The problem of finding conditions for the reconstruction of a linear combination of the vector of unknown inputs is tackled. Appropriate definitions are given in order to address formally the analysis. Thus, necessary and sufficient conditions are obtained for two cases, when the reconstruction may be achieved in finite time and when it is possible to do it asymptotically. The results are applied for the fault detection, with the respective deduction of analogous conditions. (C) 2015 Elsevier Ltd. All rights reserved.