  The Gearhart-Koshy acceleration for the Kaczmarz method for linear systems is
a line-search with the unusual property that it does not minimize the residual,
but the error. Recently one of the authors generalized the this acceleration
from a line-search to a search in affine subspaces.
  In this paper, we demonstrate that the affine search is a Krylov space method
that is neither a CG-type nor a MINRES-type method, and we prove that it is
mathematically equivalent with a more canonical Gram-Schmidt-based method. We
also investigate what abstract property of the Kaczmarz method enables this
type of algorithm, and we conclude with a simple numerical example.
