Given the matrices of a linear state space representation, we find an expression for a universal left annihilator of the matrix [GRAPHICS] and hence derive kernel representations for the input-output behaviour and by duality the controllable part. More generally in the discrete case we derive representations for the L-completion for different values of L, and the sub-behaviours of trajectories reachable in a given time interval. The representations are in certain trim canonical forms, which are intimately connected with structure indices. As a by-product of the state elimination procedure, we obtain a minimal state space representation for a given behaviour in terms of an arbitrary one.