For each of the following functions K : X X!R, determine whether or not K is a valid kernel. In each case, justify your answer by either nding a vector mapping : X!Rd for some suitable d 2 Z+ such that K(x; x0) = (x)>(x0) 8x; x0, or explaining why such a mapping cannot exist. (a) K(x; x0) = c K1(x; x0), where c > 0 (b) K(x; x0) = K1(x; x0) + K2(x; x0) (c) K(x; x0) = K1(x; x0) K2(x; x0) (d) K(x; x0) = K1(x; x0) K2(x; x0) (e) K(x; x0) = K1(f(x); f(x0)), where f : X!X is any function.