when solutions exist, is possessed by every matrix Ag which satisfies (1). The condition for solutions to exist is AAgy = y and when this is satisfied x = Agy is a particular solution of (5). Ag is a generalized inverse (g.i.) of A. A matrix A' which satisfies (1) and (2) is a reflexive g.i. A matrix A' which satisfies (1), (2) and (3) is a normalized g.i. Aw satisfying (1), (2) and (4) is a weak g.i. while the unique solution At of all four equations is the pseudoinverse or Moore-Penrose inverse. It is the purpose of this note to show that the classification can be made geometrically and that the various forms in which the different inverses can be expressed are given by making suitable choices of base vectors.