Every point of the affine subspace "A" is the intersection of "A" with a linear subspace of "L". However, some subspaces of "L" are parallel to "A"; in some sense, they intersect "A" at infinity. The set of all linear subspaces of a linear space is, by definition, a projective space. And the affine subspace "A" is embedded into the projective space as a proper subset. However, the projective space itself is homogeneous. A straight line in the projective space corresponds to a linear subspace of the (n+1)-dimensional linear space. More generally, a projective subspace of the projective space corresponds to a linear subspace of the (n+1)-dimensional linear space, and is isomorphic to the projective space.