The construction of the general solution sequence of row-finite linear systems is accomplished by implementing -ad infinitum- the Gauss-Jordan algorithm under a rightmost pivot elimination strategy. The algorithm generates a basis (finite or Schauder
) of the homogeneous solution space for row-finite systems. The infinite Gaussian elimination part of the algorithm solves linear difference equations with variable coefficients of regular order, including equations of constant order and of ascending order. The general solution thus obtained can be expressed as a single Hessenbergian.
The Gauss-Jordan elimination algorithm is extended to reduce a row-finite $omegatimesomega$ matrix to lower row-reduced form, founded on a strategy of rightmost pivot elements. Such reduced matrix form preserves row equivalence, unlike the dominant (
upper) row-reduced form. This algorithm provides a constructive alternative to an earlier existence and uniqueness result for Quasi-Hermite forms based on the axiom of countable choice. As a consequence, the general solution of an infinite system of linear equations with a row-finite coefficient $omegatimesomega$ matrix is fully constructible.
