A solution to the problem of detecting and identifying control system component failures in linear time-invariant systems is given using the geometric concept of an unobservability subspace. Conditions are developed under which it is possible to design a causal linear processor that can be used to detect and uniquely identify a component failure in a linear time-invariant system, assuming either i) the components can fail simultaneously, or ii) the components can fail only one at a time. Explicit design algorithms are provided when those conditions are satisfied. In addition to the time domain solvability conditions, the frequency domain interpretation of the results are given, and connection is drawn with the results already available in the literature.