Math 223, Section 101—Homework #10
due Friday, December 2, 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 6.2, pp. 352–3, #1(a)–(d),(f),(g)
2. Friedberg, Insel, and Spence, Section 6.2, p. 353, #2(b),(c),(e),(g),(k)
3. Friedberg, Insel, and Spence, Section 6.2, p. 354, #4
4. Friedberg, Insel, and Spence, Section 6.2, p. 354, #19(a),(b). Hint: first find a basis for W ,
then an orthogonal basis.
5. Friedberg, Insel, and Spence, Section 6.5, p. 392, #1(c)–(e) and #2(a),(d)
6. Friedberg, Insel, and Spence, Section 6.5, p. 393, #5
7. Friedberg, Insel, and Spence, Section 6.5, p. 394, #17
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 6.2, p. 353, #2(f),(h). The term “Fourier coefficient”
is defined on page 348.
R1
2. Define an inner product on P (R) by the formula hf (x), g(x)i = 0 f (t)g(t) dt.
(a) Use the Gram–Schmidt process on the linearly independent set R = {1, x, x2 } to
produce an orthogonal basis for P2 (R).
(b) Use the Gram–Schmidt process on the linearly independent set S = {x2 , x, 1} to
produce an orthogonal basis for P2 (R).
3. Friedberg, Insel, and Spence, Section 6.2, pp. 354–5, #6 and #13(a)–(c). In both problems,
you may assume that V is finite-dimensional.
4. Define a linear operator T : M2×2 (R) → M2×2 (R) by the formula T ac db = ac db .
Find an orthonormal set of eigenvectors for T . (Here we use the Frobenius inner product
on M2×2 (R).)
5. Friedberg, Insel, and Spence, Section 6.5, p. 392, #2(c),(e)
6. Friedberg, Insel, and Spence, Section 6.5, p. 394, #11
7. Let A = 32 20 .
(a) Find an orthogonal matrix Q such that Q−1 AQ is a diagonal matrix.
(b) Find eight distinct orthogonal matrices Q such that Q−1 AQ is a diagonal matrix. (You
can include the one from part (a).)
(continued on next page)
8. (a) Prove that every
number has a square root. (Hint: given real numbers x and
q complex
p
y, define u = 12 (x + x2 + y 2 ) and v = y/2u, and calculate (u + iv)2 .)
(b) Let A ∈ Mn×n (C) be a unitary matrix. Prove that there exists a unitary matrix B ∈
Mn×n (C) such that B 2 = A. (Hint: diagonalize.)
Bonus question. Let V be a complex vector space, let W be a complex inner product space, and
let T, U be two linear transformations from V to W . Suppose that hT (x), U (x)i = 0 for all x ∈ V .
(a) Show that R(T ) ⊆ R(U )⊥ and R(U ) ⊆ R(T )⊥ . (Hint: consider hT (x + y), U (x + y)i and
hT (x + iy), U (x + iy)i.)
(b) Suppose that x and y are vectors in V such that T (x) = U (y). Show that x ∈ N (T ) and
y ∈ N (U ).
This is the last homework of the course—congratulations on your hard work!