Prob 3. Let TEC ( V ). Show that ( 0 , 20 ) : = ( T'U , 20 ) is an inner product on I' if and only if I' is positive ( per our definition of positivity ) . Prob 4 . We already know ( how ? ) that the operator I = _ DZ is nonnegative on the space V : = span ( 1 , cos * , sin 20 ) over IR, with the inner product ( 8 . 9 ) : = / 8 ( 10 ) 9 ( 20 ) do . Find ( a ) its square root operator VI ; ( b ) an example of a self - adjoint operator R. * VI such that R. 2 _ I ; ( C ) an example of a non - self - adjoint operator S' such that S* S' = I . Prob 5 . Let II and Iz be normal operators on an ~- dimensional inner product space V. Suppose both have ~ distinct eigenvalues 1 1 . .... An. Show that there is an isometry SEC ( V ) such that I1 = 5* I'2S . Prob 6 . Find the singular values of the operator I' ( P2 ( [' ) : P ( 20 ) # acp' ( 20 ) + 2 20 2 p" ( 20 ) if the inner* product on Pz ( [' ) is defined as ( P , 9 ) : = / P ( 2 ) 2 ( 20 ) doc