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
MA 511
Final Exam
A. Eremenko
NAME
1. Let
2
4
A=
2
1
1
0
2
3 .
1
a) Factor A = L U , where L and U are lower and upper triangular, respectively.
b) Find dimensions of the four fundamental subspaces.
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,
Tru
MA 511 Prof. Jaroslaw Wlodarczyk
Spring 2008 Khalil Yousef
Homework 1
Problem Set 1.2: 4) 3 by 2 system Solution:
x + 2y = 2 x y =2 y =1
As we can see the system is not resolvable because not all the
Traces of elements of the modular group
Walter Bergweiler and Alex Eremenko
January 2, 2012
Let
A=
12
01
and B =
10
2 1
.
These two matrices generate the free group which is called (2), the principal
Trigonometric identity
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
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
Dan/1 Che/j
WW \ l
m/ Sci 5 (Sezm ze) liar/l, m3?)
l l
. 6. What 3 by 3 matrices represent the transformations that
(a) project every vector onto the x-y plane?
(b) r
2. If a 3 by 3 matrix has detA = 1, nd deteA), det(A), det(A2), and det(A"). ii\[email protected]
14. True or false, with reason if true and counterexample if false:
(a) If A and B are id
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
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
Homework 9
1. a) For which positive integers m is it possible that 2m and 2m+1 have
equal sums of digits in decimal system? Hint: Every number has the same
residue modulo 3, as sum of its digits in de
Homework 8
1. Expand the following functions into partial fractions:
z4
,
z3 1
1
.
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 le
Homework 5
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
Homework 4
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 d
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
MATH 511, First exam, Fall 2002
Name:
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.
1
2. Tell whether these matrices are invertible, and if yes, nd the
Determinants
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