Certain connections between the geometric theory of Wonham and Morse and the Rosenbrock— Wolovich—Forney polynomial matrix approaches to linear system theory are indicated. The results of Warren and Eckberg (1975), worked by Vardulakis (1977) into an algorithm which constructively exhibits all controllability subspaces of a given controllable pair (A, B) in canonical form, are extended and turned into a theoretical tool of considerable power. A new theory is developed which translates certain statements of the geometric theory into a succinct algebraic form