We formulate a novel learning algorithm for output zeroing of linear finite-dimensional, control systems. As in classical control systems theory, we start from the knowledge of a nominal plant to develop a feedback algorithm that achieves the control objective by means of successive trials on the plant. Algorithm convergence in the face of linear plant perturbations is proved, and performance in the face of small nonlinear perturbations is discussed. The proposed algorithm does not require output differentiation, and is based upon the learning of the initial conditions that allow the output to remain identically zero, while the state of the system, dynamically extended, freely evolves complying with an internal stability constraint. Implementation of this algorithm requires state initialization at an arbitrary point of the state space. Therefore, for those systems for which direct state initialization is not feasible, we develop a learning procedure that automatically accomplishes this task. By means of a control input generated by the algorithm, the state of the system is steered, during an initial phase, from a point where it is easily initialized to the point from which the output zeroing task starts. An illustrative example is included.