” copies of Rk into Rn for any k < n, not just for k = n− 1. We will call these copies k-dimensional affine subspaces. We cannot directly visualize them when n > 3, but we can easily understand and even axiomatize them. Indeed, Definition 1.9 below captures their essence in just two conditions.
