Problem 3. Throughout this problem, we fix a matrix A ? Fn,nwith the property that A = A?. (If F = R, then A is calledsymmetric. If F = C, then A is called Hermitian.) For u, v ? Fn,1,define [u, v] = v? Au. (a) Let Show that K is a subspace of Fn,1.K:={u?Fn,1 :[u,v]=0forallv?Fn,1}. (b) Suppose X is a subspace ofFn,1 with the property that [v,v] > 0 for all nonzero v ? X. (1)Show that [?, ?] defines an inner product on X. (c) Suppose Y is asubspace of Fn,1 with the property that [v,v] < 0 for allnonzero v ? Y. (2) If X is a subspace with property (1), prove thatX + K + Y is a direct sum, where K is defined in part (a).
