1. Show that for every g : Rn →R and every vector space of functions F as defined earlier, VCdim(POS(F +g))= VCdim(POS(F)).

2. (**) For every linearly closed family of real valued functions F, the VCdimension of the corresponding class POS(F) equals the linear dimension of F (as a vector space). Hint: Let f1, . . ., fd be a basis for the vector space F. Consider the mapping x _→ ( f1(x), . . ., fd (x)) (from Rn to Rd ). Note that thismapping induces a matching between functions over Rn of the form POS( f ) and homogeneous linear halfspaces in Rd (the VC-dimension of the class of homogeneous linear halfspaces is analyzed in Chapter 9).