Math 223, Section 101—Homework #8
due Friday, November 18, 2011 at the beginning of class
Practice problems. Don’t turn in these problems. Most of these problems are easy, and most of
them have answers in the back of the book, so they will be useful to you to check your understanding of the material.
1. Friedberg, Insel, and Spence, Section 5.1, p. 256, #1(a)–(f),(h)–(k). In (h), change F and
F n to R and Rn .
2. Friedberg, Insel, and Spence, Section 5.1, pp. 256–7, #2(a),(c),(e) and #3(a),(c)
3. Friedberg, Insel, and Spence, Section 5.1, pp. 257–8, #4(a),(b),(f),(h)–(j) and #5
4. Friedberg, Insel, and Spence, Section 5.2, p. 279, #1(a)–(g). Hint for (c): the zero vector.
5. Friedberg, Insel, and Spence, Section 5.2, pp. 279–280, #2(a),(c),(e),(g) and #3(a),(c)–(e)
6. Friedberg, Insel, and Spence, Section 5.2, p. 280, #7. Hint: follow the method of Example
7 in that section.
Homework problems. Write up and hand in solutions to these problems. Please have your solutions to these problems in the correct order on your pages. Write clearly and legibly, in complete
sentences; the solutions you hand in will be better if you think about how to best phrase your answer before you begin to write it. Please staple the pages together (unstapled solutions will not be
accepted).
1. Friedberg, Insel, and Spence, Section 5.1, pp. 257–8, #3(d) and #4(g)
2. (a) Write down a 4 × 4 matrix A whose eigenvalues are 1, 7, π, and e.
(b) Suppose that Q is an invertible matrix and D is a diagonal matrix of the same size.
Prove that each column of Q is an eigenvector for the matrix B = QDQ−1 .
(c) Find a 4 × 4 matrix C with four distinct eigenvalues (your choice) such that
     
 
2
2
0
1
1
2 4 2
  ,   ,   , and  
0
0 1 1
3
6
1
2
are all eigenvectors for C.
3. (a) Define T : P3 (R) → P3 (R) by T (p(x)) = x3 p(−1/x). Prove that T does not have
any eigenvectors. (Hint: find the characteristic polynomial of T with respect to the
standard ordered basis for P3 (R).)
(b) Define T : P3 (C) → P3 (C) by T (p(x)) = x3 p(−1/x). Find an eigenvector for T .
(P3 (C) is the set of polynomials of degree at most 3, where complex numbers are
allowed as the coefficients.)
4. Friedberg, Insel, and Spence, Section 5.1, pp. 258–9, #11
5. Friedberg, Insel, and Spence, Section 5.1, p. 259, #12. Hint: why does Q−1 AQ − tI equal
Q−1 (A − tI)Q?
(continued on next page)
6. Friedberg, Insel, and Spence, Section 5.2, pp. 279–280, #2(d),(f) and #3(f)
7. Friedberg, Insel, and Spence, Section 5.2, p. 282, #17 and #18. These problems look
intimidating, but each part has an extremely short proof.
8. (a) Let V be a finite-dimensional vector space (over R) whose dimension is odd. Prove
that every linear operator on V has an eigenvector.
(b) For every even integer n, give an example of a linear operator on Rn that has no
eigenvectors.
Bonus question. Friedberg, Insel, and Spence, Section 5.3, p. 312, #22 and #23