Problem set 8 Math 110 Lec 2 Due: Never. This problem set consists of 8 short problems and is just for “fun”. 1,2,3,4 can be solved already. 5,6 can be solved after 8/8. 7,8 can be solved after 8/9. 1. Let P 2 ( R ) denote the space of real polynomials of degree at most 2, together with the inner product h f, g i := Z 1 0 f ( x ) g ( x ) d x. Consider the map S : P 2 ( R ) -→ P 2 ( R ) , p ( x ) 7→ p (1 - x ) . Is S an isometry? 2. (Axler 7.C.2) Suppose T is a positive operator on V . Suppose v, w ∈ V are such that Tv = w and Tw = v. Prove that v = w. 3. We have seen several times that, if S and T are diagonalizable operators that commute ( ST = TS ) then S and T are simultaneously diagonalizable. No such result holds for the Jordan normal form. Let A = 1 1 - 1 - 1 ∈ C 2 , 2 . It is clear that A and - A commute. Prove that there is no basis with respect to which A and - A are simultaneously in Jordan normal form. 4. Let T ∈ L ( V ) be a normal operator on a finite-dimensional complex vector space V .