  In an unnormalized Krylov subspace framework for solving symmetric systems of
linear equations, the orthogonal vectors that are generated by a Lanczos
process are not necessarily on the form of gradients. Associating each
orthogonal vector with a triple, and using only the three-term recurrences of
the triples, we give conditions on whether a symmetric system of linear
equations is compatible or incompatible. In the compatible case, a solution is
given and in the incompatible case, a certificate of incompatibility is
obtained. In particular, the case when the matrix is singular is handled.
  We also derive a minimum-residual method based on this framework and show how
the iterates may be updated explicitly based on the triples, and in the
incompatible case a minimum-residual solution of minimum Euclidean norm is
obtained.
