The paper makes the first steps towards a behavioral theory of LPV state-space representations with only affine dependence on the scheduling signal, by characterizing minimality of such state-space representations. We show that minimality is equivalent to observability, and that minimal realizations of the same behavior are isomorphic. Finally, we establish a formal relationship between minimality of LPV state-space representations with an affine dependence on the scheduling signal and minimality of LPV state-space representations with a dynamic and meromorphic dependence on the scheduling signal.