  Matrices can be augmented by adding additional columns such that a
partitioning of the matrix in blocks of rows defines mutually orthogonal
subspaces. This augmented system can then be solved efficiently by a sum of
projections onto these subspaces. The equivalence to the original linear system
is ensured by adding additional rows to the matrix in a specific form. The
resulting solution method is known as the augmented block Cimmino method. Here
this method is extended to full rank underdetermined systems and to
overdetermined systems. In the latter case, rows of the matrix, not columns,
must be suitably augmented. The article presents an analysis of these methods.
