1 Linear Algebra Professor M. Zuker
Problem Set 9
Chapter 7
Note: Z1, Z2 etc. refer to my own problems, which may be very close to
what is in the text.
19. Prove or disprove: the identity operator on F2has infinitely many self-adjoint
square roots.
21. Prove or give a counterexample: if S∈ L(V) and there exists an orthonormal
basis (e1, . . . , en) of Vsuch that kSejk= 1 for each ej, then Sis an isometry.
22. Prove that if S∈ L(R3) is an isometry, then there exists a nonzero vector
x∈R3such that S2x=x.
Hint: What are the possible eigenvalues of S?
23. Define T∈ L(R3) by
T(x1, x2, x3) = (x3,2x1,3x2).
(a) Compute the singular values of T.
(b) Find (explicitly) an isometry S∈ L(F3) such that T=S√T∗T.
