The space of equivalence classes of vectors under non-zero scalar multiplication. Elements are sets of the form kv: k != 0, k scalar, v != o, v a vector where O is the origin. v is a representative member of this equivalence class. The projective plane of a vector space is the collection of its 1-dimensional subspaces. The properties of the vector space induce a topology and notions of smoothness on the projective plane. A projective plane is in no meaningful sense a plane and would therefore be (but isn't) better described as a "projective space". |