About systems of vectors and subspaces in finite dimensional space recovering vector-signal

The subject of this paper are the systems of vectors and subspaces in finite dimensional spaces admitting the recovery of an unknown vector-signal by modules of measurements. We analyze the relationship between the properties of recovery by modules of measurements and recovery by norms of projections and the properties of alternative completeness in Euclidean and unitary spaces. The theorem on ranks of one linear operator is considered, the result of which in some cases can be regarded as another criterion for the possibility to restore a vector-signal. As a result of this work, the equivalence of the alternative completeness property and the statement of the rank theorem for Euclidean space is proved. It is shown that the rank theorem in the real case can be extended to the systems of subspaces.
The questions about the minimum number of vectors admissible for reconstruction by modules of measurements are considered. The results available at the moment are presented, which are summarized in the form of a table for spaces of dimension less than 10. Also the known results to the question of the minimum number of subspaces admitting reconstruction by the norms of projections are briefly given.