Among many other things, Oberst proved in his fundamental paper that controllability is equivalent to minimality in a transfer equivalence class. In this paper we directly and explicitly describe minimal systems in terms of transfer functions. A transfer function is a purely algebraic object that is associated to a linear differential system and that determines in an explicit form a certain "trivial" part of the system. Trajectories belonging to this part are called transfer trajectories. It is shown that a linear differential system is controllable if and only if every one of its trajectories can be obtained from a transfer trajectory by using a finite number of differentiations.