7.2 Application to economics: Leontief Model
|
|
- Shona Hoover
- 7 years ago
- Views:
Transcription
1 7 Application to economics: Leontief Model Wassil Leontief won the Nobel prize in economics in 97 The Leontief model is a model for the economics of a whole countr or region In the model there are n industries producing n different products such that the input equals the output or, in other words, consumption equals production One distinguishes two models: open model: some production consumed internall b industries, rest consumed b eternal bodies Problem: Find production level if eternal demand is given closed model: entire production consumed b industries Problem: Find relative price of each product The open Leontief Model Let the n industries denoted b S, S,, S n The echange of products can be described b an input-output graph Here, a ij denotes the number of units produced b industr S i necessar to produce one unit b industr S j and b i is the number of eternall demanded units of industr S i Eample: Primitive model of the econom of Kansas in the 9 th centur
2 The following equations are satisfied: Production of Total output Internal consumption + Eternal Demand farming industr (in tons: horse industr: + (in km horse rides In general, let,,, n, be the total output of industr S, S,, S n, respectivel Then a + a + + a n + b a + a + + a n + b n a n + a n + + a nn n + b n, since a ij j is the number of units produced b industr S i and consumed b industr S j The total consumption equals the total production for the product of each industr S i Let a a n, B b, X A a n a nn b n n A is called the input-output matri, B the eternal demand vector and X the production level vector The above sstem of linear equations is equivalent to the matri equation X AX + B In the open Leontief model, A and B determine X from this matri equation We can transform this equation as follows: I n X AX B (I n AX B X (I n A B are given and the problem is to if the inverse of the matri I n A eists ((I n A is then called the Leontief inverse For a given realistic econom, a solution obviousl must eist For our eample we have: ( 5 5 A, B ( 8,,, X (
3 We obtain therefore the solution X (I A B (( ( ( 5 5 8,, ( ( ,, ( ( 5 8, 9 95, (,,, ie,, tons wheat and Million( km horse ride 7, If the eternal demand changes, e B, we get, 5 ( (I A B ( ( ( 5 7, 9, , 5, 5 ie, one doesn t need to recompute (I A One difficult with the model: How to determine the matri A from a given econom? Tpicall, X is known, B is known and (a ij j i,j, b n n is known One takes therefore the matri (a ij j i,j, n and divides the j-th column b j for j,, n to get A Eample: An econom has the two industries R and S The current consumption is given b the table consumption R S eternal Industr R production 5 5 Industr S production 6 Assume the new eternal demand is units of R and units of S Determine the new production levels Solution: ( The total production ( is ( units for R and units for S We obtain 5 ( 5 X, B, A 6, and B The solution is X (I A B ( ( ( The new production levels are 7 and 7 for R and S, respectivel b n,
4 The closed Leontief Model The closed Leontief model can be described b the matri equation X AX, ie, there is no eternal demand The matri I n A is usuall not invertible (Otherwise, the onl solution would be X The input-output graph looks now as follows: There is onl internal consumption Eample: Etended model of the econom of Kansas in the 9 th centur including labor The corresponding matri equation is: z z
5 If X is a solution, also t X for ever t > is a solution (Usuall, one gets a one parameter famil of solutions If, we can assume, b choosing the appropriate parameter t One obtains then the solution,, 9 66, z For this computation, it is important to use rational numbers (ie, fractions as matri entries since otherwise the approimation to the matri I n A usuall will be invertible and onl the trivial uninteresting solution,, and z will eist This is also the reason, wh the entr a has large numerator and denominator In a closed econom, the absolute units of output are less interesting More important is the relative consumption of a product We can normalize therefore the matri A such that the sum of ever row is This is a matri Ã, such that à The recipe is: Divide the i-th row of A b the i-th component of A For our eample, we have A (that is the sum of the i-th row 5 8, leading to the matri à 7 8, à The entries of the matri à (ã i,j i, j,, n have the following meaning: ã ij is the relative consumption of the product of industr S i b industr S j Market prices The consumption of products is regulated b prices All income of an industr is used for buing other (or the own products, ie, income equals ependiture Let P (p,, p n the price vector; p i is the relative price of the product of industr S i We can draw the flow of mone into the input-output graph, the mone flows in echange for the products: 5
6 One has p a p + a p + + ã n p n p ã p + ã p + + a n p p n ã n p + ã n p n + + ã nn p n, since ã ij p i is the amount paid b industr S j for products produced b industr S i The total income of industr S j equals the total price S j has to pa to all other industries Again, one can write this as a matri equation: P A P This equation can be transformed in the following wa P I n P à P (I n à (,, The matri I n à is (similar as I n A not invertible, since (I n à One can show that this implies that there is also a solution P Since with P also t P for t > is a solution, onl the relative price between the different products has a well-defined meaning Eample (continued: Assume p $, One gets p $ 6 $ 69 and p $ $ 9677 We can compare these relative prices with the production levels measured b the original units and obtain the following relative prices per unit: p / for one ton of wheat, p / 69 6 for km horse ride, and p /z for one man-ear Since the above matri equation for P is not of the usual form which we have studied so far, we make a final modification We define à (ãi,j i, j,, n, where ã i,j ã j,i 6
7 This gives us (just b switching the rôle of rows and columns the price equation P à P, where ã i,j is now the relative consumption of industr S j b industr S i, so that the sum of each column is, and P p p n is the price column vector In the tetbook, our matri à is again denoted b A and our P is denoted b X The price equation is therefore X A X However, one has to keep in mind that this matri A is different from the input-output matri A we used in the open Leontief model! Eample: Let A Compute all wages, given that the wages for the rd product is $, Solution: Let X be the different wages with z, We have to z solve X AX (I AX, This sstem of linear equations for and has the solution, and, 5 The wages for the first and second product are therefore $,, and $, 5, respectivel 7
MAT188H1S Lec0101 Burbulla
Winter 206 Linear Transformations A linear transformation T : R m R n is a function that takes vectors in R m to vectors in R n such that and T (u + v) T (u) + T (v) T (k v) k T (v), for all vectors u
More informationx y The matrix form, the vector form, and the augmented matrix form, respectively, for the system of equations are
Solving Sstems of Linear Equations in Matri Form with rref Learning Goals Determine the solution of a sstem of equations from the augmented matri Determine the reduced row echelon form of the augmented
More informationExample 1: Model A Model B Total Available. Gizmos. Dodads. System:
Lesson : Sstems of Equations and Matrices Outline Objectives: I can solve sstems of three linear equations in three variables. I can solve sstems of linear inequalities I can model and solve real-world
More informationFor supervised classification we have a variety of measures to evaluate how good our model is Accuracy, precision, recall
Cluster Validation Cluster Validit For supervised classification we have a variet of measures to evaluate how good our model is Accurac, precision, recall For cluster analsis, the analogous question is
More informationMATRIX ALGEBRA AND SYSTEMS OF EQUATIONS. + + x 2. x n. a 11 a 12 a 1n b 1 a 21 a 22 a 2n b 2 a 31 a 32 a 3n b 3. a m1 a m2 a mn b m
MATRIX ALGEBRA AND SYSTEMS OF EQUATIONS 1. SYSTEMS OF EQUATIONS AND MATRICES 1.1. Representation of a linear system. The general system of m equations in n unknowns can be written a 11 x 1 + a 12 x 2 +
More informationQuestion 2: How do you solve a matrix equation using the matrix inverse?
Question : How do you solve a matrix equation using the matrix inverse? In the previous question, we wrote systems of equations as a matrix equation AX B. In this format, the matrix A contains the coefficients
More informationLecture 2 Matrix Operations
Lecture 2 Matrix Operations transpose, sum & difference, scalar multiplication matrix multiplication, matrix-vector product matrix inverse 2 1 Matrix transpose transpose of m n matrix A, denoted A T or
More informationGraphing Linear Equations
6.3 Graphing Linear Equations 6.3 OBJECTIVES 1. Graph a linear equation b plotting points 2. Graph a linear equation b the intercept method 3. Graph a linear equation b solving the equation for We are
More informationMATRIX ALGEBRA AND SYSTEMS OF EQUATIONS
MATRIX ALGEBRA AND SYSTEMS OF EQUATIONS Systems of Equations and Matrices Representation of a linear system The general system of m equations in n unknowns can be written a x + a 2 x 2 + + a n x n b a
More informationSolving Systems of Linear Equations With Row Reductions to Echelon Form On Augmented Matrices. Paul A. Trogdon Cary High School Cary, North Carolina
Solving Sstems of Linear Equations With Ro Reductions to Echelon Form On Augmented Matrices Paul A. Trogdon Car High School Car, North Carolina There is no more efficient a to solve a sstem of linear equations
More informationDATA ANALYSIS II. Matrix Algorithms
DATA ANALYSIS II Matrix Algorithms Similarity Matrix Given a dataset D = {x i }, i=1,..,n consisting of n points in R d, let A denote the n n symmetric similarity matrix between the points, given as where
More information5.2 Inverse Functions
78 Further Topics in Functions. Inverse Functions Thinking of a function as a process like we did in Section., in this section we seek another function which might reverse that process. As in real life,
More information15.1. Exact Differential Equations. Exact First-Order Equations. Exact Differential Equations Integrating Factors
SECTION 5. Eact First-Order Equations 09 SECTION 5. Eact First-Order Equations Eact Differential Equations Integrating Factors Eact Differential Equations In Section 5.6, ou studied applications of differential
More information3 Optimizing Functions of Two Variables. Chapter 7 Section 3 Optimizing Functions of Two Variables 533
Chapter 7 Section 3 Optimizing Functions of Two Variables 533 (b) Read about the principle of diminishing returns in an economics tet. Then write a paragraph discussing the economic factors that might
More informationDirect Methods for Solving Linear Systems. Matrix Factorization
Direct Methods for Solving Linear Systems Matrix Factorization Numerical Analysis (9th Edition) R L Burden & J D Faires Beamer Presentation Slides prepared by John Carroll Dublin City University c 2011
More information{ } Sec 3.1 Systems of Linear Equations in Two Variables
Sec.1 Sstems of Linear Equations in Two Variables Learning Objectives: 1. Deciding whether an ordered pair is a solution.. Solve a sstem of linear equations using the graphing, substitution, and elimination
More information7.3 Solving Systems by Elimination
7. Solving Sstems b Elimination In the last section we saw the Substitution Method. It turns out there is another method for solving a sstem of linear equations that is also ver good. First, we will need
More informationAffine Transformations
A P P E N D I X C Affine Transformations CONTENTS C The need for geometric transformations 335 C2 Affine transformations 336 C3 Matri representation of the linear transformations 338 C4 Homogeneous coordinates
More informationData Mining Cluster Analysis: Basic Concepts and Algorithms. Clustering Algorithms. Lecture Notes for Chapter 8. Introduction to Data Mining
Data Mining Cluster Analsis: Basic Concepts and Algorithms Lecture Notes for Chapter 8 Introduction to Data Mining b Tan, Steinbach, Kumar Clustering Algorithms K-means and its variants Hierarchical clustering
More informationIntroduction to Matrices for Engineers
Introduction to Matrices for Engineers C.T.J. Dodson, School of Mathematics, Manchester Universit 1 What is a Matrix? A matrix is a rectangular arra of elements, usuall numbers, e.g. 1 0-8 4 0-1 1 0 11
More informationSECTION 7-4 Algebraic Vectors
7-4 lgebraic Vectors 531 SECTIN 7-4 lgebraic Vectors From Geometric Vectors to lgebraic Vectors Vector ddition and Scalar Multiplication Unit Vectors lgebraic Properties Static Equilibrium Geometric vectors
More informationInner products on R n, and more
Inner products on R n, and more Peyam Ryan Tabrizian Friday, April 12th, 2013 1 Introduction You might be wondering: Are there inner products on R n that are not the usual dot product x y = x 1 y 1 + +
More information160 CHAPTER 4. VECTOR SPACES
160 CHAPTER 4. VECTOR SPACES 4. Rank and Nullity In this section, we look at relationships between the row space, column space, null space of a matrix and its transpose. We will derive fundamental results
More informationSystems of Linear Equations: Solving by Substitution
8.3 Sstems of Linear Equations: Solving b Substitution 8.3 OBJECTIVES 1. Solve sstems using the substitution method 2. Solve applications of sstems of equations In Sections 8.1 and 8.2, we looked at graphing
More informationLESSON EIII.E EXPONENTS AND LOGARITHMS
LESSON EIII.E EXPONENTS AND LOGARITHMS LESSON EIII.E EXPONENTS AND LOGARITHMS OVERVIEW Here s what ou ll learn in this lesson: Eponential Functions a. Graphing eponential functions b. Applications of eponential
More informationT ( a i x i ) = a i T (x i ).
Chapter 2 Defn 1. (p. 65) Let V and W be vector spaces (over F ). We call a function T : V W a linear transformation form V to W if, for all x, y V and c F, we have (a) T (x + y) = T (x) + T (y) and (b)
More information2.7 Applications of Derivatives to Business
80 CHAPTER 2 Applications of the Derivative 2.7 Applications of Derivatives to Business and Economics Cost = C() In recent ears, economic decision making has become more and more mathematicall oriented.
More informationC3: Functions. Learning objectives
CHAPTER C3: Functions Learning objectives After studing this chapter ou should: be familiar with the terms one-one and man-one mappings understand the terms domain and range for a mapping understand the
More informationLinear Inequality in Two Variables
90 (7-) Chapter 7 Sstems of Linear Equations and Inequalities In this section 7.4 GRAPHING LINEAR INEQUALITIES IN TWO VARIABLES You studied linear equations and inequalities in one variable in Chapter.
More informationExponential Functions
Eponential Functions Deinition: An Eponential Function is an unction that has the orm ( a, where a > 0. The number a is called the base. Eample:Let For eample (0, (, ( It is clear what the unction means
More informationTo Be or Not To Be a Linear Equation: That Is the Question
To Be or Not To Be a Linear Equation: That Is the Question Linear Equation in Two Variables A linear equation in two variables is an equation that can be written in the form A + B C where A and B are not
More information1 Introduction. Linear Programming. Questions. A general optimization problem is of the form: choose x to. max f(x) subject to x S. where.
Introduction Linear Programming Neil Laws TT 00 A general optimization problem is of the form: choose x to maximise f(x) subject to x S where x = (x,..., x n ) T, f : R n R is the objective function, S
More informationChapter 7. Matrices. Definition. An m n matrix is an array of numbers set out in m rows and n columns. Examples. ( 1 1 5 2 0 6
Chapter 7 Matrices Definition An m n matrix is an array of numbers set out in m rows and n columns Examples (i ( 1 1 5 2 0 6 has 2 rows and 3 columns and so it is a 2 3 matrix (ii 1 0 7 1 2 3 3 1 is a
More informationhttps://people.richland.edu/james/lecture/m116/matrices/applications.html
Date: 15.05.2014 Teacher: Ezgi Çallı Number of Students: 19 Grade Level: 11 Time Frame: 45 minutes DETERMINANT 1. Goal(s) Students will be able to develop an understanding about the concept of determinants.
More informationNotes on Determinant
ENGG2012B Advanced Engineering Mathematics Notes on Determinant Lecturer: Kenneth Shum Lecture 9-18/02/2013 The determinant of a system of linear equations determines whether the solution is unique, without
More informationFINAL EXAM REVIEW MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
FINAL EXAM REVIEW MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Determine whether or not the relationship shown in the table is a function. 1) -
More information1. a. standard form of a parabola with. 2 b 1 2 horizontal axis of symmetry 2. x 2 y 2 r 2 o. standard form of an ellipse centered
Conic Sections. Distance Formula and Circles. More on the Parabola. The Ellipse and Hperbola. Nonlinear Sstems of Equations in Two Variables. Nonlinear Inequalities and Sstems of Inequalities In Chapter,
More informationAddition and Subtraction of Vectors
ddition and Subtraction of Vectors 1 ppendi ddition and Subtraction of Vectors In this appendi the basic elements of vector algebra are eplored. Vectors are treated as geometric entities represented b
More informationMinimizing Probing Cost and Achieving Identifiability in Probe Based Network Link Monitoring
Minimizing Probing Cost and Achieving Identifiability in Probe Based Network Link Monitoring Qiang Zheng, Student Member, IEEE, and Guohong Cao, Fellow, IEEE Department of Computer Science and Engineering
More informationQuadratic Equations and Functions
Quadratic Equations and Functions. Square Root Propert and Completing the Square. Quadratic Formula. Equations in Quadratic Form. Graphs of Quadratic Functions. Verte of a Parabola and Applications In
More informationMath 312 Homework 1 Solutions
Math 31 Homework 1 Solutions Last modified: July 15, 01 This homework is due on Thursday, July 1th, 01 at 1:10pm Please turn it in during class, or in my mailbox in the main math office (next to 4W1) Please
More informationSTRUCTURE OF THE WORLD ECONOMY
STRUCTURE OF THE WORLD ECONOMY Outline of a Simple Input-Output Formulation* Nobel Memorial Lecture, December 11, 1973 by WA SSILY LE ONTIEF Harvard University, Cambridge, Massachusetts, USA. I The world
More informationMore Equations and Inequalities
Section. Sets of Numbers and Interval Notation 9 More Equations and Inequalities 9 9. Compound Inequalities 9. Polnomial and Rational Inequalities 9. Absolute Value Equations 9. Absolute Value Inequalities
More informationRecall the basic property of the transpose (for any A): v A t Aw = v w, v, w R n.
ORTHOGONAL MATRICES Informally, an orthogonal n n matrix is the n-dimensional analogue of the rotation matrices R θ in R 2. When does a linear transformation of R 3 (or R n ) deserve to be called a rotation?
More informationSection V.2: Magnitudes, Directions, and Components of Vectors
Section V.: Magnitudes, Directions, and Components of Vectors Vectors in the plane If we graph a vector in the coordinate plane instead of just a grid, there are a few things to note. Firstl, directions
More informationCOMPANY BALANCE MODELS AND THEIR USE FOR PROCESS MANAGEMENT
COMPANY BALANCE MODELS AND THEIR USE FOR PROCESS MANAGEMENT Mária Mišanková INTRODUCTION Balance model i in general tem of equation motl linear and the goal i to find value of required quantit from pecified
More informationa 11 x 1 + a 12 x 2 + + a 1n x n = b 1 a 21 x 1 + a 22 x 2 + + a 2n x n = b 2.
Chapter 1 LINEAR EQUATIONS 1.1 Introduction to linear equations A linear equation in n unknowns x 1, x,, x n is an equation of the form a 1 x 1 + a x + + a n x n = b, where a 1, a,..., a n, b are given
More informationName Date. Break-Even Analysis
Name Date Break-Even Analsis In our business planning so far, have ou ever asked the questions: How much do I have to sell to reach m gross profit goal? What price should I charge to cover m costs and
More information6. Cholesky factorization
6. Cholesky factorization EE103 (Fall 2011-12) triangular matrices forward and backward substitution the Cholesky factorization solving Ax = b with A positive definite inverse of a positive definite matrix
More informationOperation Count; Numerical Linear Algebra
10 Operation Count; Numerical Linear Algebra 10.1 Introduction Many computations are limited simply by the sheer number of required additions, multiplications, or function evaluations. If floating-point
More informationConstrained Optimization: The Method of Lagrange Multipliers:
Constrained Optimization: The Method of Lagrange Multipliers: Suppose the equation p(x,) x 60x 7 00 models profit when x represents the number of handmade chairs and is the number of handmade rockers produced
More informationData Mining Cluster Analysis: Basic Concepts and Algorithms. Lecture Notes for Chapter 8. Introduction to Data Mining
Data Mining Cluster Analsis: Basic Concepts and Algorithms Lecture Notes for Chapter 8 Introduction to Data Mining b Tan, Steinbach, Kumar Tan,Steinbach, Kumar Introduction to Data Mining 4/8/4 What is
More informationTrading Networks with Price-Setting Agents
Trading Networks with Price-Setting Agents Larr Blume Dept. of Economics Cornell Universit, Ithaca NY lb9@cs.cornell.edu David Easle Dept. of Economics Cornell Universit, Ithaca NY dae3@cs.cornell.edu
More informationRoots of Equations (Chapters 5 and 6)
Roots of Equations (Chapters 5 and 6) Problem: given f() = 0, find. In general, f() can be any function. For some forms of f(), analytical solutions are available. However, for other functions, we have
More informationUniversity of Lille I PC first year list of exercises n 7. Review
University of Lille I PC first year list of exercises n 7 Review Exercise Solve the following systems in 4 different ways (by substitution, by the Gauss method, by inverting the matrix of coefficients
More informationSECTION 5-1 Exponential Functions
354 5 Eponential and Logarithmic Functions Most of the functions we have considered so far have been polnomial and rational functions, with a few others involving roots or powers of polnomial or rational
More informationv w is orthogonal to both v and w. the three vectors v, w and v w form a right-handed set of vectors.
3. Cross product Definition 3.1. Let v and w be two vectors in R 3. The cross product of v and w, denoted v w, is the vector defined as follows: the length of v w is the area of the parallelogram with
More informationSolving Linear Systems, Continued and The Inverse of a Matrix
, Continued and The of a Matrix Calculus III Summer 2013, Session II Monday, July 15, 2013 Agenda 1. The rank of a matrix 2. The inverse of a square matrix Gaussian Gaussian solves a linear system by reducing
More informationCHAPTER 10 SYSTEMS, MATRICES, AND DETERMINANTS
CHAPTER 0 SYSTEMS, MATRICES, AND DETERMINANTS PRE-CALCULUS: A TEACHING TEXTBOOK Lesson 64 Solving Sstems In this chapter, we re going to focus on sstems of equations. As ou ma remember from algebra, sstems
More information2.1 Three Dimensional Curves and Surfaces
. Three Dimensional Curves and Surfaces.. Parametric Equation of a Line An line in two- or three-dimensional space can be uniquel specified b a point on the line and a vector parallel to the line. The
More informationZeros of Polynomial Functions. The Fundamental Theorem of Algebra. The Fundamental Theorem of Algebra. zero in the complex number system.
_.qd /7/ 9:6 AM Page 69 Section. Zeros of Polnomial Functions 69. Zeros of Polnomial Functions What ou should learn Use the Fundamental Theorem of Algebra to determine the number of zeros of polnomial
More information7. LU factorization. factor-solve method. LU factorization. solving Ax = b with A nonsingular. the inverse of a nonsingular matrix
7. LU factorization EE103 (Fall 2011-12) factor-solve method LU factorization solving Ax = b with A nonsingular the inverse of a nonsingular matrix LU factorization algorithm effect of rounding error sparse
More informationExponential and Logarithmic Functions
Chapter 6 Eponential and Logarithmic Functions Section summaries Section 6.1 Composite Functions Some functions are constructed in several steps, where each of the individual steps is a function. For eample,
More informationOptimization in R n Introduction
Optimization in R n Introduction Rudi Pendavingh Eindhoven Technical University Optimization in R n, lecture Rudi Pendavingh (TUE) Optimization in R n Introduction ORN / 4 Some optimization problems designing
More informationLecture 2 Linear functions and examples
EE263 Autumn 2007-08 Stephen Boyd Lecture 2 Linear functions and examples linear equations and functions engineering examples interpretations 2 1 Linear equations consider system of linear equations y
More informationZero and Negative Exponents and Scientific Notation. a a n a m n. Now, suppose that we allow m to equal n. We then have. a am m a 0 (1) a m
0. E a m p l e 666SECTION 0. OBJECTIVES. Define the zero eponent. Simplif epressions with negative eponents. Write a number in scientific notation. Solve an application of scientific notation We must have
More informationNotes on Factoring. MA 206 Kurt Bryan
The General Approach Notes on Factoring MA 26 Kurt Bryan Suppose I hand you n, a 2 digit integer and tell you that n is composite, with smallest prime factor around 5 digits. Finding a nontrivial factor
More informationIntroduction to Matrix Algebra
Psychology 7291: Multivariate Statistics (Carey) 8/27/98 Matrix Algebra - 1 Introduction to Matrix Algebra Definitions: A matrix is a collection of numbers ordered by rows and columns. It is customary
More informationINVESTIGATIONS AND FUNCTIONS 1.1.1 1.1.4. Example 1
Chapter 1 INVESTIGATIONS AND FUNCTIONS 1.1.1 1.1.4 This opening section introduces the students to man of the big ideas of Algebra 2, as well as different was of thinking and various problem solving strategies.
More informationClient Based Power Iteration Clustering Algorithm to Reduce Dimensionality in Big Data
Client Based Power Iteration Clustering Algorithm to Reduce Dimensionalit in Big Data Jaalatchum. D 1, Thambidurai. P 1, Department of CSE, PKIET, Karaikal, India Abstract - Clustering is a group of objects
More informationPolynomial Degree and Finite Differences
CONDENSED LESSON 7.1 Polynomial Degree and Finite Differences In this lesson you will learn the terminology associated with polynomials use the finite differences method to determine the degree of a polynomial
More informationLinear Equations in Linear Algebra
1 Linear Equations in Linear Algebra 1.5 SOLUTION SETS OF LINEAR SYSTEMS HOMOGENEOUS LINEAR SYSTEMS A system of linear equations is said to be homogeneous if it can be written in the form A 0, where A
More informationLecture 3: Finding integer solutions to systems of linear equations
Lecture 3: Finding integer solutions to systems of linear equations Algorithmic Number Theory (Fall 2014) Rutgers University Swastik Kopparty Scribe: Abhishek Bhrushundi 1 Overview The goal of this lecture
More informationMathematics Course 111: Algebra I Part IV: Vector Spaces
Mathematics Course 111: Algebra I Part IV: Vector Spaces D. R. Wilkins Academic Year 1996-7 9 Vector Spaces A vector space over some field K is an algebraic structure consisting of a set V on which are
More information1 Introduction to Matrices
1 Introduction to Matrices In this section, important definitions and results from matrix algebra that are useful in regression analysis are introduced. While all statements below regarding the columns
More information11th National Convention on Statistics (NCS) EDSA Shangri-La Hotel October 4-5, 2010
11th National Convention on Statistics (NCS) EDSA Shangri-La Hotel October 4-5, 2010 INPUT-OUTPUT MULTIPLIER ANALYSIS FOR MAJOR INDUSTRIES IN THE PHILIPPINES by Madeline B. Dumaua For additional information,
More information8. Linear least-squares
8. Linear least-squares EE13 (Fall 211-12) definition examples and applications solution of a least-squares problem, normal equations 8-1 Definition overdetermined linear equations if b range(a), cannot
More informationUsing row reduction to calculate the inverse and the determinant of a square matrix
Using row reduction to calculate the inverse and the determinant of a square matrix Notes for MATH 0290 Honors by Prof. Anna Vainchtein 1 Inverse of a square matrix An n n square matrix A is called invertible
More informationGENERATING SETS KEITH CONRAD
GENERATING SETS KEITH CONRAD 1 Introduction In R n, every vector can be written as a unique linear combination of the standard basis e 1,, e n A notion weaker than a basis is a spanning set: a set of vectors
More informationα = u v. In other words, Orthogonal Projection
Orthogonal Projection Given any nonzero vector v, it is possible to decompose an arbitrary vector u into a component that points in the direction of v and one that points in a direction orthogonal to v
More informationUse order of operations to simplify. Show all steps in the space provided below each problem. INTEGER OPERATIONS
ORDER OF OPERATIONS In the following order: 1) Work inside the grouping smbols such as parenthesis and brackets. ) Evaluate the powers. 3) Do the multiplication and/or division in order from left to right.
More informationPhysics 53. Kinematics 2. Our nature consists in movement; absolute rest is death. Pascal
Phsics 53 Kinematics 2 Our nature consists in movement; absolute rest is death. Pascal Velocit and Acceleration in 3-D We have defined the velocit and acceleration of a particle as the first and second
More informationSome Lecture Notes and In-Class Examples for Pre-Calculus:
Some Lecture Notes and In-Class Examples for Pre-Calculus: Section.7 Definition of a Quadratic Inequality A quadratic inequality is any inequality that can be put in one of the forms ax + bx + c < 0 ax
More information1 Sets and Set Notation.
LINEAR ALGEBRA MATH 27.6 SPRING 23 (COHEN) LECTURE NOTES Sets and Set Notation. Definition (Naive Definition of a Set). A set is any collection of objects, called the elements of that set. We will most
More informationSolutions to Homework Section 3.7 February 18th, 2005
Math 54W Spring 5 Solutions to Homeork Section 37 Februar 8th, 5 List the ro vectors and the column vectors of the matrix The ro vectors are The column vectors are ( 5 5 The matrix ( (,,,, 4, (5,,,, (
More information15.062 Data Mining: Algorithms and Applications Matrix Math Review
.6 Data Mining: Algorithms and Applications Matrix Math Review The purpose of this document is to give a brief review of selected linear algebra concepts that will be useful for the course and to develop
More informationLIMITS AND CONTINUITY
LIMITS AND CONTINUITY 1 The concept of it Eample 11 Let f() = 2 4 Eamine the behavior of f() as approaches 2 2 Solution Let us compute some values of f() for close to 2, as in the tables below We see from
More informationSection 1.1 Linear Equations: Slope and Equations of Lines
Section. Linear Equations: Slope and Equations of Lines Slope The measure of the steepness of a line is called the slope of the line. It is the amount of change in y, the rise, divided by the amount of
More informationMath 152, Intermediate Algebra Practice Problems #1
Math 152, Intermediate Algebra Practice Problems 1 Instructions: These problems are intended to give ou practice with the tpes Joseph Krause and level of problems that I epect ou to be able to do. Work
More informationALGEBRA 1 SKILL BUILDERS
ALGEBRA 1 SKILL BUILDERS (Etra Practice) Introduction to Students and Their Teachers Learning is an individual endeavor. Some ideas come easil; others take time--sometimes lots of time- -to grasp. In addition,
More informationSUNY ECC. ACCUPLACER Preparation Workshop. Algebra Skills
SUNY ECC ACCUPLACER Preparation Workshop Algebra Skills Gail A. Butler Ph.D. Evaluating Algebraic Epressions Substitute the value (#) in place of the letter (variable). Follow order of operations!!! E)
More informationOrthogonal Projections
Orthogonal Projections and Reflections (with exercises) by D. Klain Version.. Corrections and comments are welcome! Orthogonal Projections Let X,..., X k be a family of linearly independent (column) vectors
More informationPolynomials. Jackie Nicholas Jacquie Hargreaves Janet Hunter
Mathematics Learning Centre Polnomials Jackie Nicholas Jacquie Hargreaves Janet Hunter c 26 Universit of Sdne Mathematics Learning Centre, Universit of Sdne 1 1 Polnomials Man of the functions we will
More information8 Square matrices continued: Determinants
8 Square matrices continued: Determinants 8. Introduction Determinants give us important information about square matrices, and, as we ll soon see, are essential for the computation of eigenvalues. You
More information6. The given function is only drawn for x > 0. Complete the function for x < 0 with the following conditions:
Precalculus Worksheet 1. Da 1 1. The relation described b the set of points {(-, 5 ),( 0, 5 ),(,8 ),(, 9) } is NOT a function. Eplain wh. For questions - 4, use the graph at the right.. Eplain wh the graph
More informationDirect Variation. 1. Write an equation for a direct variation relationship 2. Graph the equation of a direct variation relationship
6.5 Direct Variation 6.5 OBJECTIVES 1. Write an equation for a direct variation relationship 2. Graph the equation of a direct variation relationship Pedro makes $25 an hour as an electrician. If he works
More information1 Maximizing pro ts when marginal costs are increasing
BEE12 Basic Mathematical Economics Week 1, Lecture Tuesda 12.1. Pro t maimization 1 Maimizing pro ts when marginal costs are increasing We consider in this section a rm in a perfectl competitive market
More informationPolynomial and Synthetic Division. Long Division of Polynomials. Example 1. 6x 2 7x 2 x 2) 19x 2 16x 4 6x3 12x 2 7x 2 16x 7x 2 14x. 2x 4.
_.qd /7/5 9: AM Page 5 Section.. Polynomial and Synthetic Division 5 Polynomial and Synthetic Division What you should learn Use long division to divide polynomials by other polynomials. Use synthetic
More informationDERIVATIVES AS MATRICES; CHAIN RULE
DERIVATIVES AS MATRICES; CHAIN RULE 1. Derivatives of Real-valued Functions Let s first consider functions f : R 2 R. Recall that if the partial derivatives of f exist at the point (x 0, y 0 ), then we
More informationMathematics 31 Pre-calculus and Limits
Mathematics 31 Pre-calculus and Limits Overview After completing this section, students will be epected to have acquired reliability and fluency in the algebraic skills of factoring, operations with radicals
More information