...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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...Birzeit University MBA Program Managerial Accounting BUSA 631 Spring 2013 Case 3 The Rohr Company’s old equipment for making subassemblies is worn out. The company is considering two courses of action: (a) Completely replacing the old equipment with new equipment or (b) Buying subassemblies from a reliable outside supplier, who has quoted a unit price of $1 on a 7-year contract for a minimum of 50,000 units per year. Production was 60,000 units in each of the past 2 years. Future needs for the next 7 years are not expected to fluctuate beyond 50,000 to 70,000 units per year. Cost records for the past 2 years reveal the following unit costs of manufacturing the subassembly: | Per Unit | Direct Materials | $0.30 | Direct Labor | 0.35 | Variable Overhead | 0.10 | Fixed Overhead (including $0.10 depreciation and $0.10 for direct departmental fixed overhead) | 0.25 | Total Cost | $1.00 | | | The new equipment will cost $188,000 cash, will last 7 years, and will have a disposal value of $20,000. The current disposal value of the old equipment is $10,000. The sales representative for the new equipment has summarized her position as follows: The increase in machine speeds will reduce direct labor and variable overhead by $0.35 per unit. Consider last year’s experience of one of your major competitors with identical equipment. It produced 100,000 units under operating conditions very comparable to yours and showed the following unit costs...
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...VECTORS (I) Introduction Quantities with magnitude but without direction are called scalars. Examples: distance, speed, and mass. Quantities with both magnitude and direction are called vectors. Examples: displacement, velocity, and weight. Geometrical Representation of Vectors A vector can be represented with a line segment with an arrowhead indicating its specific → direction. Thus the displacement from the point A to the point B is the vector AB , represented by the straight line AB in the direction from A to B. B A Magnitude of a vector The magnitude of the vector AB , denoted by AB , is the length (or distance) AB. Negative Vectors The negative vector – a has the same magnitude as a but is in opposite direction of a . Thus → → → → – BA = AB . Zero Vectors A zero vector is any vector of zero magnitude, denoted by 0 . [Note: 0 ≠ 0] Unit Vector ˆ A unit vector is a vector with magnitude 1. A unit vector in the direction of a is denoted by a ˆ where a = a . a Page 1 Position Vector Position vectors give the location of points with respect to a fixed point of reference (commonly known called the origin). Usually it is equivalent to the coordinates of a point. Equal Vectors → → Two vectors are equal if they have the same magnitude and direction. Thus if AB = PQ , then AB = PQ and AB // PQ . → → → → Example: The position vectors of A, B and C are a = 2i + 3j – 4k, b = 5i – j + 2k and c = 11i + j + 14k. Find the position...
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... 10 a. Geology and Glaciology 10-11 b. Vibration Analysis 11-12 c. Tensor of Moment of Inertia 12 d. Stress Tensor 12 e. Basic Reproduction Number. 12 6. Conclusion 13 7. References 13 3 Abstract In abstract linear algebra, these concepts are naturally extended to more general situations,...
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...Matrix Calculations Using Calculators The calculators most commonly used by Foundation Studies students at UTAR are Casio fx-350MS, fx-570MS, and fx-570ES. Casio fx-350MS has no matrix function and cannot be used to perform matrix calculation. Both Casio fx-570MS and Casio fx-570ES have matrix mode. Casio fx-570ES can display fully a 3×3 matrix with all its 9 elements visible. Casio fx-570MS has only a 2-line display and can only show matrix elements one at a time. Hence the operations for matrix computation are different for these two series of calculators. The general procedures in matrix calculations are as follows: (1) Enter Matrix mode of the calculators. (2) Assign a variable to store the matrix. There are 3 variables available: MatA, MatB, and MatC. (3) Select the dimension or order (1×1 to 3×3) of the matrix. (4) Input the elements of the matrix; the data will be automatically saved in the matrix variable assigned. (5) Exit the matrix input or edit mode by pressing the coloured AC key. (6) Press SHIFT MATRIX or SHIFT MAT (Matrix function is at numeric key 4) to recall the stored matrix and perform matrix calculations as needed. (a) Casio fx-570ES (a) Entering a matrix. 1. 2. 3. 4. Press the MODE key. Select 6:Matrix mode. A display Matrix? will be shown to let us select one of the 3 possible matrix variables allowable. Let us choose 1:MatA by pressing 1. The next display permits us to select the order (m×n) of the matrix MatA. Let us choose 1: 3×3 by pressing 1. An...
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