1. Let U = span(v1 , v2 , v3 , v4 ) where
A = 1
the vi are the columns of the matrix
4 0 2 1
2 1 2 3 .
2 1 4 4
Reduce the spanning list to a basis of U .
2. Let u1 , u2 be the rst two rows of the matrix A above. Expand u1 , u2 to
a basis of R5 .
Question 3: How do you find a basic feasible solution?
The initial simplex tableau allows us to calculate the locations of the corner points as
well as any other points where the lines corresponding to the equations cross or cross
the axes. Lets start wit
A Characterization of the Corner-Point Solutions
Consider the previous example again. Our first step is to rewrite that problem as:
2x1 +3x2 +s1
x1 , x2 0 and s
CNF Satisfiability Problem
Andrew Makhorin <[email protected]>
The Satisfiability Problem (SAT) is a classic combinatorial problem. Given a Boolean formula
of n variables
f (x1 , x2 , . . . , xn ),
this problem is to find such v
More information from http:/www.researchandmarkets.com/reports/2178088/
The Traveling Salesman Problem. A Guided Tour of Combinatorial
Optimization. Wiley Series in Discrete Mathematics & Optimization
Provides an indepth treatment of
Solutions for 5.2 and 5.3
Problem 7. Consider R = Q[x]/(x2 3). Each element of R can be written in the form [ax + b].
(Why?) Determine the rules for addition and multiplication of congruence classes.
Lecture 5. Basic Feasible Solution
Department of Industrial Engineering and Logistics
Shanghai Jiao Tong University
Mar 18, 2010
Mar 18 2010
Copyright Feng Chen 2004-2010. All rights reserved.
min z cT x
CSE 6311 : Analysis of Randomized Kth Smallest Selection
We will give two analysis in this notes. First, we will show that the expected number of
operations (or comparions) is linear for randomized Kth smallest algorithm. Secondly, we
will show the number
Algebra II Homework Five
1. (VIII.4.1) Let R be a commutative ring with identity and I a finitely generated ideal of
R. Let C be a submodule of an R-module A. Assume that for each r I there exists a
positive integer m (depending on r) such that rm A C. Sh
Basic Counting Principles
Consider a multistep process in which
Step 1 has n1 possible outcomes,
Step 2 has n2 possible outcomes,
Step r has nr possible outcomes.
Then, the entire process has n1 n2 nr possible outcomes.
Midterm # 1
All questions are equal value.
1. Are the following subspaces of R3 ? Justify your answer.
(a) cfw_(a, b, c) R3 : abc = 0
(b) cfw_(a, b, c) R3 : a + b + c = a + b = 0
(c) cfw_(a, b, c) R3 : a b c
2. Let V = P2 (R) be the vector space
INSTRUCTORS NAME (PLEASE CIRCLE):
C. JONES (1A)
B. MONSON (2A)
D. BARCLAY (3A)
S. BURGOYNE (4A)
J. THOMPSON (5A)
R. MCKELLAR (6A)
T. JONES (7A)
B. MONSON (8A)
DEPARTMENT OF MATHEMATICS & STATISTICS
1. Let T : C4 P 3 (C) be given by
T (a, b, c, d) = (a b)x3 + (a b + c d)x2 + dx + b
(a) Find M(T ) using the usual bases (e1 , . . . , e4 ) and (1, x, . . . , x3 ).
(b) Find a basis for the range of T .
(c) Find a basis for the null space of T .
Chapter 10, Field Extensions
You are assumed to know Section 10.1. Everything you have learned in linear algebra
applies regardless of what the field of scalars is. In particular, the definitions of vector
space, linear independence, basis and dimension a