Homework 10, due 10/23/2015
1. The last sentence of Artin Exercise 4.4.2(a) is false as stated, but it is
nearly true. Explain what goes wrong, and then do all of 4.4.2, replacing
4.4.2(a) with your corrected statement. (Hint: 4.4.2(a) does not specify
the field of scalars of the vector space.)
2. If T : V → V is a linear operator on a vector space V , and if W is a
subspace of V , then we say W is T -invariant, or T -stable, if T (W ) ⊂ W
(we saw examples of this general notion in our discussion of generalized
eigenspaces). Show that if W1 and W2 are T -invariant subspaces of V ,
then the subspace W1 + W2 (= {w1 + w2 : w1 ∈ W1 , w2 ∈ W2 }, i.e., the
span of W1 and W2 ) is T -invariant. Also show that W1 ∩W2 is T -invariant.
3. (Artin 5.7) Do a matrix A and its transpose At have the same eigenvalues? The same eigenvectors? Either prove the positive assertion or give a
counter-example.
4. (a) Determine the Jordan form of the matrix


1 1 0
0 1 0 .
0 1 1
(b) What are the possible Jordan forms for a matrix with complex entries
whose characteristic polynomial is (t + 2)2 (t − 5)2 ? Does your answer
change if your matrix has entries in a field other than C?
1