Math 601 Homework

• Homework #1 (Due Friday, Feb. 6)
• p. 674: 9.33, 9.35, 9.40
• p. 440: 7.10, 7.13, 7.14 (these 3 problems on p. 440 are postponed)
• I. If the characteristic polynomial of a matrix is X^2 (X-1)^3, find all possible Jordan forms for the matrix.
• II. Let V be a finite dimensional vector space over a field F and let T be a linear transformation from V to V.
  (a) Suppose that every nonzero v in V is an eigenvector of T. Show that T is a scalar multiple of the identity.
  (b) A cyclic vector for T is a vector v in V such that {v, Tv, (T^2)v, ...} spans V. Suppose that every nonzero vector v in V is a cyclic vector for T. Show that the characteristic polynomial of T must be irreducible over F.
  (c) Suppose that V is 2-dimensional over F and that the characteristic polynomial of T is irreducible over F. Show that every nonzero v in V is a cyclic vector for T.

• Homework #2 (Due Monday, Feb. 16)
• p. 440: 7.10, 7.13, 7.14
• p. 486: 7.40, 7.43, 7.49, 7.50
• I. Let M be an n by n complex matrix. Show that det(e^M)=e^(trace(M)).

• Homework #3 (Due Monday, Feb 23)

• Homework #4 (Due Wednesday, March 3)

• Homework #5 (Due Friday, March 12)
• p. 604: 8.46, 8.47, 8.49, 8.50
• One more problem

Midterm #1: Wednesday, March 17