a. Let : Rn Rn be self-adjoint with matrix A = (aij), so that aij = aji. If f (x) = = aij xixj, show

a. Let Ί: Rn → Rn be self-adjoint with matrix A = (aij), so that aij = aji. If f (x) = =Σ aij xixj, show that Dkf (x) = 2 Σj = 1 akjxj. By considering the maximum of on Sn-1 show that there is xЄSn-1 and ^ ЄR with Tx = ^x.
b. If V = {yЄRn: = 0}, show that Ί(v) CV and Ί: V and Ί: V → V is self-adjoint.
c. Show that Ί has a basis of eigenvectors.