Verify that the coefficients Aij in the important expression $(7\mathrm{.}94)$ for the kinetic energy of any $$natural$$ system are symmetric; that is$,$ Aij $=$ Aj i $.($b$)$ Prove that for any n variables v$1,...,$ vn $$v Ajkvjvk $=2$ Aijvj $.$ i j$,$k j $[$Hint: Start with the case that n $=2,$ for which you can write out the sums in full. Notice that you need the result of part $($a$).]$ This identity is useful in many areas of physics; we needed it to prove the expression $(7\mathrm{.}96)$ for the generalized momentum pi$.$