Answers and some solutions for the midterm, 511 (Spring 2011)
1. Statements B and D are true. To see that D is true, notice that columns
of AT B are linear combinations of columns of AT , but AT has m columns
and m < n. So the dimension of the column spac
a) Factor A = L U , where L and U are lower and upper triangular, respectively.
b) Find dimensions of the four fundamental subspaces.
c) Find a basis for the column space of A.
d) Find a ba
Math 511, Spring 2012 Midterm exam solutions
1. Circle the letters corresponding to the statements which are true for all
m n matrices A and B , where m < n:
A. Ax = 0 has innitely many solutions,
True. Because m < n
B. Ax = b has solutions for every b Rm
Here is a short proof of the sin-product identity without almost any computation. It is due to Mario Bonk from U. Michigan.
The polynomial P (x) = xn1 + xn2 + . + 1 has the n-th roots of unity
dierent from 1 as zeros. Put x = y + 1,
Review for Final Exam.
The exam will be Wednesday May 5, 7:009:00 PM in HORT 117.
The exam will cover homework assignments 1014 and the assigned reading.
Here are some review problems. You should also study the homework, especially the harder problems.
Traces of elements of the modular group
Walter Bergweiler and Alex Eremenko
January 2, 2012
and B =
These two matrices generate the free group which is called (2), the principal
congruence subgroup of level 2.
With arbitrary integers
MA EH \Aom wovk \
Problems 10-19 study elimination on 3 by 3 systems (and possible failure).
10. Reduce this system to upper triangular form by two row operations:
2. If a 3 by 3 matrix has detA = 1, nd deteA), det(A), det(A2), and det(A"). ii\iiiiiiiiiiiiiiiiiiiiii@
14. True or false, with reason if true and counterexample if false:
(a) If A and B are identical except that bu = 2a. 1, then detB = 2detA.
WW \ l
m/ Sci 5 (Sezm ze) liar/l, m3?)
. 6. What 3 by 3 matrices represent the transformations that
(a) project every vector onto the x-y plane?
(b) reect every vector through the x-y plane?
(c) rotate the
Proof that t exp(it) : R T is surjective
Another proof is in Ahlfors, p. 45, or Whittaker-Watson, vol. 1, Appendix. All other authors seem to rely on the facts about trigonometric
functions proved in high school.
A topological group G is a set with a grou
Chasing a U-boat. Solution
We are looking for a parametrized curve, whose speed at every point is k
times the distance from the point to the origin. In the original problem
k = 2. It is natural to use polar coordinates r and . Let us choose as
fX (x)dx = fX (t) + fX (t), t 0,
and f|X | (t) = 0 if t < 0.
(Using the ordinary change of the variable formula here is wrong: rst
|x| is not dierentiable, second, it is not one-to-one.)
2. (t) = t (Change of the Variable formula).
f|X | (t) =
1. Expand the following functions into partial fractions:
z (z + 1)2 (z + 2)3
2. If Q is a polynomial with distinct roots 1 , . . . , n , and if P is a
polynomial of degree less than n, prove that
P (k )
P (z )
Q(z ) k=1 Q (
1. A projectile of mass m is red vertically from a cannon with initial speed
v0 . The air resistance has magnitude kv 2 , where k is a constant. It is clear
that the projectile will accent for some time, until it reaches some maximal
1. Parachute. There are two forces acting on a parachute with a parachutist:
the force of gravity and the air resistance. Suppose that the air resistance is
kv 2 , where v is the speed of descent, and k is a positive constant.
a) In which units
Discrete Fourier Transform
Problems with star are just for fun: they are not a part of the homework,
and their solutions are not used in the text.
We begin by recalling prerequisites: some facts about integers and complex numbers.
1. Arithmetic modulo N .
MATH 511, First exam, Fall 2002
1. Compute the rank of the matrix
2 1 37
3 2 1 7
1 4 57
and nd a basis of its column space.
2. Tell whether these matrices are invertible, and if yes, nd the inverses.
A = 1 1 1 ,
1. Permutations. Suppose that (j1 , j2 , . . . , jn ) is a permutation of (1, 2, . . . , n),
that is each jk is one of the integers between 1 and n, and every such integer occurs exactly once. Every permutation can be obtained starting from
String with beads
Let us consider a massless string of length L stretched with force .
Suppose that the ends of the string are xed and n beads of mass m are
placed with equal spacing
between them. Denoting by xj the displacement of the j -th