The well-established concept of subspace algorithms has led us to introduce a new structured problem formulation for Electrical Impedance Tomography. Whereas conventional output-least squares methods can be regarded as minimizing a certain error norm, solutions are recovered here as the minimizers of a closely related residual norm problem. An iterative solution scheme is shown to lead to a sequence of structured sparse matrix problems, the conditioning of which appears to be far more favorable than typically observed in output least squares.