Veroneseans, power subspaces and independence

Results are proved indicating that the Veronese map v_d often increases independence of both sets of points and sets of subspaces. For example, any d+1 Veronesean points of degree d are independent. Similarly, the dth power map on the space of linear forms of a polynomial algebra also often increases independence of both sets of points and sets of subspaces. These ideas produce d+1-independent families of subspaces in a natural manner.