k(x,z) = $(x)T() Mercer's theorem gives a necessary and sufficient condition for a function k to be a kernel function: its corresponding kernel matrix K has to be symmetric and positive semidefinite. Suppose that ki(x, z) and k2(x,z) are two valid kernels. For each of the cases below, state whether k is also a valid kernel. If it is, prove it. If it is not give a counterexample. You can use either Mercer's theorem, or the definition of a kernel as needed to prove it. ki(x,x) (c) (10 points) k(x,z) = where ki(x,x) > 0 for any . Vk1(x,x)k1 (2,7) k(x,z) = $(x)T() Mercer's theorem gives a necessary and sufficient condition for a function k to be a kernel function: its corresponding kernel matrix K has to be symmetric and positive semidefinite. Suppose that ki(x, z) and k2(x,z) are two valid kernels. For each of the cases below, state whether k is also a valid kernel. If it is, prove it. If it is not give a counterexample. You can use either Mercer's theorem, or the definition of a kernel as needed to prove it. ki(x,x) (c) (10 points) k(x,z) = where ki(x,x) > 0 for any . Vk1(x,x)k1 (2,7)