...Excel and most other Windows-based programs. In this document the symbol >> represents the Matlab prompt in the command window. Commands following it indicate that you should type these in; Matlab commands or responses are printed in Bookman Font. Pushing the Enter key causes the command to be executed. Matlab has extensive help resources, available under the "Help" tab of the MATLAB window, or type >> help topic. Hint Use the ( key to recall previous commands, which you can then edit or re-execute. Type a letter or part command followed by the ( key, and the last instance of a matching command will be recalled. 1. Basic input and display Matlab stands for MATrix LABoratory, and was originally written to perform matrix algebra[2]. It is very efficient at doing this, as well as having the capabilities of any normal programming language. In most instances,...
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...high-risk decisions as a manager can be based. One also needs to know how to analyze the research findings. The study of quantitative techniques provides one with the knowledge and skills needed to solve the problems and the challenges of a fast-paced decisionmaking environment. Managers make decisions on a day to day basis and it is necessary for them to be able to analyze the data so as to be able to make optimal decisions. This module has ten lesson which cover matrix algebra, markov analysis, Linear programming, differentiation, applications of differentiation to cost, revenue and profit functions, integral calculus, inventory models, sampling and estimation theory, hypothesis testing and chi-square tests. iii MODULE OBJECTIVES By the end of the course, the student should be able to:- 1. Perform various operations on matrices matrix algebra, 2. Apply the concept of matrices in solving simultaneous equations, input-output analysis and markov analysis, 3. Formulate and solve Linear programming using the graphical and simplex method 4. Differentiate various functions and apply to cost, revenue and profit functions 5. Apply...
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...Lecture Notes on Mathematics for Economists Chien-Fu CHOU September 2006 Contents Lecture 1 Lecture 2 Lecture 3 Lecture 4 Lecture 5 Lecture 6 Lecture 7 Lecture 8 Lecture 9 Lecture 10 Static Economic Models and The Concept of Equilibrium Matrix Algebra Vector Space and Linear Transformation Determinant, Inverse Matrix, and Cramer’s rule Differential Calculus and Comparative Statics Comparative Statics – Economic applications Optimization Optimization–multivariate case Optimization with equality constraints and Nonlinear Programming General Equilibrium and Game Theory 1 5 10 16 25 36 44 61 74 89 1 1 Static Economic Models and The Concept of Equilibrium Here we use three elementary examples to illustrate the general structure of an economic model. 1.1 Partial market equilibrium model A partial market equilibrium model is constructed to explain the determination of the price of a certain commodity. The abstract form of the model is as follows. Qd = D(P ; a) Qd : Qs : P: a: Qs = S(P ; a) Qd = Qs , quantity demanded of the commodity quantity supplied to the market market price of the commodity a factor that affects demand and supply D(P ; a): demand function S(P ; a): supply function Equilibrium: A particular state that can be maintained. Equilibrium conditions: Balance of forces prevailing in the model. Substituting the demand and supply functions, we have D(P ; a) = S(P ; a). For a given a, we can solve this last equation to obtain the equilibrium...
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...1 When you do not specify an output variable, MATLAB uses the variable ans, short for answer, to store the results of your calculation. sin(a) ans = 0.8415 If you end a statement with a semicolon, MATLAB performs the computation, but suppresses the display of output in the Command Window. sin(a); At any time you want to know the active variables you can use: KMLIM TCI2261 2012/2013 Arrays MATLAB is an abbreviation for "matrix laboratory."While other programming languages mostly work with numbers one at a time, MATLAB is designed to operate primarily on whole matrices and arrays. All MATLAB variables are multidimensional arrays, no matter what type of data. A matrix is a two-dimensional array often used for linear algebra. Row vector: comma or space separated values between brackets Column vector: semicolon separated values between brackets Matrices Matrix...
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...coordinates of the image are (8, -1). Example 2: A rectangle has coordinates (1, 1), (4, 1), (4, 3) and (1, 3). Find the coordinates of the image of the rectangle under the transformation represented by the matrix . Solution: You could find the image of each vertex in turn by finding , etc. However, it is more efficient to multiply the transformation matrix by a rectangular matrix containing the coordinates of each vertex: . So the image has coordinates (2, 0), (11, -3), (9, -1) and (0, 2). The diagram below shows the object and the image: Any transformation that can be represented by a 2 by 2 matrix, , is called a linear transformation. 1.1 Transforming the unit square The square with coordinates O(0, 0), I(1, 0), J(0, 1) and K(1, 1) is called the unit square. Suppose we consider the image of this square under a general linear transformation as represented by the matrix : . We therefore can notice the following things: * The origin O(0, 0) is mapped to itself; * The image of the point I(1, 0) is (a, c), i.e. the first column of the transformation matrix; * The image of the point J(0, 1) is (b, d), i.e. the second column of the transformation matrix; * The image of the point K(1, 1) is (a + b, c+ d), i.e. the result of finding the sum of the entries in each row of the matrix. Example: Find the image of the...
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...MATH 3330 INFORMATION SHEET FOR FINAL EXAM FALL 2011 FINAL EXAM will be in PKH 103 at 2:00-4:30 pm on Tues Dec 13 • See above for date, time and location of FINAL EXAM. Recall from the first-day handout that any student not obtaining a positive score on the FINAL EXAM will not pass this class. • The material covered will be the same as that covered on the homework from the start of the semester through Dec 6 (but not §6.3) inclusive. (Homework is listed at my website: www.uta.edu/math/vancliff/T/F11 .) • My remaining office hours are: 3:30-4:20 pm on Thurs Dec 8 and 3:30-5:30 pm on Mon Dec 12. • This test will be, in part, multiple choice, but you do NOT need to bring a scantron form. There will be several choices of answer per multiple-choice question and, for each, only one answer will be the correct one. You should do rough work on the test or on paper provided by me. No calculator is allowed. No notes or cards are allowed. BRING YOUR MYMAV ID CARD WITH YOU. • When I write a test, I look over the lecture notes and homework which have already been assigned, and use them to model about 85% of the test problems (and most of them are fair game). You should expect between 30 and 40 questions in total. • A good way to review is to go over the homework problems you have not already done & make sure you understand all the homework well by 48 hours prior to the test. You should also look over the past tests/midterms and understand those fully. In addition, this...
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............3, 7 cofactor ........................4 coordinate vector ..........9 Cramer's rule................1 determinant...............2, 5 diagonal matrix .............6 diagonalizable...............8 dimension .....................6 dot product ...................8 eigenbasis ....................7 eigenspace...................7 eigenvalue ....................7 eigenvector...................7 geometric multiplicity....7 identity matrix ...............4 image ...........................6 inner product................9 inverse matrix...............5 inverse transformation..4 invertible.......................4 isomorphism.................4 kernal ...........................6 Laplace expansion by minors .....................8 linear independence.....6 linear transformation.....4 lower triangular.............6 norm .......................... 10 nullity............................ 8 orthogonal ................ 7, 9 orthogonal diagonalization ................................ 8 orthogonal projection.... 7 orthonormal.................. 7 orthonormal basis ........ 7 pivot columns............... 7 quadratic form.............. 9 rank.............................. 3 reduced row echelon form ................................ 3 reflection ...................... 8 row operations ............. 3 rref ................................3 similarity .......................8 simultaneous equations 1 singular.........................8 skew-symmetric............6 span .............................6...
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...2001 Submitted by Arun Avudainaygam LINEAR AND ADAPTIVE LINEAR MULTIUSER DETECTION IN CDMA SYSTEMS Project Website: http://arun-10.tripod.com/mud/mud.html SECTION 0 Introduction Multiuser detection is a technology that spawned in the early 80’s. It has now developed into an important, full-fledged field in multi-access communications. Multiuser Detection (MUD) is the intelligent estimation/demodulation of transmitted bits in the presence of Multiple Access Interference (MAI). MAI occurs in multi-access communication systems (CDMA/ TDMA/ FDMA) where simultaneously occurring digital streams of information interfere with each other. Conventional detectors based on the matched filter just treat the MAI as additive white gaussian noise (AWGN). However, unlike AWGN, MAI has a nice correlative structure that is quantified by the cross-correlation matrix of the signature sequences. Hence, detectors that take into account this correlation would perform better than the conventional matched filter-bank. MUD is basically the design of signal processing algorithms that run in the black box shown in figure 0.1. These algorithms take into account the correlative structure of the MAI. 0.1 Overview of the project This project investigates a couple of different approaches to linear multiuser detection in CDMA systems. Linear MUDs are detectors that operate linearly on the received signal statistic i.e., they perform only linear transformations on the received statistic....
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...TUTOR CLASS Content: Covered all chapter 3, 4, 5. No: * Linear Approximation * Linear programming * Leontief model Chapter 3: INTRODUCTION TO LINEAR EQUATIONS ANDMATRICES (Gaussian elimination) 1. Solve the following system x1-4x2+x3=5-2x1-5x2+3x3=33x1+2x2-x3=1 Using: A, Gauss elimination method B, Cramer’s rule. (chapter 4) Chapter 4: EIGENVECTORS AND EIGENVALUES (Determinants, Cramer’s rule, Matrix Inverse, Eigenvectors, Eigenvalues) 2. Let A b the matrix defined by A=3 4 -1 04 -1 0 3-1 1 2 3 A, Applying the row operation 4 R4+R2→R2 to A, what is the resulting matrix B we get? B, Use row operations to transform A to the form C where C=0 a b c0 d e f0 g h i-1 1 2 3 And where the letters a, b, c, d, e, f, g, h, g and I are the numbers that you have to find out (depending on your row operations). Find a relation between the determinant of A and determinant of C. C, Evaluate the determinant of A, detA, by using the properties of determinants and cofactor expansions along rows and columns of your choice. 3. ( Matrix Inverse) Recall that if a matrix A is invertible, then the unique solution of the linear system of equations AX= b is given by X= A-1b Given the linear system of equations x+2y-9z =1-2x-4y+19z=0-y+2z =1 A, Find the inverse of the matrix A of coefficient of the linear system above, where A= 1 2 -9-2 -4 190...
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...SUPLEMEN Pemodelan Sistem / Pengolahan Sinyal / Metode Kuantitatif TUTORIAL SINGKAT MATLAB oleh: Judi Prajetno Sugiono Sekolah Tinggi Teknik Surabaya (2005, 2008, 2011) judi@stts.edu ©2005 p. 1 of 40 MATLAB Short Tutorial Reserve word (don’t used it as variable's name) · · · · · ans pi nan inf eps Special sign · · · · · % [] ; ' : line comment begin - end of matrix row separation, or not echoed command if place in the end of a statement begin - end of string indexing sign Variable is assume as matrix % empty matrix A=[] A = [] % matrix 1x1 or a constant A=[0] A = 0 % same with A=0 A = 0 % complex number: use i or j to express imaginary part z=3+4j z = 3.0000 + 4.0000i Entry a matrix % use as column separation and or as row separation A=[1 2 3; 4 5 6; 7 8 9] A = 1 4 7 2 5 8 3 6 9 Last saved by jpsugiono 9/23/2011 judi@stts.edu ©2005 p. 2 of 40 How to point element of matrix % A(row,column) A(1,3) ans = 3 % sign use as get all row or column A(2,:) ans = 4 5 6 % sign use as get from m to n cell in row or colomn A(1:2, 2:3) ans = 2 5 3 6 row and column vector % row vector a=[0 1 2 3 4 5] a = 0 1 2 3 4 5 % column vector b=[0; 1; 3; 4; 5] b = 0 1 3 4 5 % Shortcut to build a vector % init:step:final a=0:0.2:1 a = 0 0.2000 0.4000 0.6000 String % begin and end with < ' >, and act like a matrix of character ...
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...1) Prior to analysis, the data from a survey is arranged into a(n) ________. A) information graph B) data graph C) information matrix D) data matrix E) data table Answer: D Diff: 1 Page Ref: 266 AACSB: Analytic Skills Objective: LO 1 2) ________ refers to the creation of a computer file that holds the raw data taken from all of the acceptable completed questionnaires. A) Data entry B) Data coding C) Data matrix D) Data filing E) Data surveying Answer: A Diff: 2 Page Ref: 266 AACSB: Use of IT Objective: LO 1 3) ________ is the identification of code values that are associated with the possible responses for each question on the questionnaire. A) Data entry B) Data coding C) Data matrix D) Data filing E) Data surveying Answer: B Diff: 2 Page Ref: 226 AACSB: Analytic Skills Objective: LO 1 4) Errors that pertain to anything in the research process except sample size are known as: A) summarization errors B) non-sampling errors C) field errors D) generalization errors E) none of the above; all errors are derived from sample size Answer: B Diff: 2 Page Ref: 226 AACSB: Analytic Skills Objective: LO 3 5) Fieldworkers who cheat by submitting bogus questionnaires are an example of: A) intentional fieldworker error B) unintentional fieldworker error C) intentional respondent error D) unintentional respondent error E) none of the above; fieldworkers do not cheat Answer: A Diff: 1 Page Ref: 227 AACSB: ...
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...Republic Polytechnic A114 Mathematics II Worksheet 03: Points on the Same Plane Discussion 1. Discuss the characteristics of a flat plane. How would a flat plane look like? Give some examples of a flat plane. Is define as a surface with no thickness. 2. What is a straight line? Give some examples of a straight line. Y=mx+c, it’s a line where 2 points meet. a) Given any two points in space, how would you find the vector from one point to another point? b) What is the relationship of this vector with the straight line that passes through the two points? 3. What do you understand by the term “normal vector to a flat plane”? What is/are the angle/angle(s) between the normal vector and the flat plane? Describe the relationship between any line on a plane with respect to the normal vector of the flat plane. Exploring and Understanding 4. Figure 1 Figure 1 above shows three points (i.e. A, B and C) lying on the same plane. a) From the information given, find vectors , and . = (0 3 -5) = ( b) From what you have discussed in Q2 and Q3, what is the relationship between the normal vector to the flat plane and the three vectors , and ? The normal vector is always perpendicular to the flat plane. c) What can you conclude about any vectors that lie on the plane with the normal vector to the plane? All perpendicular to the plane. Dot Product 5. Similar to multiplication of two...
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...1.0 Introduction In this era of information technology, civil engineers rely heavily on software to perform their design tasks. Unfortunately, most commercial structural analysis packages are closed-source, which means that the operations that the program performs cannot be inspected by the user. Moreover, such software packages are invariably very pricey, and, hence, are generally not affordable for students and smaller engineering firms. The objective of this design project was to design a structural analysis program that would be free of charge and available to all. This computer program was to be open source and well commented, so that its users could comprehend the operations performed in the analysis of a given structure. To accomplish these objectives, the generalized stiffness method of structural analysis was implemented into a computer algorithm. This algorithm, called “TrussT Structural Analysis”, is a collection of visual basic modules embedded in a Microsoft Excel document using Visual Basic for Applications (VBA). This design report outlines the theory behind TrussT Structural Analysis, as well as the methods by which that theory was implemented into computer algorithms. The first two sections of this report present the theory of the generalized stiffness method of structural analysis and its implementation into a computer algorithm. The following sections present the procedures by which the stiffness method was modified to incorporate the analysis...
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...Page Reference Manual Compound List File List Theory: The trace of a field element | You probably already heard of the trace of a matrix. It is defined as the sum of the diagonal elements of a square matrix. What is so special about that? Well, as explained by that link, the trace is basis invariant. Let L : K be a finite field extension, like is a field extension of 2. Given some linear transformation A: L L you can write that linear transformation in the form of a matrix equation (because it is linear) after chosing some basis for L that associates L with a vector space. This equation is linear in x because y is fixed. We already saw discussed one such basis for our field , elsewhere: (1, t, t2, t3, ..., tm-1). Lets work out a simple example using this polynomial basis. Let m = 4, using reduction polynomial t4 + t + 1. Let y = t, some element of our field 24. The linear transformation that sends x yx is then given by, Atx = tx. The trace of the matrix should be independent of the chosen basis, so that we might as well talk about "the trace of the linear transformation", or even about "the trace of y", for y uniquely determines this transformation. Multiplying with t means that this transformation sends 1 to t, t to t2, ... and tm-2 to tm-1. Only the transformation of the last member of our polynomial basis is a little different: tm-1 ttm-1 = tk + 1. Writing out the matrix for the given example gives therefore, where xi is the coefficient corresponding...
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...SCHAUM’S outlines SCHAUM’S outlines Linear Algebra Fourth Edition Seymour Lipschutz, Ph.D. Temple University Marc Lars Lipson, Ph.D. University of Virginia Schaum’s Outline Series New York Chicago San Francisco Lisbon London Madrid Mexico City Milan New Delhi San Juan Seoul Singapore Sydney Toronto Copyright © 2009, 2001, 1991, 1968 by The McGraw-Hill Companies, Inc. All rights reserved. Except as permitted under the United States Copyright Act of 1976, no part of this publication may be reproduced or distributed in any form or by any means, or stored in a database or retrieval system, without the prior written permission of the publisher. ISBN: 978-0-07-154353-8 MHID: 0-07-154353-8 The material in this eBook also appears in the print version of this title: ISBN: 978-0-07-154352-1, MHID: 0-07-154352-X. All trademarks are trademarks of their respective owners. Rather than put a trademark symbol after every occurrence of a trademarked name, we use names in an editorial fashion only, and to the benefit of the trademark owner, with no intention of infringement of the trademark. Where such designations appear in this book, they have been printed with initial caps. McGraw-Hill eBooks are available at special quantity discounts to use as premiums and sales promotions, or for use in corporate training programs. To contact a representative please e-mail us at bulksales@mcgraw-hill.com. TERMS OF USE This is a copyrighted work and The McGraw-Hill Companies,...
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