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Chapter 10. Trace and Determinant
                       246
                          Exercise 19 fails on
                          infinite-dimensional
                                                   if
                                                                            ∗
                        inner-product spaces,  19.  Suppose V is an inner-product space and T ∈L(V). Prove that
                                                                           T v ≤ Tv
                          leading to what are
                                                   for every v ∈ V, then T is normal.
                           called hyponormal
                       operators, which have a  20.  Prove or give a counterexample: if T ∈L(V) and c ∈ F, then
                        well-developed theory.     det(cT) = c dim V  det T.
                                              21.  Prove or give a counterexample: if S, T ∈L(V), then det(S+T) =
                                                   det S + det T.
                                              22.  Suppose A is a block upper-triangular matrix
                                                                                        
                                                                             A 1      ∗
                                                                                .       
                                                                                 .       ,
                                                                      A =         .
                                                                                        
                                                                             0       A m
                                                   where each A j along the diagonal is a square matrix. Prove that
                                                                    det A = (det A 1 )...(det A m ).

                                                                                                            n
                                              23.  Suppose A is an n-by-n matrix with real entries. Let S ∈L(C )
                                                                           n
                                                   denote the operator on C whose matrix equals A, and let T ∈
                                                                                 n
                                                       n
                                                   L(R ) denote the operator on R whose matrix equals A. Prove
                                                   that trace S = trace T and det S = det T.
                                              24.  Suppose V is an inner-product space and T ∈L(V). Prove that

                                                                          det T  ∗  = det T.
                                                                                       √
                                                   Use this to prove that |det T|= det T T, giving a different
                                                                                          ∗
                                                   proof than was given in 10.37.
                                              25.  Let a, b, c be positive numbers. Find the volume of the ellipsoid

                                                                $            3  x 2  y 2  z 2   %
                                                                 (x,y,z) ∈ R :     +    +    < 1
                                                                                a 2  b 2  c 2
                                                                       3
                                                   by finding a set Ω ⊂ R whose volume you know and an operator
                                                           3
                                                   T ∈L(R ) such that T(Ω) equals the ellipsoid above.
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