Within the algebraic analysis approach to linear systems theory, we investigate the equivalence problem of linear functional systems, i.e., the problem of characterizing when all the solutions of two linear functional systems are in a one-to- one correspondence. To do that, we first provide a new characterization of isomorphic finitely presented modules in terms of inflation of their presentation matrices. We then prove several isomorphisms which are consequences of the unimodular completion problem. We then use these isomorphisms to complete and refine existing results concerning Serre’s reduction problem. Finally, different consequences of these results are given. All the results obtained here are algorithmic for rings for which Gröbner basis techniques exist and the computations can be performed by the Maple packages OreModules and OreMorphisms or the Mathematica package OreAlgebraicAnalysis.