1.2 Solving a System of Linear Equations


 Jeffry Wilcox
 5 years ago
 Views:
Transcription
1 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 is of the form >< > a 11 x 1 + a 1 x + + a 1n x n = b 1 a 1 x 1 + a x + + a n x n = b a m1 x 1 + a m x + + a mn x n = b m Some of the coe cients a ij can be zero. In fact, the more coe cients are zero, the easier the system will be to solve. The easiest system to solve is of the form a 11 x 1 = b 1 >< a x = b a x = b > a mn x n = b m Its corresponding augmented matrix is a b 1 0 a 0 b 0 0 a 0 b a mn b m To solve, we simply divide by the coe cient in front of each variable. This is illustrated below. < x 1 = 10 Example 0 Solve x = 9 x = The solution is x 1 = 10 =, x = 9 =, x =. When all the coe cients in the system above are 1, the system in the above example is said to be in reduced rowechelon form. The precise de nition is given below. De nition 1 Consider a system of linear equations. 1. The system is said to be in reduced rowechelon form if it satis es the four properties below
2 CHAPTER 1. SYSTEMS OF LINEAR EQUATIONS AND MATRICES (a) If a row does not consist entirely of zeros, then the rst nonzero number is a 1. This number is called the leading 1. (b) The rows consisting entirely of zeros, if there are any, are at the bottom of the system. (c) If two consecutive rows contain a leading 1, then the leading 1 of the higher row is further to the left than the leading 1 of the lower row. (d) Each column that contains a leading 1 contains 0 everywhere else.. The system is said to be in rowechelon form if the properties a  c are satis ed. Thus, a system in reduced rowechelon form is necessarily in rowechelon form. Remark The same de nitions apply to the augmented matrix of a system. Remark Since we can switch back and forth between a system of linear equations and its corresponding augmented matrix, we can either work on a system, or on its corresponding augmented matrix. We will illustrate both. Example The matrices below are in reduced rowechelon form Example The matrices below are in rowechelon form A system in rowechelon form is also easy to solve. We illustrate how with an example.
3 1.. SOLVING A SYSTEM OF LINEAR EQUATIONS 9 Example Consider the system < x 1 +x x = 10 x +x = x = Its corresponding augmented matrix is Such a system is also fairly easy to solve. The method used is called backsubstitution. Since we know x =, we can replace x by its value in the second equation to obtain x + = or x =. Now that we know x and x, we can replace in the rst equation to obtain x 1 + = 10 or x 1 =. It is called back substitution because we work backward. We start by nding the value of the last variable, then the next to the last and so on, until we nd the value of the rst variable. At this point, we are done. Remark In order to be able to do backsubstitution, it is not necessary that a ii = 1 in the i th equation. De nition Two systems are said to be equivalent if they have the same solution set. Similarly, two augmented matrices are said to be equivalent if they are the augmented matrices of two systems having the same solution set. You remember for elementary mathematics, to solve a given equation in one variable say x, we transform it into an equation of the form x = S where S is then the solution we are after. To transform it, we use certain transformations which do not change the solution set of the equation, in other words, we obtain equivalent equations until we have one of the form x = S. You will recall that the transformations that can be applied when solving an equation are 1. Add the same number on both sides of the equation.. Multiply both sides by the same nonzero number. We follow a similar procedure to solve systems of equations. There are actually two procedures we can use. They are outlined below, then explored thoroughly in the next two subsections. The rst technique, is called Gaussian elimination. Algorithm 9 (Gaussian Elimination) To solve a system of linear equations, follow the steps below 1. Transform its augmented matrix into an equivalent augmented matrix in rowechelon form.
4 10 CHAPTER 1. SYSTEMS OF LINEAR EQUATIONS AND MATRICES. Use backsubstitution to nish solving. The second technique is called GaussJordan elimination Algorithm 0 (GaussJordan Elimination) To solve a system of linear equations, follow the steps below 1. Transform its augmented matrix into an equivalent augmented matrix in reduced rowechelon form.. The solution is given by the last column of the augmented matrix in reduced rowechelon form. With either technique, since the systems are equivalent, they will have the same solution set. Usually, getting from a system to an equivalent system in rowechelon form or reduced rowechelon form requires several steps, each step producing an equivalent system. The question is which transformations can we apply to a system so that it is transformed into a system in row echelon form or reduced rowechelon form. The key is that the new system should be equivalent to the original one, so they have the same solutions. It would not help to get a system which is simpler to solve, but which does not have the same solutions as the original one. There are three transformations which can be applied, to a system that will produce an equivalent system. For simplicity, let us label the equations of a system E 1, E,, E m. We will use the same names for the rows of the corresponding augmented matrix. Proposition 1 Each of the following operations on a system of linear equations produces an equivalent system 1. Equation E i can be multiplied by a nonzero constant, with the result used in place of E i. This operation is denoted (E i )! (E i ). Equation E j can be multiplied by any constant and added to equation E i, with the result used in place of E i. This operation is denoted (E j + E i )! (E i ). Equations E i and E j can be transposed in order. This operation is denoted (E i ) $ (E j ) Remark The same operations can be applied to the augmented matrix of a system. Simply replace the term equation by row in the above proposition. Remark These operations are called elementary row operations. Let us look at some examples to illustrate how these transformations are done.
5 1.. SOLVING A SYSTEM OF LINEAR EQUATIONS 11 Example Consider If we perform (E 1 )! (E ), we obtain Example Consider If we perform 1 E! (E ), we obtain This is useful when we want to obtain a leading 1 on a given row. Example Consider If we perform ( E 1 + E )! (E ), we obtain This is useful when we want to make an entry equal to 0. In this case, we made the rst entry of the second row equal to 0. We now look at each method to solve a system in greater detail. 1.. Gaussian Elimination Gaussian elimination was brie y outlined above. It has two main steps. First, we get a system (or its augmented matrix) in rowechelon form. Then, we use back substitution to nish solving. We have already explained back substitution. The rst step is always done in a very orderly fashion. In fact, it is very easy to implement on a computer. It always proceeds as follows (the algorithm below works for both a system and its augmented matrix. The augmented matrix part appears in parentheses)
6 1 CHAPTER 1. SYSTEMS OF LINEAR EQUATIONS AND MATRICES Remark Proceed from the rst equation (row) to the last. Look at the rst equation (row), make sure no equation (row) below it has a leading entry further to the left. If one does, switch the two. Eliminate all the entries below the leading entry of the rst equation (row). Repeat the procedure for equation (row), then, In general, we look at the i th equation (row), make sure the coe cient for x i is not 0. If it is, you will have to nd an equation in which it is not, and interchange the two equations. Then, eliminate x i from all the equations below the i th equation. We do this for i = 1; ; ; n 1 if the system has n equations. Recall that Gaussian elimination is a systematic procedure which transforms a system into an equivalent system in rowechelon form. We illustrate it with examples. Example Consider the system E 1 x 1 +x +x = >< E x 1 +x x +x = 1 E x 1 x x +x = > E x 1 +x +x x = (1.) Its corresponding augmented matrix is We follow the algorithm described above. We begin with the rst row. Its leading entry is in column 1. The goal is to set to 0 all the other entries in column 1 and rows . In order to achieve this, we perform (E E 1 )! (E ), (E E 1 )! (E ) and (E + E 1 )! (E ). The resulting augmented matrix is Next, we work on the second row. First, we make its leading entry 1 by performing ( E 1 )! (E 1 ). The resulting augmented matrix is
7 1.. SOLVING A SYSTEM OF LINEAR EQUATIONS 1 Then we make the entries below the leading entry of the second row equal to 0 by performing (E + E )! (E ) and (E E )! (E ). The resulting augmented matrix is We actually were lucky. In the process, we did part of the next step. Looking at row, we already set the entry below it to 0. We just have 1 to set the leading entry of row to 1 by performing E! E. The resulting augmented matrix is We are on the last row, the only thing to do here is to set its leading entry 1 to 1 by performing 1 E! (E ). The resulting augmented matrix is Now, we see that the matrix is in rowechelon form. The corresponding system is > < > E 1 x 1 +x +x = E x +x +x = E x + 1 x = 1 E x = 1 (1.) The system in (1) has the same solutions as the original system (1). We nish solving the system in (1) using backsubstitution. From E, we get x = 1 We can now use E to nd x as follows x + 1 x = 1 x = 1 (1 x ) x = 0 since x = 1
8 1 CHAPTER 1. SYSTEMS OF LINEAR EQUATIONS AND MATRICES We now continue with E. Finally, using E 1 yields x x x = x = (x + x ) x = x 1 + x + x = x 1 = x x + x 1 = 1 The solutions of the system in (1) and therefore of the system in (1) are x 1 = 1; x = ; x = 0; and x = 1 This is an example in which the procedure produced a unique solution, we had a consistent system. Following the same procedure, how will we detect if we have a system with no solutions, or one with an in nite number of solutions? The next two examples illustrate this. To also illustrate that the same procedure can be applied to system as well as augmented matrices, we do the next example using the system. < Example 9 Solve the system x 1 + x + x = x 1 + x + x = x 1 + x + x = To eliminate x 1 from E and E, we perform (E E 1 )! (E ) and (E E 1 )! (E ). The resulting system is < x 1 + x + x = x = x = The next step would be to eliminate x from E. This was done in the previous step. The only thing left is to set the leading coe cient of the second equation to 1 by performing ( E )! (E ). The corresponding system is < x 1 + x + x = x = x = This system has an in nite number of solutions given by x = and x 1 +x = or x 1 = x. The free variable is x. In parametric notation, the solution set is < x 1 = t x = t x =
9 1.. SOLVING A SYSTEM OF LINEAR EQUATIONS 1 < Example 0 Solve the system x 1 + x + x = x 1 + x + x = x 1 + x + x = To eliminate x 1 from E and E, we perform (E E 1 )! (E ) and (E E 1 )! (E ). The resulting system is < x 1 + x + x = x = x = The next step would be to eliminate x from E. This was done in the previous step. The only thing left is to set the leading coe cient of the second equation to 1 by performing ( E )! (E ). The corresponding system is x 1 + x + x = x = x = This system has no solutions because the last two equations are in contradiction. Remark 1 It is important to notice that the Gaussian elimination procedure is always done in a very orderly fashion. In fact, it is very easy to implement on a computer. It always proceeds as follows Look at the rst equation, make sure the coe cient for x 1 is not 0. If it is, you will have to nd an equation in which it is not, and interchange the two equations. Then, eliminate x 1 from all the equations below the rst equation. Next, look at the second equation, make sure the coe cient for x is not 0. If it is, you will have to nd an equation in which it is not, and interchange the two equations. Then, eliminate x from all the equations below the second equation. The procedure continues the same way. In general, we look at the i th equation, make sure the coe cient for x i is not 0. If it is, you will have to nd an equation in which it is not, and interchange the two equations. Then, eliminate x i from all the equations below the i th equation. We do this for i = 1; ; ; n 1 if the system has n equations. 1.. GaussJordan Elimination Here, we will work with augmented matrices. Recall that GaussJordan elimination is a systematic procedure which transforms a system into an equivalent system in reduced rowechelon form. To obtain a reduced rowechelon form, we rst obtain a rowechelon form, then we go further. We need to make the columns containing the leading entries equal to 0, except for the leading entry of course. We illustrate it with an example.
10 1 CHAPTER 1. SYSTEMS OF LINEAR EQUATIONS AND MATRICES Example Solve the system x 1 +x +x = >< x 1 +x x +x = 1 x 1 x x +x = > x 1 +x +x x = Its corresponding augmented matrix is This is the system we did above. We will take it from its rowechelon form Now, we start from the bottom and work our way up. There is nothing to do on the last row. For row, we make the th entry 0 by performing E 1 E! (E ). The resulting augmented matrix is For row, we make the rd and th entry 0. Let s do them one at a time. First, we perform (E E )! (E ). We obtain Next, we perform (E E ) to obtain
11 1.. SOLVING A SYSTEM OF LINEAR EQUATIONS 1 Finally row 1. We make entries and equal to 0 by performing rst (E 1 E )! (E 1 ), this gives us then (E 1 E )! (E 1 ), this gives us The solution is now read from the last column. Thus, we see that x 1 = 1, x =, x = 0, and x = Homogeneous Linear Systems We nish this section by looking at a special type of system, homogeneous systems. These system as we will see, always have at least one solution. De nition (Homogeneous System) A system of linear equations is said to be homogeneous if the constant terms are always 0. In other words, a homogeneous system is of the form a 11 x 1 + a 1 x + + a 1n x n = 0 >< a 1 x 1 + a x + + a n x n = 0 > a m1 x 1 + a m x + + a mn x n = 0 De nition The solution x i = 0 for i = 1,,, n is called the trivial solution. Other solutions are called nontrivial solutions. Remark A homogeneous system always has at least the trivial solution. So, a homogeneous system is always consistent. Remark Another important fact about homogeneous systems is that the elementary transformations will not alter the last column of their augmented matrix. Thus, when we transform a homogeneous system in rowechelon form or reduced rowechelon form, we still have a homogeneous system. It turns out that a homogeneous system of linear equations either has only the trivial solution, or has in nitely many solutions. We state this result without proof. We will prove it later in the course. If the homogeneous system has more equations than unknowns, then it will have in nitely many solutions.
12 1 CHAPTER 1. SYSTEMS OF LINEAR EQUATIONS AND MATRICES Example Solve < x 1 +x +x = 0 x 1 +x x +x = 0 x 1 x x +x = 0 The corresponding augmented matrix is To set the 1st entries in rows and, we perform (E (E E 1 )! E. We obtain E 1 )! E and Set the leading entry of row to 1 by performing ( E )! E. We obtain Set the second entry in row to 0 by performing (E + E )! E. We obtain Set the leading entry in row to 1 by performing 1 E! E. We obtain The corresponding system is < x 1 +x +x = 0 x +x +x = 0 x + 1 x = 0 We can use back substitution. Letting x be the free variable and setting x = t, we get from equation that x = t. From equation, we get 1 x = x x = 1 t t = t
13 1.. SOLVING A SYSTEM OF LINEAR EQUATIONS 19 Therefore x 1 = x x = t t = t Thus, the solution in para,etric form is x 1 = >< x = x = > x = t 1.. Concept Review t t 1 t Know what a system in rowechelon form and reduced rowechelon form is. Be able to solve a system in rowechelon form, using backsubstitution. Know the transformations which produce equivalent systems. Be able to perform Gaussian elimination to transform a system of linear equations into an equivalent system in rowechelon form. Be able to perform GaussJordan elimination to transform a system of linear equations into an equivalent system in reduced rowechelon form. Be able to solve a system of linear equation using either Gaussian elimination or GaussJordan elimination. 1.. Problems On pages 191, do # 1,,,,,,, 1, 1, 1, 1, 19, 0,,,,.
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 informationSection 8.2 Solving a System of Equations Using Matrices (Guassian Elimination)
Section 8. Solving a System of Equations Using Matrices (Guassian Elimination) x + y + z = x y + 4z = x 4y + z = System of Equations x 4 y = 4 z A System in matrix form x A x = b b 4 4 Augmented Matrix
More informationMATH10212 Linear Algebra. Systems of Linear Equations. Definition. An ndimensional 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 0534405967. Systems of Linear Equations Definition. An ndimensional vector is a row or a column
More informationSolving 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
More informationLecture 1: Systems of Linear Equations
MTH Elementary Matrix Algebra Professor Chao Huang Department of Mathematics and Statistics Wright State University Lecture 1 Systems of Linear Equations ² Systems of two linear equations with two variables
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 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.
More informationSystems 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
More informationDecember 4, 2013 MATH 171 BASIC LINEAR ALGEBRA B. KITCHENS
December 4, 2013 MATH 171 BASIC LINEAR ALGEBRA B KITCHENS The equation 1 Lines in twodimensional space (1) 2x y = 3 describes a line in twodimensional space The coefficients of x and y in the equation
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 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 information5.5. Solving linear systems by the elimination method
55 Solving linear systems by the elimination method Equivalent systems The major technique of solving systems of equations is changing the original problem into another one which is of an easier to solve
More informationRow 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 inclass presentation
More informationPartial Fractions Decomposition
Partial Fractions Decomposition Dr. Philippe B. Laval Kennesaw State University August 6, 008 Abstract This handout describes partial fractions decomposition and how it can be used when integrating rational
More informationSYSTEMS OF EQUATIONS AND MATRICES WITH THE TI89. by Joseph Collison
SYSTEMS OF EQUATIONS AND MATRICES WITH THE TI89 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
More informationLinear Equations ! 25 30 35$ & " 350 150% & " 11,750 12,750 13,750% MATHEMATICS LEARNING SERVICE Centre for Learning and Professional Development
MathsTrack (NOTE Feb 2013: This is the old version of MathsTrack. New books will be created during 2013 and 2014) Topic 4 Module 9 Introduction Systems of to Matrices Linear Equations Income = Tickets!
More informationMethods for Finding Bases
Methods for Finding Bases Bases for the subspaces of a matrix Rowreduction 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,
More informationSystems of Linear Equations
Chapter 1 Systems of Linear Equations 1.1 Intro. to systems of linear equations Homework: [Textbook, Ex. 13, 15, 41, 47, 49, 51, 65, 73; page 11]. Main points in this section: 1. Definition of Linear
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 informationSolving 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
More informationSolutions 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
More informationRepresentation of functions as power series
Representation of functions as power series Dr. Philippe B. Laval Kennesaw State University November 9, 008 Abstract This document is a summary of the theory and techniques used to represent functions
More information2.2/2.3  Solving Systems of Linear Equations
c Kathryn Bollinger, August 28, 2011 1 2.2/2.3  Solving Systems of Linear Equations A Brief Introduction to Matrices Matrices are used to organize data efficiently and will help us to solve systems of
More information2x + y = 3. Since the second equation is precisely the same as the first equation, it is enough to find x and y satisfying the system
1. Systems of linear equations We are interested in the solutions to systems of linear equations. A linear equation is of the form 3x 5y + 2z + w = 3. The key thing is that we don t multiply the variables
More information5 Homogeneous systems
5 Homogeneous systems Definition: A homogeneous (homojeen ius) system of linear algebraic equations is one in which all the numbers on the right hand side are equal to : a x +... + a n x n =.. a m
More informationSolutions of Linear Equations in One Variable
2. Solutions of Linear Equations in One Variable 2. OBJECTIVES. Identify a linear equation 2. Combine like terms to solve an equation We begin this chapter by considering one of the most important tools
More information4.5 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
More information1.5 SOLUTION SETS OF LINEAR SYSTEMS
12 CHAPTER 1 Linear Equations in Linear Algebra 1.5 SOLUTION SETS OF LINEAR SYSTEMS Many of the concepts and computations in linear algebra involve sets of vectors which are visualized geometrically as
More information1 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.
More informationAbstract: 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
More informationPractical 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
More informationContinued Fractions and the Euclidean Algorithm
Continued Fractions and the Euclidean Algorithm Lecture notes prepared for MATH 326, Spring 997 Department of Mathematics and Statistics University at Albany William F Hammond Table of Contents Introduction
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 informationMATH 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 ndimensional column
More informationReduced 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
More informationMATH2210 Notebook 1 Fall Semester 2016/2017. 1 MATH2210 Notebook 1 3. 1.1 Solving Systems of Linear Equations... 3
MATH0 Notebook Fall Semester 06/07 prepared by Professor Jenny Baglivo c Copyright 009 07 by Jenny A. Baglivo. All Rights Reserved. Contents MATH0 Notebook 3. Solving Systems of Linear Equations........................
More informationChapter 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
More information5.1 Radical Notation and Rational Exponents
Section 5.1 Radical Notation and Rational Exponents 1 5.1 Radical Notation and Rational Exponents We now review how exponents can be used to describe not only powers (such as 5 2 and 2 3 ), but also roots
More informationHomogeneous systems of algebraic equations. A homogeneous (homogeen ius) system of linear algebraic equations is one in which
Homogeneous systems of algebraic equations A homogeneous (homogeen ius) system of linear algebraic equations is one in which all the numbers on the right hand side are equal to : a x + + a n x n = 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 informationNotes on Determinant
ENGG2012B Advanced Engineering Mathematics Notes on Determinant Lecturer: Kenneth Shum Lecture 918/02/2013 The determinant of a system of linear equations determines whether the solution is unique, without
More informationAu = = = 3u. Aw = = = 2w. so the action of A on u and w is very easy to picture: it simply amounts to a stretching by 3 and 2, respectively.
Chapter 7 Eigenvalues and Eigenvectors In this last chapter of our exploration of Linear Algebra we will revisit eigenvalues and eigenvectors of matrices, concepts that were already introduced in Geometry
More information7 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
More informationSolution to Homework 2
Solution to Homework 2 Olena Bormashenko September 23, 2011 Section 1.4: 1(a)(b)(i)(k), 4, 5, 14; Section 1.5: 1(a)(b)(c)(d)(e)(n), 2(a)(c), 13, 16, 17, 18, 27 Section 1.4 1. Compute the following, if
More informationSOLVING LINEAR SYSTEMS
SOLVING LINEAR SYSTEMS Linear systems Ax = b occur widely in applied mathematics They occur as direct formulations of real world problems; but more often, they occur as a part of the numerical analysis
More informationLAB 11: MATRICES, SYSTEMS OF EQUATIONS and POLYNOMIAL MODELING
LAB 11: MATRICS, SYSTMS OF QUATIONS and POLYNOMIAL MODLING Objectives: 1. Solve systems of linear equations using augmented matrices. 2. Solve systems of linear equations using matrix equations and inverse
More informationNOTES 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
More informationLecture Notes 2: Matrices as Systems of Linear Equations
2: Matrices as Systems of Linear Equations 33A Linear Algebra, Puck Rombach Last updated: April 13, 2016 Systems of Linear Equations Systems of linear equations can represent many things You have probably
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 informationSolution of Linear Systems
Chapter 3 Solution of Linear Systems In this chapter we study algorithms for possibly the most commonly occurring problem in scientific computing, the solution of linear systems of equations. We start
More informationDERIVATIVES AS MATRICES; CHAIN RULE
DERIVATIVES AS MATRICES; CHAIN RULE 1. Derivatives of Realvalued 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 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 informationName: Section Registered In:
Name: Section Registered In: Math 125 Exam 3 Version 1 April 24, 2006 60 total points possible 1. (5pts) Use Cramer s Rule to solve 3x + 4y = 30 x 2y = 8. Be sure to show enough detail that shows you are
More informationArithmetic and Algebra of Matrices
Arithmetic and Algebra of Matrices Math 572: Algebra for Middle School Teachers The University of Montana 1 The Real Numbers 2 Classroom Connection: Systems of Linear Equations 3 Rational Numbers 4 Irrational
More informationVector 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
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 information10.2 Systems of Linear Equations: Matrices
SECTION 0.2 Systems of Linear Equations: Matrices 7 0.2 Systems of Linear Equations: Matrices OBJECTIVES Write the Augmente Matrix of a System of Linear Equations 2 Write the System from the Augmente Matrix
More informationLecture notes on linear algebra
Lecture notes on linear algebra David Lerner Department of Mathematics University of Kansas These are notes of a course given in Fall, 2007 and 2008 to the Honors sections of our elementary linear algebra
More information10.2 ITERATIVE METHODS FOR SOLVING LINEAR SYSTEMS. The Jacobi Method
578 CHAPTER 1 NUMERICAL METHODS 1. ITERATIVE METHODS FOR SOLVING LINEAR SYSTEMS As a numerical technique, Gaussian elimination is rather unusual because it is direct. That is, a solution is obtained after
More informationThese axioms must hold for all vectors ū, v, and w in V and all scalars c and d.
DEFINITION: A vector space is a nonempty set V of objects, called vectors, on which are defined two operations, called addition and multiplication by scalars (real numbers), subject to the following axioms
More informationLinear 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
More information1 Review of Least Squares Solutions to Overdetermined Systems
cs4: introduction to numerical analysis /9/0 Lecture 7: Rectangular Systems and Numerical Integration Instructor: Professor Amos Ron Scribes: Mark Cowlishaw, Nathanael Fillmore Review of Least Squares
More informationII. Linear Systems of Equations
II. Linear Systems of Equations II. The Definition We are shortly going to develop a systematic procedure which is guaranteed to find every solution to every system of linear equations. The fact that such
More informationMatrices 2. Solving Square Systems of Linear Equations; Inverse Matrices
Matrices 2. Solving Square Systems of Linear Equations; Inverse Matrices Solving square systems of linear equations; inverse matrices. Linear algebra is essentially about solving systems of linear equations,
More informationCAPM, Arbitrage, and Linear Factor Models
CAPM, Arbitrage, and Linear Factor Models CAPM, Arbitrage, Linear Factor Models 1/ 41 Introduction We now assume all investors actually choose meanvariance e cient portfolios. By equating these investors
More informationSequences. A sequence is a list of numbers, or a pattern, which obeys a rule.
Sequences A sequence is a list of numbers, or a pattern, which obeys a rule. Each number in a sequence is called a term. ie the fourth term of the sequence 2, 4, 6, 8, 10, 12... is 8, because it is the
More information1 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
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 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 informationSolving simultaneous equations using the inverse matrix
Solving simultaneous equations using the inverse matrix 8.2 Introduction The power of matrix algebra is seen in the representation of a system of simultaneous linear equations as a matrix equation. Matrix
More information3. Mathematical Induction
3. MATHEMATICAL INDUCTION 83 3. Mathematical Induction 3.1. First Principle of Mathematical Induction. Let P (n) be a predicate with domain of discourse (over) the natural numbers N = {0, 1,,...}. If (1)
More informationElementary Matrices and The LU Factorization
lementary Matrices and The LU Factorization Definition: ny matrix obtained by performing a single elementary row operation (RO) on the identity (unit) matrix is called an elementary matrix. There are three
More information36 CHAPTER 1. LIMITS AND CONTINUITY. Figure 1.17: At which points is f not continuous?
36 CHAPTER 1. LIMITS AND CONTINUITY 1.3 Continuity Before Calculus became clearly de ned, continuity meant that one could draw the graph of a function without having to lift the pen and pencil. While this
More informationI. GROUPS: BASIC DEFINITIONS AND EXAMPLES
I GROUPS: BASIC DEFINITIONS AND EXAMPLES Definition 1: An operation on a set G is a function : G G G Definition 2: A group is a set G which is equipped with an operation and a special element e G, called
More informationLinear 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
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 informationLinearly Independent Sets and Linearly Dependent Sets
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 inclass presentation
More informationSample Problems. Practice Problems
Lecture Notes Quadratic Word Problems page 1 Sample Problems 1. The sum of two numbers is 31, their di erence is 41. Find these numbers.. The product of two numbers is 640. Their di erence is 1. Find these
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 informationLS.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
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 informationHomogeneous equations, Linear independence
Homogeneous equations, Linear independence 1. Homogeneous equations: Ex 1: Consider system: B" #B# œ! B" #B3 œ! B B œ! # $ Matrix equation: Ô " #! Ô B " Ô! "! # B # œ! œ 0Þ Ð3Ñ Õ! " " ØÕB Ø Õ! Ø $ Homogeneous
More informationMAT 200, Midterm Exam Solution. a. (5 points) Compute the determinant of the matrix A =
MAT 200, Midterm Exam Solution. (0 points total) a. (5 points) Compute the determinant of the matrix 2 2 0 A = 0 3 0 3 0 Answer: det A = 3. The most efficient way is to develop the determinant along the
More information( ) which must be a vector
MATH 37 Linear Transformations from Rn to Rm Dr. Neal, WKU Let T : R n R m be a function which maps vectors from R n to R m. Then T is called a linear transformation if the following two properties are
More informationLinear Equations and Inequalities
Linear Equations and Inequalities Section 1.1 Prof. Wodarz Math 109  Fall 2008 Contents 1 Linear Equations 2 1.1 Standard Form of a Linear Equation................ 2 1.2 Solving Linear Equations......................
More information1 Lecture: Integration of rational functions by decomposition
Lecture: Integration of rational functions by decomposition into partial fractions Recognize and integrate basic rational functions, except when the denominator is a power of an irreducible quadratic.
More information8.2. Solution by Inverse Matrix Method. Introduction. Prerequisites. Learning Outcomes
Solution by Inverse Matrix Method 8.2 Introduction The power of matrix algebra is seen in the representation of a system of simultaneous linear equations as a matrix equation. Matrix algebra allows us
More informationLinear Algebra Notes for Marsden and Tromba Vector Calculus
Linear Algebra Notes for Marsden and Tromba Vector Calculus ndimensional Euclidean Space and Matrices Definition of n space As was learned in Math b, a point in Euclidean three space can be thought of
More informationPractice with Proofs
Practice with Proofs October 6, 2014 Recall the following Definition 0.1. A function f is increasing if for every x, y in the domain of f, x < y = f(x) < f(y) 1. Prove that h(x) = x 3 is increasing, using
More informationSecond Order Linear Nonhomogeneous Differential Equations; Method of Undetermined Coefficients. y + p(t) y + q(t) y = g(t), g(t) 0.
Second Order Linear Nonhomogeneous Differential Equations; Method of Undetermined Coefficients We will now turn our attention to nonhomogeneous second order linear equations, equations with the standard
More informationSection 1.7 22 Continued
Section 1.5 23 A homogeneous equation is always consistent. TRUE  The trivial solution is always a solution. The equation Ax = 0 gives an explicit descriptions of its solution set. FALSE  The equation
More informationMATH 423 Linear Algebra II Lecture 38: Generalized eigenvectors. Jordan canonical form (continued).
MATH 423 Linear Algebra II Lecture 38: Generalized eigenvectors Jordan canonical form (continued) Jordan canonical form A Jordan block is a square matrix of the form λ 1 0 0 0 0 λ 1 0 0 0 0 λ 0 0 J = 0
More informationLinear 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
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 informationThe Determinant: a Means to Calculate Volume
The Determinant: a Means to Calculate Volume Bo Peng August 20, 2007 Abstract This paper gives a definition of the determinant and lists many of its wellknown properties Volumes of parallelepipeds are
More informationThe last three chapters introduced three major proof techniques: direct,
CHAPTER 7 Proving NonConditional Statements The last three chapters introduced three major proof techniques: direct, contrapositive and contradiction. These three techniques are used to prove statements
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 information8 Divisibility and prime numbers
8 Divisibility and prime numbers 8.1 Divisibility In this short section we extend the concept of a multiple from the natural numbers to the integers. We also summarize several other terms that express
More information0.8 Rational Expressions and Equations
96 Prerequisites 0.8 Rational Expressions and Equations We now turn our attention to rational expressions  that is, algebraic fractions  and equations which contain them. The reader is encouraged to
More informationLeastSquares Intersection of Lines
LeastSquares Intersection of Lines Johannes Traa  UIUC 2013 This writeup derives the leastsquares solution for the intersection of lines. In the general case, a set of lines will not intersect at a
More information