A recursive approach to the solution of abstract linear equations and the tau method

Earlier work of Ortiz [1,2] and his collaborators [3,4]is generalized and extended for a recursive approach to the solution of abstract linear equations. Two families of vectors are simultaneously generated by means of Noether bases in the context of well-ordered bases. They are the families of Ortiz canonical vectors and residual vectors associated with every linear mapping on infinite-dimensional vector spaces. The latter would serve to determine a necessary and sufficient condition which ensures the existence of solutions and the former to provide a direct representation of a solution. Such a representation of a preimage vector admits the same scalar-cooredinates and the same index as its corresposnding image vector, omitting the scalars of nonaccessible oindices. It is shown that every linear mapping is uniquely associated with a pair whose components are a family of cosets of Ortiz canonical vectors and a family of residual vectors, for any given well-ordered basis of its codomain space. the terms of the above-mentioned families are reproduced by self-starting recursive relations in the context of standard bases. Our abstract results show that the recursive relations between the elements of the families mentioned above are conveniently generated through matrices in row echelon form. The former makes possible the recursive construction of the solution for an extensive class of linear operator equations, including equations determined by operators of infinite kernel index and deficiency. Several examples from different fields of applications, such as algebraic systems and partial differential equations with bivariate polynomial coefficients, are used to illustrate our method.