# 3.1 Simplex Method for Problems in Feasible Canonical Form

Save this PDF as:

Size: px
Start display at page:

Download "3.1 Simplex Method for Problems in Feasible Canonical Form"

## Transcription

1 Chapter SIMPLEX METHOD In this chapter, we put the theory developed in the last to practice We develop the simplex method algorithm for LP problems given in feasible canonical form and standard form We also discuss two methods, the M-Method and the Two-Phase Method, that deal with the situation that we have an infeasible starting basic solution Simplex Method for Problems in Feasible Canonical Form The Simplex method is a method that proceeds from one BFS or extreme point of the feasible region of an LP problem expressed in tableau form to another BFS, in such a way as to continually increase (or decrease) the value of the objective function until optimality is reached The simplex method moves from one extreme point to one of its neighboring extreme point Consider the following LP in feasible canonical form, ie its right hand side vector b : max subject to x = c T x { Ax b, x Its initial tableau is x x x s x n x n+ x n+r x n+m b x n+ a a a s a n b x n+ a a a s a n b x n+r a r a r a rs a rn b r x n+m a m a m a ms a mn b r x c c c s c n Here x n+i, i =, m are the slack variables The original variables x i, i =,, n are called the structural or decision variables Since all b i, we can read off directly from the tableau a starting

2 Chapter SIMPLEX METHOD BFS given by [,,,, b, b,, b m ] T, ie all structural variables x j are set to zero Note that this corresponds to the origin of the n-dimensional subspace R n of R n+m In matrix form, the original constraint Ax b has be augmented to [ x [A I] = Ax + Ix xs] s = b () Here x s is the vector of slack variables Since the columns of the augmented matrix [A I] that correspond to the slack variables {x n+i } m i= is an identity matrix which is clearly invertible, the slack variables {x n+i } m i= are basic We denote by B the set of current basic variables, ie B = {x n+i} m i= The set of non-basic variables, ie {x i } n i= will be denoted by N Consider now the process of replacing an x r B by an x s N We say that x r is to leave the basis and x s is to enter the basis Consequently after this operation, x r becomes non-basic, ie x r N and x s becomes basic, ie x s B This of course amounts to a different (selection of columns of matrix A to give a different) basis B We shall achieve this change of basis by a pivot operation (or simply called a pivot) This pivot operation is designed to maintain an identity matrix as the basis in the tableau at all time Pivot Operation with Respect to the Element a rs Once we have decided to replace x r B by x s N, the a rs in the tableau will be called the pivot element We will see later that the feasibility condition implies that a rs > The r-th row and the s-th column of the tableau are called the pivot row and the pivot column respectively The rules to update the tableau are: (a) In pivot row, a rj a rj /a rs for j =, n + m (b) In pivot column, a rs, a is for i =, m, i r (c) For all other elements, a ij a ij a rj a is /a rs Graphically, we have j s i a ij a is r a rj a rs becomes j s i a ij a rj a is /a rs r a rj /a rs Or, simply, a b c d becomes a bc d c d Notice that this pivot operation is simply the Gaussian elimination such that variable x s is eliminated from all m + but the r-th equation, and in the r-th equation, the coefficient of x s is equal to In fact, Rule (a) above amounts to normalization of the pivot row such that the pivot element becomes Rule (b) above amounts to eliminations of all the entries in the pivot column except the pivot element Rule (c) is to compute the Schur s complement for the remaining entries in the tableau

3 Simplex Method for Problems in Feasible Canonical Form Example Consider The initial tableau is given by Tableau : x + x x + x 4 = x x + x + x = x + x x + x 6 = x x x x 4 x x 6 b x 4 x x 6 B The current basic solution is [,,,,, ] T which is clearly feasible Suppose we choose a, as our pivot element Then after one pivot operation, we have Tableau : x x x x 4 x x 6 b x x 7 x 6 6 B We note that the current basic solution is [,,,, 7, 6] T which is infeasible Using the new (, ) entry as pivot, we have Tableau : x x x x 4 x x 6 b x x x B The current basic solution is [8/, 7/,,,, 9/] T and is feasible Finally, let us eliminate the last slack variable x 6 by replacing it by x Tableau 4: x x x x 4 x x 6 b x x 4 x 9 B 4 The current basic solution is [, 4, 9,,, ] T which is infeasible and degenerate Thus we see that one cannot choose the pivot arbitrarily It has to be chosen according to some feasibility criterion There are three important observations that we should note here First the pivot operations which amounts to elementary row operations on the tableaus, are being recorded in the tableaus at the columns that correspond to the slack variables In the example above, one can easily check that Tableau i is obtained from Tableau by pre-multiplying Tableau by the matrix formed by the columns of x 4, x and x 6 in Tableau i In the tableaus, the inverse of these matrices are computed

4 4 Chapter SIMPLEX METHOD and are denoted by B i Let the columns in Tableau be denoted as usual by a j and the columns in Tableau i be denoted by y j, then since [a, a,, a m+n ] = [A I] = Bi [B i A B i ] = B i [y, y,, y m+n ], it is clear that B i y j = a j Comparing this with equation (), we see that B i are the change of basis matrices from Tableau i to Tableau Our second observation is the following one Since the last column in Tableau is given by b, the last column in Tableau i, which we denote by y = (y,, y m ) T, will be given by B i y = b Since B i is invertible, y gives the basic variables of the current basic solution, ie the basic solution x i B corresponding to B i is given by x i B = y = B i b () For this reason, the last column of the tableau, ie y, is called the solution column The third observation is that the columns of B i are the columns of the initial tableau For example, the columns of B are the first, second and the sixth columns of Tableau In fact, Tableau is obtained by moving (via elementary row operations) the identity matrix in Tableau to the first, second and the sixth columns in Tableau It indicates that in each iteration of the simplex method, we are just choosing different selection of columns of the augmented matrix to give a different basic matrix B In particular, the solution obtained in each tableau is indeed the basic solution to our original augmented matrix system () In fact, Tableau means that 8/ 7/ [A I] = B 8/ 7/ = = b, 9/ 9/ ie the current solution is given by [8/, 7/,,,, 9/] T For Tableau 4, since B 4 = A, we have [I A ] [ ] A b = A b, which is equivalent to A (A b) + I = b, ie the current solution in Tableau 4 is given by [A b, ] T In the following, we consider the criteria that guarantee the feasibility and optimality of the solutions Feasibility Condition Suppose that the entering variable x s has been chosen according to some optimality conditions, ie the pivot column is the s-th column Then the leaving basic variable x r must be selected as the basic variable corresponding to the smallest positive ratio of the values of the current right hand side to the current positive constraint coefficients of the entering non-basic variable x s

5 Simplex Method for Problems in Feasible Canonical Form To determine row r x s y Ratio y r y rs = min i { y i y is y is > } y s y y y s y s y y y s y is y i y i y is y ms y m y m y ms This follows directly from equation (4) and the fact that the current basic solution x B defined in (4) is given by the solution column y here, see () above Optimality Condition For simplicity, we consider a maximization problem We first denote the entries in the row that correspond to x by y j The (m +, m + n + )-th entry in the tableau is denoted by y We will show in the next section that y j = (c j z j ), j =,, n + m, () the negation of the reduced cost coefficients that appeared in Theorem 6 Here z j is defined in () Moreover, we will show also that y = c T Bx B, (4) ie y is the current objective function value associated with the current BFS in the tableau Thus according to Theorem 6, the entering variable x s N can be selected as a non-basic variable x s having a negative coefficient Usual choices are the first negative y s or the most negative y s If all coefficients y j are non-negative, then by Theorem 7, an optimal solution has been reached 4 Summary of Computation Procedure Once the initial tableau has been constructed, the simplex procedure calls for the successive iteration of the following steps Testing of the coefficients of the objective function row to determine whether an optimal solution has been reached, ie, whether the optimality condition that all coefficients are nonnegative in that row is satisfied If not, select a currently non-basic variable x s to enter the basis For example, the first negative coefficient or the most negative one Then determine the currently basic variable x r to leave the basis using the feasibility condition, ie select x r where y r /y rs = min i {y i /y is y is > } 4 Perform a pivot operation with pivot row corresponding to x r and pivot column corresponding to x s Return to

6 6 Chapter SIMPLEX METHOD Example Consider the LP problem: Max x = x + x + x x + x + x x + x + x Subject to x + x + x 6 x, x, x By adding slack variables x 4, x and x 6, we have the following initial tableau Tableau : Initial tableau, current BFS is x = [,,,,, 6] T and x = x x x x 4 x x 6 b Ratio x 4 = x = x = x We choose x as the entering variable to illustrate that any nonbasic variable with negative coefficient can be chosen as entering variable The smallest ratio is given by x 4 row Thus x 4 is the leaving variable Tableau : Current BFS is x = [,,,,, ] T and x = x x x x 4 x x 6 b Ratio x = x = x 6 x Tableau : Current BFS is x = [,,,,, ] T and x = 4 x x x x 4 x x 6 b Ratio x x x 6 4 x 7 4 Tableau 4: Optimal tableau, optimal BFS x = [/,, 8/,,, 4] T, x = 7/ x x x x 4 x x 6 b x x 8 x x 7

7 Simplex Methods for Problems in Standard Form 7 We note that the extreme point sequence that the simplex method passes through are {x 4, x, x 6 } {x, x, x 6 } {x, x, x 6 } {x, x, x 6 } Simplex Methods for Problems in Standard Form Our previous method is based upon the existence of an initial BFS to the problem It is desirable to have an identity matrix as the initial basic matrix For LP in feasible canonical form, the initial basic matrix is the matrix associated with the slack variables, and is an identity matrix Consider an LP in standard form: max subject to x = c T x { Ax = b, x where we assume that b There is no obvious initial starting basis B such that B = I m For notational simplicity, assume that we pick B as the last m (linearly independent) columns of A, ie A is of the form A = [N B] We then have for the augmented system: { Nx N + Bx B = b Multiplying by B to the first equation yields, x c T N x N c T Bx B = B Nx N + x B = B b or Hence the x equation becomes x B = B b B Nx N x c T Nx N c T B(B b B Nx N ) = Thus we have { B Nx N x (c T N c T BB N)x N + x B = B b = c T BB b Denoting z T N = ct B B N (an (n m) row vector) gives { B Nx N + x B = B b x (c T N z T N)x N = c T BB b which is called the general representation of an LP in standard form with respect to the basis B Its initial simplex tableau is then x N x B b x B B N I B b x (c T N zt N ) ct B B b

8 8 Chapter SIMPLEX METHOD We note that the j-th entry of z N is given by c T BB N j = c T BB a j = c T By j = z j where z j is defined as in () Thus in the table, we see that the entries in the x row are given by (c j z j ) for x j N and zero for x j B Thus they are the negation of the reduced cost coefficients This verifies equation () that we have assumed earlier Moreover, by (), we see that y = c T BB b = c T Bx B, which is the same as (4) We remark that x is now expressed in terms of the non-basic variables, x = c T BB b + (c j z j )x j () Hence it is easy to see that for maximization problem, the current BFS is optimal when c j z j for all j For minimization problem, the current BFS will be optimal when c j z j for all j Example Consider the following LP x j N max x = x + x subject to x + x 4 x + x = 6 x, x Putting into standard form by adding the surplus variable x, the augmented system is: x + x x = 4 x + x = 6 x x x = The simplex tableau for the problem is: Tableau : x x x b 4 6 x Here we do not have a starting identity matrix Suppose we let x and x to be our starting basic variables, then [ ] [ ] [ B =, N = and c B = ] In this case B = [ ], x = B N = [ ] [ ] = b = B b = [ ] [ ] 4 = 6 [ [ ] 8 ],

9 Simplex Methods for Problems in Standard Form 9 It is also easily check that z = c T B a j = [, ] [ and the current value of the objective function is given by Hence the starting tableau is: Tableau : c T BB b = [, ] ] [ ] = [ ] [ 4 = / 6] x x x b x 8 x x Thus x = [/, 8/, ] T is an initial BFS We can now apply the simplex method as discussed in to find the optimal solution The next iteration gives: Tableau : x x x b x 6 x 8 x 6 Thus the optimal solution is x = [6,, 8] T with x = 6 We note that if we choose x and x as our starting basis variables, then we get Tableau immediately and no iteration is required However, if x and x are chosen as starting variables, then we have Tableau : x x x b x x x Hence the starting basic solution is not feasible and we cannot use the simplex method to find our optimal solution

10 Chapter SIMPLEX METHOD The M-Method Example illustrates that the starting basic solution may sometimes be infeasible The M- method and the Two-phase method discussed in this and the next sections are methods that can find a starting basic feasible solution whenever it exists Consider again an LPP where there is no desirable starting identity matrix max subject to x = c T x { Ax = b, x where b We may add suitable number of artificial variables x a, x a,, x am to it to get a starting identity matrix The corresponding prices for the artificial variables are M for maximization problem, where M is sufficiently large The effect of the constant M is to penalize any artificial variables that will occur with positive values in the final optimal solutions Using the idea, the LPP becomes max subject to z = c T x M T x a { Ax + Im x a = b x, where x a = (x a, x a,, x am ) T and is the vector of all ones We observe that x = and x a = b is a feasible starting BFS Moreover, any solution to Ax + I m x a = b which is also a solution to Ax = b must have x a = Thus, we have to drive x a = if possible Example 4 Consider the LP in Example again max x = x + x x + x 4 subject to x + x = 6 x, x Introducing surplus variable x and artificial variables x 4 and x yields, x + x x + x 4 = 4 x + x + x = 6 x x x + Mx 4 + Mx = Now the columns corresponding to x 4 and x form an identity matrix In tableau form, we have x x x x 4 x b x 4 4 x 6 x M M Notice that in the x row, the reduced cost coefficients that correspond to the basic variables x 4 and x are not zero These nonzero entries are to be eliminated first before we have our starting tableau After eliminations of those M, we have the initial tableau:

11 4 The Two-Phase Method x x x x 4 x b x 4 4 x 6 x ( + M) ( + M) M M We note that once an artificial variable becomes non-basic, it can be dropped from consideration in subsequent calculations x x x x b x x 4 x +M +M 4M After we eliminate all the artificial variables we have x x x b x 8 x x At this point all artificial variables are dropped from the problem, and x = [/, 8/, ] T is an initial BFS Notice that this is the same as Tableau in Example After one iteration, we get the final optimal tableau x x x b x 6 x 8 x 6 Thus the optimal solution is x = (6,, 8) T with x = 6 4 The Two-Phase Method The M-method is sensitive to round-off error when being implemented on computers The two-phase method is used to circumvent this difficulty Phase I: (Search for a Starting BFS) Instead of considering the actual objective function in the M-Method z = n m c i x i M x ai, i= i=

12 Chapter SIMPLEX METHOD we maximize the function z = m x ai Since b, the initial BFS satisfies x a Notice that z and the possible maximum value of z is zero Moreover, z will be zero only if each artificial variable is zero If the maximum of z is zero, we have driven all artificial variables to zero If the maximum of z is not zero, then the artificial variables cannot be driven to zero and the original problem has no feasible solution In Phase I, we stop as soon as z becomes zero, because we know that this is the maximum value of z We need not continue until the optimality criterion is satisfied if z becomes zero before this happens During Phase I, the sequence of vectors to enter and leave the basis is the same as the M-method except when the vectors are tied In fact, the reduced cost coefficients of both methods are given by i= M-method: z j c j = M r y rj + β Phase I: z j c j = r y rj where β is a small quantity compared with M At the end of Phase I, ie when the optimality condition is satisfied or z =, we have one of the following three possibilities: (i) max z <, in this case, no feasible solution exists for our original problem, see 4 (ii) max z = and no artificial variable appears in the basis, ie we have found a BFS to the original problem (iii) max z = and one or more artificial variables appear in the basis at zero level In this case, we have also found a basic degenerate feasible solution to the original problem Degenerate solutions are discussed in 4 Phase II: (Conclude with an Optimal BFS) When Phase I ends in (ii) or (iii), we go to Phase II to find an optimal solution In Phase II, we assign the actual price c j to each structural variable and a price of zero to any artificial variables which still appear in the basis at zero level Thus the objective function to be optimized in Phase II is the actual objective function z = n c i x i i= When Phase I ends in (ii), we are back to the situation discussed in, and there should be no problem When Phase I ends in (iii), we must give special attentions to the artificial variables which appear in the basis at zero level We must make sure that the artificial variables never become positive again in Phase II We will return to this case in 46 Example Consider the following LP min x = x + 4x + 7x + x 4 + x x + x + x + x 4 + x = 7 x + x + x + x 4 + x = 6 subject to x + x + x + x 4 + x = 4 x free, x, x, x 4, x Since x is free, it can be eliminated by solving for x in terms of the other variables from the first equation and substituting everywhere else This can be done nicely using our pivot operation on the following simplex tableau:

13 4 The Two-Phase Method x x x x 4 x b Initial tableau We select any non-zero element in the first column as our pivot element this will eliminate x from all other rows:- x x x x 4 x b 7 ( ) 4 Equivalent Problem Saving the first row ( ) for future reference only, we carry on only the sub-tableau with the first row and the first column deleted There is no obvious basic feasible solution, so we use the two-phase method: After making b, we introduce artificial variables y and y to give the artificial problem:- x x x 4 x y y b c B = Initial Tableau for Phase I The cost coefficients of the artificial variables are + because we are dealing with a minimization problem Transforming (by adding the first two rows to the last row) the last row to give a tableau in canonical form, we get x x x 4 x y y b y y x 4 First tableau for Phase I

14 4 Chapter SIMPLEX METHOD which is in canonical form Recall that this is a minimization problem, entering variable is chosen with positive entry (rather than negative) in the x -row We carry out the pivot operations with the indicated pivot elements:- x x x 4 x y y b x y x Second tableau for Phase I x x x 4 x y y b x x x Final tableau for Phase I At the end of Phase I, we go back to the equivalent reduced problem (ie discarding the artificial variables y, y ):- x x x 4 x b x x x 4 c B = Initial problem for Phase II This is transform into x x x 4 x b x x x c B = Initial tableau for Phase II Pivoting as shown gives x x x 4 x b x x x 9 Final tableau for Phase II

15 4 The Two-Phase Method The solution x =, x = can be inserted in the expression ( ) for x giving x = 7 + () + () = Thus the final solution is x = [,,,, ] T with x = 9

### Chapter 6. Linear Programming: The Simplex Method. Introduction to the Big M Method. Section 4 Maximization and Minimization with Problem Constraints

Chapter 6 Linear Programming: The Simplex Method Introduction to the Big M Method In this section, we will present a generalized version of the simplex method that t will solve both maximization i and

### 1 Solving LPs: The Simplex Algorithm of George Dantzig

Solving LPs: The Simplex Algorithm of George Dantzig. Simplex Pivoting: Dictionary Format We illustrate a general solution procedure, called the simplex algorithm, by implementing it on a very simple example.

### Practical Guide to the Simplex Method of Linear Programming

Practical Guide to the Simplex Method of Linear Programming Marcel Oliver Revised: April, 0 The basic steps of the simplex algorithm Step : Write the linear programming problem in standard form Linear

### Definition of a Linear Program

Definition of a Linear Program Definition: A function f(x 1, x,..., x n ) of x 1, x,..., x n is a linear function if and only if for some set of constants c 1, c,..., c n, f(x 1, x,..., x n ) = c 1 x 1

### 3 Does the Simplex Algorithm Work?

Does the Simplex Algorithm Work? In this section we carefully examine the simplex algorithm introduced in the previous chapter. Our goal is to either prove that it works, or to determine those circumstances

### Linear Programming in Matrix Form

Linear Programming in Matrix Form Appendix B We first introduce matrix concepts in linear programming by developing a variation of the simplex method called the revised simplex method. This algorithm,

### IEOR 4404 Homework #2 Intro OR: Deterministic Models February 14, 2011 Prof. Jay Sethuraman Page 1 of 5. Homework #2

IEOR 4404 Homework # Intro OR: Deterministic Models February 14, 011 Prof. Jay Sethuraman Page 1 of 5 Homework #.1 (a) What is the optimal solution of this problem? Let us consider that x 1, x and x 3

### Systems of Linear Equations

Systems of Linear Equations Beifang Chen Systems of linear equations Linear systems A linear equation in variables x, x,, x n is an equation of the form a x + a x + + a n x n = b, where a, a,, a n and

### Linear Dependence Tests

Linear Dependence Tests The book omits a few key tests for checking the linear dependence of vectors. These short notes discuss these tests, as well as the reasoning behind them. Our first test checks

### 4.6 Linear Programming duality

4.6 Linear Programming duality To any minimization (maximization) LP we can associate a closely related maximization (minimization) LP. Different spaces and objective functions but in general same optimal

### Solving Linear Programs

Solving Linear Programs 2 In this chapter, we present a systematic procedure for solving linear programs. This procedure, called the simplex method, proceeds by moving from one feasible solution to another,

### Special Situations in the Simplex Algorithm

Special Situations in the Simplex Algorithm Degeneracy Consider the linear program: Maximize 2x 1 +x 2 Subject to: 4x 1 +3x 2 12 (1) 4x 1 +x 2 8 (2) 4x 1 +2x 2 8 (3) x 1, x 2 0. We will first apply the

### Simplex method summary

Simplex method summary Problem: optimize a linear objective, subject to linear constraints 1. Step 1: Convert to standard form: variables on right-hand side, positive constant on left slack variables for

### Standard Form of a Linear Programming Problem

494 CHAPTER 9 LINEAR PROGRAMMING 9. THE SIMPLEX METHOD: MAXIMIZATION For linear programming problems involving two variables, the graphical solution method introduced in Section 9. is convenient. However,

### Linear Programming. March 14, 2014

Linear Programming March 1, 01 Parts of this introduction to linear programming were adapted from Chapter 9 of Introduction to Algorithms, Second Edition, by Cormen, Leiserson, Rivest and Stein [1]. 1

### Sensitivity Analysis 3.1 AN EXAMPLE FOR ANALYSIS

Sensitivity Analysis 3 We have already been introduced to sensitivity analysis in Chapter via the geometry of a simple example. We saw that the values of the decision variables and those of the slack and

### 7.4 Linear Programming: The Simplex Method

7.4 Linear Programming: The Simplex Method For linear programming problems with more than two variables, the graphical method is usually impossible, so the simplex method is used. Because the simplex method

### OPRE 6201 : 2. Simplex Method

OPRE 6201 : 2. Simplex Method 1 The Graphical Method: An Example Consider the following linear program: Max 4x 1 +3x 2 Subject to: 2x 1 +3x 2 6 (1) 3x 1 +2x 2 3 (2) 2x 2 5 (3) 2x 1 +x 2 4 (4) x 1, x 2

### Linear Programming Notes V Problem Transformations

Linear Programming Notes V Problem Transformations 1 Introduction Any linear programming problem can be rewritten in either of two standard forms. In the first form, the objective is to maximize, the material

### 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

### 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

### Chapter 2 Solving Linear Programs

Chapter 2 Solving Linear Programs Companion slides of Applied Mathematical Programming by Bradley, Hax, and Magnanti (Addison-Wesley, 1977) prepared by José Fernando Oliveira Maria Antónia Carravilla A

### 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

### Duality in Linear Programming

Duality in Linear Programming 4 In the preceding chapter on sensitivity analysis, we saw that the shadow-price interpretation of the optimal simplex multipliers is a very useful concept. First, these shadow

### Lecture 6. Inverse of Matrix

Lecture 6 Inverse of Matrix Recall that any linear system can be written as a matrix equation In one dimension case, ie, A is 1 1, then can be easily solved as A x b Ax b x b A 1 A b A 1 b provided that

### Solving 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

### 7.1 Modelling the transportation problem

Chapter Transportation Problems.1 Modelling the transportation problem The transportation problem is concerned with finding the minimum cost of transporting a single commodity from a given number of sources

### Solving Systems of Linear Equations; Row Reduction

Harvey Mudd College Math Tutorial: Solving Systems of Linear Equations; Row Reduction Systems of linear equations arise in all sorts of applications in many different fields of study The method reviewed

### MATH10212 Linear Algebra. Systems of Linear Equations. Definition. An n-dimensional vector is a row or a column of n numbers (or letters): a 1.

MATH10212 Linear Algebra Textbook: D. Poole, Linear Algebra: A Modern Introduction. Thompson, 2006. ISBN 0-534-40596-7. Systems of Linear Equations Definition. An n-dimensional vector is a row or a column

### Linear Programming Problems

Linear Programming Problems Linear programming problems come up in many applications. In a linear programming problem, we have a function, called the objective function, which depends linearly on a number

### Operation Research. Module 1. Module 2. Unit 1. Unit 2. Unit 3. Unit 1

Operation Research Module 1 Unit 1 1.1 Origin of Operations Research 1.2 Concept and Definition of OR 1.3 Characteristics of OR 1.4 Applications of OR 1.5 Phases of OR Unit 2 2.1 Introduction to Linear

### SYSTEMS OF EQUATIONS

SYSTEMS OF EQUATIONS 1. Examples of systems of equations Here are some examples of systems of equations. Each system has a number of equations and a number (not necessarily the same) of variables for which

### Linear Programming I

Linear Programming I November 30, 2003 1 Introduction In the VCR/guns/nuclear bombs/napkins/star wars/professors/butter/mice problem, the benevolent dictator, Bigus Piguinus, of south Antarctica penguins

### 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 +

### Linear Programming: Theory and Applications

Linear Programming: Theory and Applications Catherine Lewis May 11, 2008 1 Contents 1 Introduction to Linear Programming 3 1.1 What is a linear program?...................... 3 1.2 Assumptions.............................

### Using the Simplex Method to Solve Linear Programming Maximization Problems J. Reeb and S. Leavengood

PERFORMANCE EXCELLENCE IN THE WOOD PRODUCTS INDUSTRY EM 8720-E October 1998 \$3.00 Using the Simplex Method to Solve Linear Programming Maximization Problems J. Reeb and S. Leavengood A key problem faced

### Solving Systems of Linear Equations. Substitution

Solving Systems of Linear Equations There are two basic methods we will use to solve systems of linear equations: Substitution Elimination We will describe each for a system of two equations in two unknowns,

### MATH 304 Linear Algebra Lecture 18: Rank and nullity of a matrix.

MATH 304 Linear Algebra Lecture 18: Rank and nullity of a matrix. Nullspace Let A = (a ij ) be an m n matrix. Definition. The nullspace of the matrix A, denoted N(A), is the set of all n-dimensional column

### Solving Systems of Linear Equations

LECTURE 5 Solving Systems of Linear Equations Recall that we introduced the notion of matrices as a way of standardizing the expression of systems of linear equations In today s lecture I shall show how

### NOTES ON LINEAR TRANSFORMATIONS

NOTES ON LINEAR TRANSFORMATIONS Definition 1. Let V and W be vector spaces. A function T : V W is a linear transformation from V to W if the following two properties hold. i T v + v = T v + T v for all

### MATH 551 - APPLIED MATRIX THEORY

MATH 55 - APPLIED MATRIX THEORY FINAL TEST: SAMPLE with SOLUTIONS (25 points NAME: PROBLEM (3 points A web of 5 pages is described by a directed graph whose matrix is given by A Do the following ( points

### Abstract: We describe the beautiful LU factorization of a square matrix (or how to write Gaussian elimination in terms of matrix multiplication).

MAT 2 (Badger, Spring 202) LU Factorization Selected Notes September 2, 202 Abstract: We describe the beautiful LU factorization of a square matrix (or how to write Gaussian elimination in terms of matrix

### 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

### 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

### Linear Programming. April 12, 2005

Linear Programming April 1, 005 Parts of this were adapted from Chapter 9 of i Introduction to Algorithms (Second Edition) /i by Cormen, Leiserson, Rivest and Stein. 1 What is linear programming? The first

### Math 313 Lecture #10 2.2: The Inverse of a Matrix

Math 1 Lecture #10 2.2: The Inverse of a Matrix Matrix algebra provides tools for creating many useful formulas just like real number algebra does. For example, a real number a is invertible if there is

### What is Linear Programming?

Chapter 1 What is Linear Programming? An optimization problem usually has three essential ingredients: a variable vector x consisting of a set of unknowns to be determined, an objective function of x to

### Linear Programming for Optimization. Mark A. Schulze, Ph.D. Perceptive Scientific Instruments, Inc.

1. Introduction Linear Programming for Optimization Mark A. Schulze, Ph.D. Perceptive Scientific Instruments, Inc. 1.1 Definition Linear programming is the name of a branch of applied mathematics that

### Row Echelon Form and Reduced Row Echelon Form

These notes closely follow the presentation of the material given in David C Lay s textbook Linear Algebra and its Applications (3rd edition) These notes are intended primarily for in-class presentation

### Diagonalisation. Chapter 3. Introduction. Eigenvalues and eigenvectors. Reading. Definitions

Chapter 3 Diagonalisation Eigenvalues and eigenvectors, diagonalisation of a matrix, orthogonal diagonalisation fo symmetric matrices Reading As in the previous chapter, there is no specific essential

### 1.2 Solving a System of Linear Equations

1.. SOLVING A SYSTEM OF LINEAR EQUATIONS 1. Solving a System of Linear Equations 1..1 Simple Systems - Basic De nitions As noticed above, the general form of a linear system of m equations in n variables

### Nonlinear Programming Methods.S2 Quadratic Programming

Nonlinear Programming Methods.S2 Quadratic Programming Operations Research Models and Methods Paul A. Jensen and Jonathan F. Bard A linearly constrained optimization problem with a quadratic objective

### 7 Gaussian Elimination and LU Factorization

7 Gaussian Elimination and LU Factorization In this final section on matrix factorization methods for solving Ax = b we want to take a closer look at Gaussian elimination (probably the best known method

### 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

### 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

### Lecture 3: Linear Programming Relaxations and Rounding

Lecture 3: Linear Programming Relaxations and Rounding 1 Approximation Algorithms and Linear Relaxations For the time being, suppose we have a minimization problem. Many times, the problem at hand can

### Math 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

### 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

### SYSTEMS OF EQUATIONS AND MATRICES WITH THE TI-89. by Joseph Collison

SYSTEMS OF EQUATIONS AND MATRICES WITH THE TI-89 by Joseph Collison Copyright 2000 by Joseph Collison All rights reserved Reproduction or translation of any part of this work beyond that permitted by Sections

### 1. LINEAR EQUATIONS. A linear equation in n unknowns x 1, x 2,, x n is an equation of the form

1. LINEAR EQUATIONS A linear equation in n unknowns x 1, x 2,, x n is an equation of the form a 1 x 1 + a 2 x 2 + + a n x n = b, where a 1, a 2,..., a n, b are given real numbers. For example, with x and

### Solutions to Math 51 First Exam January 29, 2015

Solutions to Math 5 First Exam January 29, 25. ( points) (a) Complete the following sentence: A set of vectors {v,..., v k } is defined to be linearly dependent if (2 points) there exist c,... c k R, not

### Vector and Matrix Norms

Chapter 1 Vector and Matrix Norms 11 Vector Spaces Let F be a field (such as the real numbers, R, or complex numbers, C) with elements called scalars A Vector Space, V, over the field F is a non-empty

### 2.5 Elementary Row Operations and the Determinant

2.5 Elementary Row Operations and the Determinant Recall: Let A be a 2 2 matrtix : A = a b. The determinant of A, denoted by det(a) c d or A, is the number ad bc. So for example if A = 2 4, det(a) = 2(5)

### Matrix Representations of Linear Transformations and Changes of Coordinates

Matrix Representations of Linear Transformations and Changes of Coordinates 01 Subspaces and Bases 011 Definitions A subspace V of R n is a subset of R n that contains the zero element and is closed under

### 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

### 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

### 3. Evaluate the objective function at each vertex. Put the vertices into a table: Vertex P=3x+2y (0, 0) 0 min (0, 5) 10 (15, 0) 45 (12, 2) 40 Max

SOLUTION OF LINEAR PROGRAMMING PROBLEMS THEOREM 1 If a linear programming problem has a solution, then it must occur at a vertex, or corner point, of the feasible set, S, associated with the problem. Furthermore,

### 56:171. Operations Research -- Sample Homework Assignments Fall 1992 Dennis Bricker Dept. of Industrial Engineering University of Iowa.

56:171 Operations Research -- Sample Homework Assignments Fall 1992 Dennis Bricker Dept. of Industrial Engineering University of Iowa Homework #1 (1.) Linear Programming Model Formulation. SunCo processes

### Recall that two vectors in are perpendicular or orthogonal provided that their dot

Orthogonal Complements and Projections Recall that two vectors in are perpendicular or orthogonal provided that their dot product vanishes That is, if and only if Example 1 The vectors in are orthogonal

### Lecture 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

### This exposition of linear programming

Linear Programming and the Simplex Method David Gale This exposition of linear programming and the simplex method is intended as a companion piece to the article in this issue on the life and work of George

### 4.6 Null Space, Column Space, Row Space

NULL SPACE, COLUMN SPACE, ROW SPACE Null Space, Column Space, Row Space In applications of linear algebra, subspaces of R n typically arise in one of two situations: ) as the set of solutions of a linear

### Reduced echelon form: Add the following conditions to conditions 1, 2, and 3 above:

Section 1.2: Row Reduction and Echelon Forms Echelon form (or row echelon form): 1. All nonzero rows are above any rows of all zeros. 2. Each leading entry (i.e. left most nonzero entry) of a row is in

### Sec 4.1 Vector Spaces and Subspaces

Sec 4. Vector Spaces and Subspaces Motivation Let S be the set of all solutions to the differential equation y + y =. Let T be the set of all 2 3 matrices with real entries. These two sets share many common

### University 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

### LS.6 Solution Matrices

LS.6 Solution Matrices In the literature, solutions to linear systems often are expressed using square matrices rather than vectors. You need to get used to the terminology. As before, we state the definitions

### We seek a factorization of a square matrix A into the product of two matrices which yields an

LU Decompositions We seek a factorization of a square matrix A into the product of two matrices which yields an efficient method for solving the system where A is the coefficient matrix, x is our variable

### Solutions to Linear Algebra Practice Problems 1. form (because the leading 1 in the third row is not to the right of the

Solutions to Linear Algebra Practice Problems. Determine which of the following augmented matrices are in row echelon from, row reduced echelon form or neither. Also determine which variables are free

### Methods for Finding Bases

Methods for Finding Bases Bases for the subspaces of a matrix Row-reduction methods can be used to find bases. Let us now look at an example illustrating how to obtain bases for the row space, null space,

### Similarity and Diagonalization. Similar Matrices

MATH022 Linear Algebra Brief lecture notes 48 Similarity and Diagonalization Similar Matrices Let A and B be n n matrices. We say that A is similar to B if there is an invertible n n matrix P such that

### Math 240: Linear Systems and Rank of a Matrix

Math 240: Linear Systems and Rank of a Matrix Ryan Blair University of Pennsylvania Thursday January 20, 2011 Ryan Blair (U Penn) Math 240: Linear Systems and Rank of a Matrix Thursday January 20, 2011

### 1 Eigenvalues and Eigenvectors

Math 20 Chapter 5 Eigenvalues and Eigenvectors Eigenvalues and Eigenvectors. Definition: A scalar λ is called an eigenvalue of the n n matrix A is there is a nontrivial solution x of Ax = λx. Such an x

### Linear Algebra Notes

Linear Algebra Notes Chapter 19 KERNEL AND IMAGE OF A MATRIX Take an n m matrix a 11 a 12 a 1m a 21 a 22 a 2m a n1 a n2 a nm and think of it as a function A : R m R n The kernel of A is defined as Note

### Vector Spaces 4.4 Spanning and Independence

Vector Spaces 4.4 and Independence October 18 Goals Discuss two important basic concepts: Define linear combination of vectors. Define Span(S) of a set S of vectors. Define linear Independence of a set

### LINEAR ALGEBRA W W L CHEN

LINEAR ALGEBRA W W L CHEN c W W L Chen, 1997, 2008 This chapter is available free to all individuals, on understanding that it is not to be used for financial gain, and may be downloaded and/or photocopied,

### 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

### 9.4 THE SIMPLEX METHOD: MINIMIZATION

SECTION 9 THE SIMPLEX METHOD: MINIMIZATION 59 The accounting firm in Exercise raises its charge for an audit to \$5 What number of audits and tax returns will bring in a maximum revenue? In the simplex

### 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

### Linear Maps. Isaiah Lankham, Bruno Nachtergaele, Anne Schilling (February 5, 2007)

MAT067 University of California, Davis Winter 2007 Linear Maps Isaiah Lankham, Bruno Nachtergaele, Anne Schilling (February 5, 2007) As we have discussed in the lecture on What is Linear Algebra? one of

### 56:171 Operations Research Midterm Exam Solutions Fall 2001

56:171 Operations Research Midterm Exam Solutions Fall 2001 True/False: Indicate by "+" or "o" whether each statement is "true" or "false", respectively: o_ 1. If a primal LP constraint is slack at the

### LECTURE 5: DUALITY AND SENSITIVITY ANALYSIS. 1. Dual linear program 2. Duality theory 3. Sensitivity analysis 4. Dual simplex method

LECTURE 5: DUALITY AND SENSITIVITY ANALYSIS 1. Dual linear program 2. Duality theory 3. Sensitivity analysis 4. Dual simplex method Introduction to dual linear program Given a constraint matrix A, right

### 4.5 Linear Dependence and Linear Independence

4.5 Linear Dependence and Linear Independence 267 32. {v 1, v 2 }, where v 1, v 2 are collinear vectors in R 3. 33. Prove that if S and S are subsets of a vector space V such that S is a subset of S, then

### Mathematics 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

### 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.

### 1 Determinants and the Solvability of Linear Systems

1 Determinants and the Solvability of Linear Systems In the last section we learned how to use Gaussian elimination to solve linear systems of n equations in n unknowns The section completely side-stepped

### December 4, 2013 MATH 171 BASIC LINEAR ALGEBRA B. KITCHENS

December 4, 2013 MATH 171 BASIC LINEAR ALGEBRA B KITCHENS The equation 1 Lines in two-dimensional space (1) 2x y = 3 describes a line in two-dimensional space The coefficients of x and y in the equation

### 1 VECTOR SPACES AND SUBSPACES

1 VECTOR SPACES AND SUBSPACES What is a vector? Many are familiar with the concept of a vector as: Something which has magnitude and direction. an ordered pair or triple. a description for quantities such

### Solving Systems of Linear Equations Using Matrices

Solving Systems of Linear Equations Using Matrices What is a Matrix? A matrix is a compact grid or array of numbers. It can be created from a system of equations and used to solve the system of equations.