7.2 Application to economics: Leontief Model

Save this PDF as:
 WORD  PNG  TXT  JPG

Size: px
Start display at page:

Download "7.2 Application to economics: Leontief Model"

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

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 information

x y The matrix form, the vector form, and the augmented matrix form, respectively, for the system of equations are

x 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 information

MULTIPLE REPRESENTATIONS through 4.1.7

MULTIPLE REPRESENTATIONS through 4.1.7 MULTIPLE REPRESENTATIONS 4.1.1 through 4.1.7 The first part of Chapter 4 ties together several was to represent the same relationship. The basis for an relationship is a consistent pattern that connects

More information

MATH 304 Linear Algebra Lecture 8: Inverse matrix (continued). Elementary matrices. Transpose of a matrix.

MATH 304 Linear Algebra Lecture 8: Inverse matrix (continued). Elementary matrices. Transpose of a matrix. MATH 304 Linear Algebra Lecture 8: Inverse matrix (continued). Elementary matrices. Transpose of a matrix. Inverse matrix Definition. Let A be an n n matrix. The inverse of A is an n n matrix, denoted

More information

MATRIX 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. + + 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 information

Example 1: Model A Model B Total Available. Gizmos. Dodads. System:

Example 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 information

For supervised classification we have a variety of measures to evaluate how good our model is Accuracy, precision, recall

For 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 information

Math 315: Linear Algebra Solutions to Midterm Exam I

Math 315: Linear Algebra Solutions to Midterm Exam I Math 35: Linear Algebra s to Midterm Exam I # Consider the following two systems of linear equations (I) ax + by = k cx + dy = l (II) ax + by = 0 cx + dy = 0 (a) Prove: If x = x, y = y and x = x 2, y =

More information

12.3 Inverse Matrices

12.3 Inverse Matrices 2.3 Inverse Matrices Two matrices A A are called inverses if AA I A A I where I denotes the identit matrix of the appropriate size. For example, the matrices A 3 7 2 5 A 5 7 2 3 If we think of the identit

More information

Question 2: How do you solve a matrix equation using the matrix inverse?

Question 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 information

1 Vector Spaces and Matrix Notation

1 Vector Spaces and Matrix Notation 1 Vector Spaces and Matrix Notation De nition 1 A matrix: is rectangular array of numbers with n rows and m columns. 1 1 1 a11 a Example 1 a. b. c. 1 0 0 a 1 a The rst is square with n = and m = ; the

More information

1.2 GRAPHS OF EQUATIONS

1.2 GRAPHS OF EQUATIONS 000_00.qd /5/05 : AM Page SECTION. Graphs of Equations. GRAPHS OF EQUATIONS Sketch graphs of equations b hand. Find the - and -intercepts of graphs of equations. Write the standard forms of equations of

More information

Section 5: The Jacobian matrix and applications. S1: Motivation S2: Jacobian matrix + differentiability S3: The chain rule S4: Inverse functions

Section 5: The Jacobian matrix and applications. S1: Motivation S2: Jacobian matrix + differentiability S3: The chain rule S4: Inverse functions Section 5: The Jacobian matri and applications. S1: Motivation S2: Jacobian matri + differentiabilit S3: The chain rule S4: Inverse functions Images from Thomas calculus b Thomas, Wier, Hass & Giordano,

More information

Lecture 2 Matrix Operations

Lecture 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 information

Graphing Linear Equations

Graphing 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 information

Cofactor Expansion: Cramer s Rule

Cofactor Expansion: Cramer s Rule Cofactor Expansion: Cramer s Rule MATH 322, Linear Algebra I J. Robert Buchanan Department of Mathematics Spring 2015 Introduction Today we will focus on developing: an efficient method for calculating

More information

MATRIX ALGEBRA AND SYSTEMS OF EQUATIONS

MATRIX 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 information

1.5 Elementary Matrices and a Method for Finding the Inverse

1.5 Elementary Matrices and a Method for Finding the Inverse .5 Elementary Matrices and a Method for Finding the Inverse Definition A n n matrix is called an elementary matrix if it can be obtained from I n by performing a single elementary row operation Reminder:

More information

Lecture 11. Shuanglin Shao. October 2nd and 7th, 2013

Lecture 11. Shuanglin Shao. October 2nd and 7th, 2013 Lecture 11 Shuanglin Shao October 2nd and 7th, 2013 Matrix determinants: addition. Determinants: multiplication. Adjoint of a matrix. Cramer s rule to solve a linear system. Recall that from the previous

More information

Chapter 5. The Inverse; Numerical Methods

Chapter 5. The Inverse; Numerical Methods Vector Spaces in Physics 8/6/ Chapter. The nverse Numerical Methods n the Chapter we discussed the solution of systems of simultaneous linear algebraic equations which could be written in the form C -

More information

( % . This matrix consists of $ 4 5 " 5' the coefficients of the variables as they appear in the original system. The augmented 3 " 2 2 # 2 " 3 4&

( % . This matrix consists of $ 4 5  5' the coefficients of the variables as they appear in the original system. The augmented 3  2 2 # 2  3 4& Matrices define matrix We will use matrices to help us solve systems of equations. A matrix is a rectangular array of numbers enclosed in parentheses or brackets. In linear algebra, matrices are important

More information

UNIT 2 MATRICES - I 2.0 INTRODUCTION. Structure

UNIT 2 MATRICES - I 2.0 INTRODUCTION. Structure UNIT 2 MATRICES - I Matrices - I Structure 2.0 Introduction 2.1 Objectives 2.2 Matrices 2.3 Operation on Matrices 2.4 Invertible Matrices 2.5 Systems of Linear Equations 2.6 Answers to Check Your Progress

More information

Solving Systems of Linear Equations With Row Reductions to Echelon Form On Augmented Matrices. Paul A. Trogdon Cary High School Cary, North Carolina

Solving 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 information

1 Determinants. Definition 1

1 Determinants. Definition 1 Determinants The determinant of a square matrix is a value in R assigned to the matrix, it characterizes matrices which are invertible (det 0) and is related to the volume of a parallelpiped described

More information

Using determinants, it is possible to express the solution to a system of equations whose coefficient matrix is invertible:

Using determinants, it is possible to express the solution to a system of equations whose coefficient matrix is invertible: Cramer s Rule and the Adjugate Using determinants, it is possible to express the solution to a system of equations whose coefficient matrix is invertible: Theorem [Cramer s Rule] If A is an invertible

More information

5.2 Inverse Functions

5.2 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 information

Matrix Inverse and Determinants

Matrix Inverse and Determinants DM554 Linear and Integer Programming Lecture 5 and Marco Chiarandini Department of Mathematics & Computer Science University of Southern Denmark Outline 1 2 3 4 and Cramer s rule 2 Outline 1 2 3 4 and

More information

DATA ANALYSIS II. Matrix Algorithms

DATA 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 information

z = f(x, y) = c + mx + ny

z = f(x, y) = c + mx + ny Linear Functions In single-variable calculus, we studied linear equations because the allowed us to approimate complicated curves with a fairl simple linear function. Our goal will be to use linear functions

More information

Diagonal, Symmetric and Triangular Matrices

Diagonal, Symmetric and Triangular Matrices Contents 1 Diagonal, Symmetric Triangular Matrices 2 Diagonal Matrices 2.1 Products, Powers Inverses of Diagonal Matrices 2.1.1 Theorem (Powers of Matrices) 2.2 Multiplying Matrices on the Left Right by

More information

Helpsheet. Giblin Eunson Library MATRIX ALGEBRA. library.unimelb.edu.au/libraries/bee. Use this sheet to help you:

Helpsheet. Giblin Eunson Library MATRIX ALGEBRA. library.unimelb.edu.au/libraries/bee. Use this sheet to help you: Helpsheet Giblin Eunson Library ATRIX ALGEBRA Use this sheet to help you: Understand the basic concepts and definitions of matrix algebra Express a set of linear equations in matrix notation Evaluate determinants

More information

9.2 LINEAR PROGRAMMING INVOLVING TWO VARIABLES

9.2 LINEAR PROGRAMMING INVOLVING TWO VARIABLES 86 CHAPTER 9 LINEAR PROGRAMMING 9. LINEAR PROGRAMMING INVOLVING TWO VARIABLES Figure 9.0 Feasible solutions Man applications in business and economics involve a process called optimization, in which we

More information

MATH 304 Linear Algebra Lecture 4: Matrix multiplication. Diagonal matrices. Inverse matrix.

MATH 304 Linear Algebra Lecture 4: Matrix multiplication. Diagonal matrices. Inverse matrix. MATH 304 Linear Algebra Lecture 4: Matrix multiplication. Diagonal matrices. Inverse matrix. Matrices Definition. An m-by-n matrix is a rectangular array of numbers that has m rows and n columns: a 11

More information

Q (x 1, y 1 ) m = y 1 y 0

Q (x 1, y 1 ) m = y 1 y 0 . Linear Functions We now begin the stud of families of functions. Our first famil, linear functions, are old friends as we shall soon see. Recall from Geometr that two distinct points in the plane determine

More information

Direct Methods for Solving Linear Systems. Matrix Factorization

Direct 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 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 information

15.1. Exact Differential Equations. Exact First-Order Equations. Exact Differential Equations Integrating Factors

15.1. 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 information

Applications of Linear Algebra in Economics: Input-Output and Inter-Industry Analysis. From: Lucas Davidson To: Professor Tushar Das May, 2010

Applications of Linear Algebra in Economics: Input-Output and Inter-Industry Analysis. From: Lucas Davidson To: Professor Tushar Das May, 2010 Applications of Linear Algebra in Economics: Input-Output and Inter-Industry Analysis. From: Lucas Davidson To: Professor Tushar Das May, 2010 1 1. Introduction In 1973 Wessily Leontiff won the Noble Prize

More information

Lesson 6: Linear Functions and their Slope

Lesson 6: Linear Functions and their Slope Lesson 6: Linear Functions and their Slope A linear function is represented b a line when graph, and represented in an where the variables have no whole number eponent higher than. Forms of a Linear Equation

More information

P1. Plot the following points on the real. P2. Determine which of the following are solutions

P1. Plot the following points on the real. P2. Determine which of the following are solutions Section 1.5 Rectangular Coordinates and Graphs of Equations 9 PART II: LINEAR EQUATIONS AND INEQUALITIES IN TWO VARIABLES 1.5 Rectangular Coordinates and Graphs of Equations OBJECTIVES 1 Plot Points in

More information

Quadratic Equations in One Unknown

Quadratic Equations in One Unknown 1 Quadratic Equations in One Unknown 1A 1. Solving Quadratic Equations Using the Factor Method Name : Date : Mark : Ke Concepts and Formulae 1. An equation in the form a + b + c, where a, b and c are real

More information

7.4 Applications of Eigenvalues and Eigenvectors

7.4 Applications of Eigenvalues and Eigenvectors 37 Chapter 7 Eigenvalues and Eigenvectors 7 Applications of Eigenvalues and Eigenvectors Model population growth using an age transition matri and an age distribution vector, and find a stable age distribution

More information

(a) The transpose of a lower triangular matrix is upper triangular, and the transpose of an upper triangular matrix is lower triangular.

(a) The transpose of a lower triangular matrix is upper triangular, and the transpose of an upper triangular matrix is lower triangular. Theorem.7.: (Properties of Triangular Matrices) (a) The transpose of a lower triangular matrix is upper triangular, and the transpose of an upper triangular matrix is lower triangular. (b) The product

More information

7.3 Solving Systems by Elimination

7.3 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 information

2.3 Quadratic Functions

2.3 Quadratic Functions . Quadratic Functions 9. Quadratic Functions You ma recall studing quadratic equations in Intermediate Algebra. In this section, we review those equations in the contet of our net famil of functions: the

More information

Data Mining Cluster Analysis: Basic Concepts and Algorithms. Clustering Algorithms. Lecture Notes for Chapter 8. Introduction to Data Mining

Data 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 information

Introduction to Matrices for Engineers

Introduction 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 information

3 Optimizing Functions of Two Variables. Chapter 7 Section 3 Optimizing Functions of Two Variables 533

3 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 information

160 CHAPTER 4. VECTOR SPACES

160 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 information

4. Matrix inverses. left and right inverse. linear independence. nonsingular matrices. matrices with linearly independent columns

4. Matrix inverses. left and right inverse. linear independence. nonsingular matrices. matrices with linearly independent columns L. Vandenberghe EE133A (Spring 2016) 4. Matrix inverses left and right inverse linear independence nonsingular matrices matrices with linearly independent columns matrices with linearly independent rows

More information

Contents. How You May Use This Resource Guide

Contents. How You May Use This Resource Guide Contents How You Ma Use This Resource Guide ii 9 Fractional and Quadratic Equations 1 Worksheet 9.1: Similar Figures.......................... 5 Worksheet 9.: Stretch of a Spring........................

More information

DETERMINANTS. b 2. x 2

DETERMINANTS. b 2. x 2 DETERMINANTS 1 Systems of two equations in two unknowns A system of two equations in two unknowns has the form a 11 x 1 + a 12 x 2 = b 1 a 21 x 1 + a 22 x 2 = b 2 This can be written more concisely in

More information

Affine Transformations

Affine 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 information

Rational Functions ( )

Rational Functions ( ) Rational Functions A rational function is a function of the form r P Q where P and Q are polynomials. We assume that P() and Q() have no factors in common, and Q() is not the zero polynomial. The domain

More information

Calculus and linear algebra for biomedical engineering Week 4: Inverse matrices and determinants

Calculus and linear algebra for biomedical engineering Week 4: Inverse matrices and determinants Calculus and linear algebra for biomedical engineering Week 4: Inverse matrices and determinants Hartmut Führ fuehr@matha.rwth-aachen.de Lehrstuhl A für Mathematik, RWTH Aachen October 30, 2008 Overview

More information

To 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 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 information

Direct Methods for Solving Linear Systems. Linear Systems of Equations

Direct Methods for Solving Linear Systems. Linear Systems of Equations Direct Methods for Solving Linear Systems Linear Systems of Equations Numerical Analysis (9th Edition) R L Burden & J D Faires Beamer Presentation Slides prepared by John Carroll Dublin City University

More information

Inner products on R n, and more

Inner 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 information

Linear Inequality in Two Variables

Linear 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 information

MATH 118, LECTURES 14 & 15: POLAR AREAS

MATH 118, LECTURES 14 & 15: POLAR AREAS MATH 118, LECTURES 1 & 15: POLAR AREAS 1 Polar Areas We recall from Cartesian coordinates that we could calculate the area under the curve b taking Riemann sums. We divided the region into subregions,

More information

Systems of Linear Equations: Solving by Substitution

Systems 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 information

Exponential Functions

Exponential 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 information

LESSON EIII.E EXPONENTS AND LOGARITHMS

LESSON 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 information

Facts About Eigenvalues

Facts About Eigenvalues Facts About Eigenvalues By Dr David Butler Definitions Suppose A is an n n matrix An eigenvalue of A is a number λ such that Av = λv for some nonzero vector v An eigenvector of A is a nonzero vector v

More information

Anytime plan TalkMore plan

Anytime plan TalkMore plan CONDENSED L E S S O N 6.1 Solving Sstems of Equations In this lesson ou will represent situations with sstems of equations use tables and graphs to solve sstems of linear equations A sstem of equations

More information

Inequalities and Absolute Values. Assignment Guide: EOO = every other odd, 1, 5, 9, 13, EOP = every other pair, 1, 2, 5, 6, 9, 10,

Inequalities and Absolute Values. Assignment Guide: EOO = every other odd, 1, 5, 9, 13, EOP = every other pair, 1, 2, 5, 6, 9, 10, Chapter 4 Inequalities and Absolute Values Assignment Guide: E = ever other odd,, 5, 9, 3, EP = ever other pair,, 2, 5, 6, 9, 0, Lesson 4. Page 75-77 Es. 4-20. 23-28, 29-39 odd, 40-43, 49-52, 59-73 odd

More information

1 Introduction. Linear Programming. Questions. A general optimization problem is of the form: choose x to. max f(x) subject to x S. where.

1 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 information

Mathematics 23 - Applied Matrix Algebra Supplement 1. Application: Production Planning

Mathematics 23 - Applied Matrix Algebra Supplement 1. Application: Production Planning Mathematics - Applied Matrix Algebra Supplement Application: Production Planning A manufacturer makes three different types of chemical products: A, B, and C. Each product must go through two processing

More information

Polynomial and Rational Functions

Polynomial and Rational Functions Chapter Section.1 Quadratic Functions Polnomial and Rational Functions Objective: In this lesson ou learned how to sketch and analze graphs of quadratic functions. Course Number Instructor Date Important

More information

Mathematics Notes for Class 12 chapter 3. Matrices

Mathematics Notes for Class 12 chapter 3. Matrices 1 P a g e Mathematics Notes for Class 12 chapter 3. Matrices A matrix is a rectangular arrangement of numbers (real or complex) which may be represented as matrix is enclosed by [ ] or ( ) or Compact form

More information

B such that AB = I and BA = I. (We say B is an inverse of A.) Definition A square matrix A is invertible (or nonsingular) if matrix

B such that AB = I and BA = I. (We say B is an inverse of A.) Definition A square matrix A is invertible (or nonsingular) if matrix Matrix inverses Recall... Definition A square matrix A is invertible (or nonsingular) if matrix B such that AB = and BA =. (We say B is an inverse of A.) Remark Not all square matrices are invertible.

More information

Crystallographic Directions, and Planes

Crystallographic Directions, and Planes Crstallographic Directions, and Planes Now that we know how atoms arrange themselves to form crstals, we need a wa to identif directions and planes of atoms. Wh? Deformation under loading (slip) occurs

More information

The basic unit in matrix algebra is a matrix, generally expressed as: a 11 a 12. a 13 A = a 21 a 22 a 23

The basic unit in matrix algebra is a matrix, generally expressed as: a 11 a 12. a 13 A = a 21 a 22 a 23 (copyright by Scott M Lynch, February 2003) Brief Matrix Algebra Review (Soc 504) Matrix algebra is a form of mathematics that allows compact notation for, and mathematical manipulation of, high-dimensional

More information

= [a ij ] 2 3. Square matrix A square matrix is one that has equal number of rows and columns, that is n = m. Some examples of square matrices are

= [a ij ] 2 3. Square matrix A square matrix is one that has equal number of rows and columns, that is n = m. Some examples of square matrices are This document deals with the fundamentals of matrix algebra and is adapted from B.C. Kuo, Linear Networks and Systems, McGraw Hill, 1967. It is presented here for educational purposes. 1 Introduction In

More information

Dr. Radhakant Padhi Asst. Professor Dept. of Aerospace Engineering Indian Institute of Science - Bangalore

Dr. Radhakant Padhi Asst. Professor Dept. of Aerospace Engineering Indian Institute of Science - Bangalore Lecture 9 Conversion Between State Space and Transfer Function Representations in Linear Systems I Dr. Radhakant Padhi Asst. Professor Dept. of Aerospace Engineering Indian Institute of Science - Bangalore

More information

https://people.richland.edu/james/lecture/m116/matrices/applications.html

https://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 information

Mathematics of Cryptography

Mathematics of Cryptography CHAPTER 2 Mathematics of Cryptography Part I: Modular Arithmetic, Congruence, and Matrices Objectives This chapter is intended to prepare the reader for the next few chapters in cryptography. The chapter

More information

Linear Programming

Linear Programming 97887_9.qp // 9 9. 9. 9. 9. 9.5 7:6 AM Page Linear Programming Sstems of Linear Inequalities Linear Programming Involving Two The Simple Method: Maimization The Simple Method: Minimization The Simple Method:

More information

Notes on Determinant

Notes 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 information

Filling in Coordinate Grid Planes

Filling in Coordinate Grid Planes Filling in Coordinate Grid Planes A coordinate grid is a sstem that can be used to write an address for an point within the grid. The grid is formed b two number lines called and that intersect at the

More information

SECTION 7-4 Algebraic Vectors

SECTION 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 information

Chapter 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. ( 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 information

1. 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

1. 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 information

Addition and Subtraction of Vectors

Addition 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 information

T ( a i x i ) = a i T (x i ).

T ( 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 information

FINAL 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. 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 information

Identify a pattern and find the next three numbers in the pattern. 5. 5(2s 2 1) 2 3(s 1 2); s 5 4

Identify a pattern and find the next three numbers in the pattern. 5. 5(2s 2 1) 2 3(s 1 2); s 5 4 Chapter 1 Test Do ou know HOW? Identif a pattern and find the net three numbers in the pattern. 1. 5, 1, 3, 7, c. 6, 3, 16, 8, c Each term is more than the previous Each term is half of the previous term;

More information

2.7 Applications of Derivatives to Business

2.7 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 information

C3: Functions. Learning objectives

C3: 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 information

Alex and Morgan were asked to graph the equation y = 2x + 1

Alex and Morgan were asked to graph the equation y = 2x + 1 Which is better? Ale and Morgan were asked to graph the equation = 2 + 1 Ale s make a table of values wa Morgan s use the slope and -intercept wa First, I made a table. I chose some -values, then plugged

More information

3.4 The Point-Slope Form of a Line

3.4 The Point-Slope Form of a Line Section 3.4 The Point-Slope Form of a Line 293 3.4 The Point-Slope Form of a Line In the last section, we developed the slope-intercept form of a line ( = m + b). The slope-intercept form of a line is

More information

Introduction. Introduction

Introduction. Introduction Introduction Solving Sstems of Equations Let s start with an eample. Recall the application of sales forecasting from the Working with Linear Equations module. We used historical data to derive the equation

More information

a 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.

a 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 information

v w is orthogonal to both v and w. the three vectors v, w and v w form a right-handed set of vectors.

v 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 information

Polynomial and Rational Functions

Polynomial and Rational Functions Chapter 5 Polnomial and Rational Functions Section 5.1 Polnomial Functions Section summaries The general form of a polnomial function is f() = a n n + a n 1 n 1 + +a 1 + a 0. The degree of f() is the largest

More information

Translating Points. Subtract 2 from the y-coordinates

Translating Points. Subtract 2 from the y-coordinates CONDENSED L E S S O N 9. Translating Points In this lesson ou will translate figures on the coordinate plane define a translation b describing how it affects a general point (, ) A mathematical rule that

More information

Matrix Algebra and Applications

Matrix Algebra and Applications Matrix Algebra and Applications Dudley Cooke Trinity College Dublin Dudley Cooke (Trinity College Dublin) Matrix Algebra and Applications 1 / 49 EC2040 Topic 2 - Matrices and Matrix Algebra Reading 1 Chapters

More information

Recall the basic property of the transpose (for any A): v A t Aw = v w, v, w R n.

Recall 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 information

Minimizing Probing Cost and Achieving Identifiability in Probe Based Network Link Monitoring

Minimizing 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 information