# Homework, MATH 405, Spring 2011

Homework is from our text (Lang) unless indicated.
Homework is due at the beginning of class on the day listed.
The due dates for listed problems (and problems in later chapters) will come later.

HW 1: due Monday 31 January
Chapter 1
Sec. 1: 1,5,10,11, 13
Sec. 2: 1h, 2a, 3b, 5g, 9
Sec. 4: 1

HW 2: due Monday 7 February
Chapter 2
Sec. 1 (Matrices): 5, 10
Sec. 1 (Dimension): 4
Sec. 2: 2,3,4
Sec. 3: 3a,5,6,8,9,11,13,19,32
Problem X: Suppose A is a square matrix and A^7 - 4A^5 + (sqaure root of 5)A^2 + I = 0. Show that A is invertible.

HW 3: due Friday 11 February
Chapter 3
Sec. 1: 1a, 2, 7a,8
Sec. 2: 1cdh (give proof),9, 10 [[also for #10: if P is the parallellogram spanned by v_1 and v_2, what are the vertices (corners) of P, and what are the vertices of f(P)?]], 15, 17, 18a

HW 4: due Friday 18 February
Chapter 3
Sec. 3: 4 [then read problem 8], 14, 18 [you can cite your HW problem 4, p.22]
Sec. 4: 4, 7 [give the rule for the inverse map], 11, 16, 19, 21
Sec. 5: 2, 6 [then read the Remark and problems 3,4,5]

HW 5: due Friday 25 February
Chapter 4
Sec. 2: 1ad, 2f, 5, 7, 9
Sec. 3: 1,8abc, 9, 10b, 11
Problem X: Let V denote the vector space of 10 x 10 symmetric matrices. Let W denote the vector space of 10 x 10 upper triangular matrices. Define a map T which takes a matrix A in V and produces a matrix T(A) = B in W, such that T is an isomorphism of vector spaces, V to W. No proof necessary -- just give a definition of B from A.

HW 6: due Monday 7 March
Chapter 5
Sec. 1: 2
Sec. 2: 0, 1a,2b, 3, 5, 6a, 7c (you already know dimV in 7c from an earlier assignment)

HW 7: due Friday 11 March
Chapter 5
Sec. 3: 4a, 4d, 5b
Sec. 4: 1, 3
Sec. 5: 1a, 2
Sec. 6: 3, 4, 5
Sec. 7: 2, 3cd

HW 8: due Friday 18 March
Chapter 5, Sec. 8: 1,2
Chapter 6, Sec. 2: 1
Chapter 6, Sec. 3: 1de, 2bh, 5, 6a, 8, 11

March 21-25 : Spring Break

HW *: finish by Friday April 1
Don't hand this in -- I will test at least one on a quiz or the next exam.
Chapter 6:
Sec. 5: 1b
Sec. 6: 1fh, 2fh
Sec. 9: 4
Problem X: Let A be an n x n matrix with only integers as entries and with nonzero determinant.
Consider the following conditions:
(1) The inverse of A has only integer entries.
(2) det(A) is 1 or -1.
Prove (2) implies (1).
(Optional. (1) does imply (2) -- can you prove it?)

HW 9: due Friday 8 April
7.1: 1,2,4,5,6,14
7.2: 1-5, 11
7.3: 6,7,13

HW 10: due Friday 15 April
From Chapter 8:
Sec. 1: 2, 5
[and to 5, add part c: Show that if a 2x2 matrix A has eigenvalues 1 and -1 (so, there exists a line L such that for all x, Ax is the reflection of x through L), then A has the form of A in part (a) for some angle theta. Which angles theta occur in this way? For these, what is the slope of the line L?
Sec. 2: 3a, 5ab, 11, 12, 15; and, find the characteristic polyonmial and eigenvalues of the matrices in 4a and 7a
Sec. 3: 1b, 2a, 3
Sec. 4: 3, 5, 7, 9a, 11, 16ab, 22

HW 11: due Friday 22 April
Chapter 8: Sec. 5: 3: 5, 6, 7a, 9, 10
Chapter 9: Sec. 2: 4,5.

HW 12: due Friday 29 April
Chapter 10:
Section 1: 6
Problem U: Suppose \$A\$ is an n by n matrix over the field K such that every root of its chracteristic polynomial lies in the field K. Prove that there is a complex unitary matrix U such that U^{-1}AU is upper triangular.
[Hint. You know from Chapter 10 (and can assume) that (1) A has a fan basis and (2) if the columns of U are the elements of a fan basis, chosen in order, then U^{-1}AU is upper triangular. So it suffices to take a fan basis {v_1, ... ,v_n} for A and from it produce a fan basis {u_1, ... , u_n} for A which is also an orthonormal basis.]
Chapter 11:
Sec. 6: 1,3
Sec. 1: 1, 2
Sec. 2: 1,2
[Historical note: Leibniz claimed for a time that if b is a real number, then the polynomial x^4 + b^4 has no root which is a complex number. So in 10.2.2, if you also factor the given quartic into linear terms with complex roots, then you would be constructing an explicit example proving Leibniz was wrong.]