1. Let U = span(v1 , v2 , v3 , v4 ) where
2
A = 1
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
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 t
A Characterization of the Corner-Point Solutions
Consider the previous example again. Our first step is to rewrite that problem as:
Maximize
Subject to:
4x1 +3x2
2x1 +3x2 +s1
3x1 +2x2
+s2
2x2
+s3
2x1
CNF Satisfiability Problem
Andrew Makhorin <[email protected]>
August 2011
1
Introduction
The Satisfiability Problem (SAT) is a classic combinatorial problem. Given a Boolean formula
of n variables
f (x1 ,
Brochure
More information from http:/www.researchandmarkets.com/reports/2178088/
The Traveling Salesman Problem. A Guided Tour of Combinatorial
Optimization. Wiley Series in Discrete Mathematics & Opt
Math 236
Fall 2008
Dr. Seelinger
Solutions for 5.2 and 5.3
Section 5.2
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 additi
Lecture 5. Basic Feasible Solution
Feng Chen
Department of Industrial Engineering and Logistics
Management
Shanghai Jiao Tong University
Mar 18, 2010
Mar 18 2010
1
Copyright Feng Chen 2004-2010. All r
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 Kt
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
p
Basic Counting Principles
Multiplication Principle
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
Math 3213
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,
STUDENTS NAME:
ID #:
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 MATHEM
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
MATH 3213
Assignment # 8
Due FRIDAY, Decemeber 2, 2011
1. Do 8, 9, 10, 22, 24, 25, from the text.
2. Let U = span(1, x). Find a basis for U in P3 (F) where the inner product is f, g =
1
1 f (x)g(x)dx
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 v