On checking linear dependence of parametric vectors

Abstract

Checking linear dependence of a finite number of vectors is a basic problem in linear algebra. We aim to extend the theory of linear dependence to parametric vectors where the entries are polynomials. This dependency depends on the specifications of the parameters or values of the variables in the polynomials. We propose a new method to check if parametric vectors are linearly dependent. Furthermore, this new method can also give the maximal linearly independent subset, and by which the remaining vectors are expressed in a linear combination. The new method is based on the computation of comprehensive Gröbner system for a finite set of parametric polynomials.