Consider two finite-dimensional feature transforms and and their corresponding kernels and .

(a) Define . Express the corresponding kernel of in terms of and .

(b) Consider the matrix and let be the vector representation of the matrix (say, by concatenating all the rows). Express the corresponding kernel of in terms of and .

Let and ,we have:

(c) Hence, show that if and are kernels, then so are and .

We can prove this by contradiction, for example: Assume and are kernels but is not, that means we can’t find any transform such that , however we have just found one possible transform above hence the contradiction, so one part of the above statement follows. The same goes for .