An important problem in the structure theory of linear systems is the construction of a canonical form for strongly controllable systems (i.e. surjective systems without finite zeros). The problem has been solved in principle (Morse 1973, Kronecker 1890), but without a, clear identification of the system-theoretic meaning of the construction steps. This paper proposes the concept of a ‘ train basis ’ in order to give this interpretation. The second main idea of the paper is the ‘ system associated with a sub-space ’, through which concept it is possible to translate problems concerning ‘A, B’-invariance etc. of subspaces in terms of input-output systems. In this way, we make contact with results of Willems (1980) on almost.invariance.