We study the conservative structure of linear Friedrichs systems with linear relaxation in view of the definition of well-balanced schemes. We introduce a particular global change of basis and show that the change-of-basis matrix can be used to develop a systematic treatment of well-balanced schemes in one dimension. This algebra sheds new light on a family of schemes proposed recently by L. Gosse [14]. The application to the S n model (a paradigm for the approximation of kinetic equations) for radiation is detailed. The discussion of the singular case is performed, and the 2D extension is shown to be equal to a specific multidimensional scheme proposed in [5]. This work is dedicated to the 2014 celebration of C. D. Munz' scientific accomplishments in the development of numerical methods for various problems in fluid mechanics.