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Problem Set 10
Math 423, Section 200, Spring 2015
Due: Monday, April 27.
Review Sections 8A and 8B in your textbook.
Complete the following items, staple this page to the front of your work, and turn your assignment
in at the beginning of class on Monday, April 27. One of the problems is considered extra credit.
Remember to fully justify all your answers, and provide complete details. Neatness is greatly
appreciated.
1. Let T : C2 → C2 given by T (w, z) = (z, 0). Find all generalized eigenvectors of T .
2. Let T : C2 → C2 given by T (w, z) = (−z, w). Find all the generalized eigenspaces corresponding to the distinct eigenvalues of T .
3. Let T : C3 → C3 given by T (z1 , z2 , z3 ) = (z2 , z3 , 0). Prove that there does not exist S ∈ L (C)
such that S 2 = T . In other words, prove that T does not have a square root.
4. Let V be a finite dimensional F-vector space, and N ∈ L (V) nilpotent. Prove that 0 is the only
eigenvalue of N.
5. Show that the set of nilpotent operators on V is not a subspace of L (V).
6. Let V be a finite dimensional F-vector space, and let S , T ∈ L (V) such that S T is nilpotent.
Prove that T S is nilpotent.
7.
a. Let V be a finite dimensional C-vector space. Let N ∈ L (V), and suppose that 0 is the
only eigenvalue of N. Prove that N is nilpotent.
b. Give an example of an operator T on a finite dimensional vector space over R such that
0 is the only eigenvalue of T , and T is not nilpotent.
8. Let V be a finite dimensional F-vector space, T ∈ L (V), m a positive integer, and v ∈ V such
that T m−1 v , 0, but T m v = 0. Prove that v, T v, T 2 v, . . . , T m−1 v are linearly independent.
9. Let V be a finite dimensional C-vector space, and T ∈ L (V). Prove that T has a basis consisting of eigenvectors of T if and only if every generalized eigenvector of T is an eigenvector of
T.
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10. Let V be a finite dimensional vector space over F and let T, S ∈ L (V). Assume that S is
invertible. Prove that T and S −1 T S have the same eigenvalues with the same multiplicities.
11. Let V be a finite dimensional C-vector space, and T ∈ L (V). Prove that there exist D, N ∈
L (V) such that T = D + N, D is diagonalizable, N is nilpotent, and DN = ND.
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Through the course of this assignment, I have followed the Aggie
Code of Honor. An Aggie does not lie, cheat or steal or tolerate
those who do.
Signed:
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