The present paper is devoted to discussing Gale duality from the Z-linear
algebraic point of view. This allows us to isolate the class of Q-factorial
complete toric varieties whose class group is torsion free, here called poly
weighted spaces (PWS), as an interesting generalization of weighted projective
spaces (WPS).Comment: 29 pages: revised version to appear in Linear Algebra and Its
Applications. Major changes: the paper has been largely rewritten following
refree's comments. In particular, main geometric results have been
anticipated giving rise to the motivational Section
