Hamel-isomorphic images of the unit ball

In this article we consider linear isomorphisms over the field of rational numbers between linear spaces R and R. We prove that if f is such an isomorphism, then the image by f of the unit disk is a strictly non-measurable subset of the real line, which is strongly non-similar to any classical nonmeasurable subset of reals. We also show the consistency and independence of the proposition that all images of bounded measurable subsets of the plane via a such mapping are non-measurable.