Part 1
Foundations
Chapter 1
Sets and vector spaces
We assume some familiarity with basic notions from set theory and linear
algebra. For example the reader should be comfortable working with nite
dim
Calulus
Rie Mathematis Tournament 2000
1. Find the slope of the tangent at the point of inetion of y = x3
3x2 + 6x + 2000.
2. Karen is attempting to limb a rope that is not seurely fastened. If she pu
Harvard-MIT Mathematics Tournament
February 19, 2005
Team Round A
Disconnected Domino Rally [175]
On an infinite checkerboard, the union of any two distinct unit squares is called a (disconnected) dom
Harvard-MIT Mathematics Tournament
March 15, 2003
Individual Round: Calculus Subject Test
1. A point is chosen randomly with uniform distribution in the interior of a circle of radius
1. What is its e
Harvard-MIT Mathematics Tournament
February 19, 2005
Team Round B
Disconnected Domino Rally [150]
On an infinite checkerboard, the union of any two distinct unit squares is called a (disconnected) dom
HMMT 1998: Calculus Solutions
1. Problem: Farmer Tim is lost in the densely-forested Cartesian plane. Starting from the origin he walks
a sinusoidal path in search of home; that is, after t minutes he
Harvard-MIT Mathematics Tournament
March 15, 2003
Individual Round: Calculus Subject Test Solutions
1. A point is chosen randomly with uniform distribution in the interior of a circle of radius
1. Wha
Power Question - Coloring Graphs
Perhaps you have heard of the Four Color Theorem (if not, dont panic!), which essentially says that any
map (e.g. a map of the United States) can be colored with four
Calculus Solutions
Harvard-MIT Math Tournament
February 27, 1999
Problem C1 [3 points]
Find all twice differentiable functions f (x) such that f (x) = 0, f (0) = 19, and f (1) = 99.
Solution: Since f
Algebra: Assignment 8
Due on Wednesday, December 5, 2012
Brundan 1:00pm
A digital copy of this document can be found at http:/pages.uoregon.edu/raies
Dan Raies
Last edited December 6, 2012
Contents
Ex
Algebra: Assignment 7
Due on Wednesday, November 21, 2012
Brundan 1:00pm
A digital copy of this document can be found at http:/pages.uoregon.edu/raies
Dan Raies
Last edited November 25, 2012
Contents
Algebra Midterm 1
Fall 2012
Work out if each of the following statements is TRUE or FALSE.
Justify your answer carefully by supplying a PROOF or a COUNTEREXAMPLE.
1. Let Vecf pRq be the category of ni
Modules review
True or False?
1. Let V W X and V W Y be decompositions of a left
R-module V as direct sums of submodules. Then X Y .
2. Let V W X and V W Y be decompositions of a left
R-module V as di
Sample homework
solutions 1
1.2.3 Let V be a nite dimensional vector space and W V be a subspace. Show that W is nite dimensional and any basis of W can be
extended to a basis of V . Deduce that dim W
Abstract Algebra: Assignment 1
Due on Friday, October 5, 2012
Brundan 1:00pm
A digital copy of this document can be found at http:/pages.uoregon.edu/raies
Dan Raies
Last edited November 21, 2012
Conte
Abstract Algebra: Assignment 2
Due on Friday, October 12, 2012
Brundan 1:00pm
A digital copy of this document can be found at http:/pages.uoregon.edu/raies
Dan Raies
Last edited November 21, 2012
Cont
Abstract Algebra: Assignment 3
Due on Friday, October 19, 2012
Brundan 1:00pm
A digital copy of this document can be found at http:/pages.uoregon.edu/raies
Dan Raies
Last edited November 21, 2012
Cont
Algebra: Assignment 4
Due on Friday, October 26, 2012
Brundan 1:00pm
A digital copy of this document can be found at http:/pages.uoregon.edu/raies
Dan Raies
Last edited November 21, 2012
Contents
Exer
Algebra: Assignment 5
Due on Firday, November 2, 2012
Brundan 1:00pm
A digital copy of this document can be found at http:/pages.uoregon.edu/raies
Dan Raies
Last edited November 21, 2012
Contents
Exer
1998 Power Question Solutions
I. Graphs, total of 20 points
a. completely correct gets 1 point, total of 6 points
i. yes. vertices A,B,C,D, edges AB,AC,AD,BD
ii. no. A and B are connected twice
iii. y