Show that T PP defined by T p x p x p x is an isomorphism Since the dimensions of the domain and codomain are the same it suffices to show that T is one to one Let p x a a x ax Then p x p x a a a 2a x x 1 x But then the coefficient of x 1 is 0 an 1 so that an 1 If p x p x 0 then the coefficient of x is Select so that a a for all i Thus p x so that T is one to one and is thus an isomorphism We continue this process determinim