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Problem Set 4
Math 423, Section 200, Spring 2015
Due: Friday, February 20.
Review Chapter 5B and Section 5C 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 Friday, February 20. One of the problems is considered extra credit.
Remember to fully justify all your answers, and provide complete details. Neatness is greatly
appreciated.
1. Suppose that T ∈ L (V) and n is a positive integer such that T n = 0. (In this case, T is called
nilpotent.) Prove that id − T is invertible and
(id − T )−1 = id + T + T 2 + · · · + T n−1
2. Let T be a nilpotent operator on V. Show that 0 is an eigenvalue of T , and that T has no other
eigenvalues.
3. Suppose that T ∈ L (V). Prove that 9 is an eigenvalue of T 2 if and only if 3 or −3 is an
eigenvalue of T .
4. Let T : C3 → C3 given by
T (z1 , z2 , z3 ) = (−z1 − z2 − 2z3 , 8z1 − 11z2 − 8z3 , 10z1 + 11z2 + 7z3 ).
Find a basis of C3 such that the matrix of T with respect to this basis is upper triangular. Hint:
The eigenvalues of T are 1 and −3.
5. This exercise is to verify that if a matrix is not upper triangular, one cannot determine the
eigenvalues by just reading the coefficients on the diagonal.
a. Give an example of an invertible 2 × 2 matrix that has only zeros along the diagonal.
b. Give an example of a 2 × 2 matrix which is not invertible, but its diagonal entries are
nonzero.
6. Let T : R2 → R2 given by T (x, y) = (5x + 2y, −3x). Find a square root of T , that is, find find
a transformation S such that S 2 = T . Hint: This would be easy if T were given in diagonal
form.
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7. Suppose that T ∈ L (V) is invertible, and λ ∈ F r {0}. Prove that E(λ, T ) = E(1/λ, T −1 ).
8. Suppose R, T ∈ L (F3 ) each have 2, 6, 7 as eigenvalues. Prove that there exists an invertible
operator S ∈ L (F3 ) such that R = S −1 T S .
9. Find R, T ∈ L (F4 ) such that each have 2, 6, 7 as their only eigenvalues, and there does not
exist an invertible S ∈ L (F4 ) such that R = S −1 T S .
10. Suppose that the eigenvalues of T ∈ L (C3 ) are 6 and 7, and
√ T is not diagonalizable. Show
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that there exists (x, y, z) ∈ C such that T (x, y, z) = (17 + 8x, 5 + 8y, 2π + 8z).
11. Suppose that V is a finite dimensional C-vector space, and T ∈ L (V). Prove that T is diagonalizable if and only if
V = ker(T − λid) ⊕ range(T − λid)
for every λ ∈ C.
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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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