Linear algebra has applications in science, computer science, social science, business, and other fields. Engineering is filled with visible applications of linear algebra, with examples all around us of infrastructure that affects our everyday lives Web15/08/ · Download Linear Algebra Solution Manual Jeffrey Holt File Type Pdf Download Linear Algebra Solution Manual Jeffrey Holt File Type free pdf, Download Linear Linear WebHolt's Linear Algebra with Applications, Second Edition, blends computational and conceptual topics throughout to prepare students for the rigors of conceptual thinking in Web[REQUEST] Linear Algebra with Applications by Jeffrey Holt (2nd Edition) ISBN: 3 3 3 comments Best Add a Comment [deleted] • 3 yr. ago Found it: Web5/12/ · Mathematics Books Collection of Refere ... read more

Log in with Facebook Log in with Google. Remember me on this computer. Enter the email address you signed up with and we'll email you a reset link. Need an account? Click here to sign up. Download Free PDF. Holt Linear Algebra with Applications 1st c txtbk 1. Azwad Arman. Continue Reading Download Free PDF. Related Papers. STUDENT SOLUTIONS MANUAL Elementary Linear Algebra with Applications NINTH EDITION Prepared by. Download Free PDF View PDF. solutions elementary linear algebra. A first course in linear algebra. Linear Algebra Geometric Approach. STUDENT SOLUTIONS MANUAL Elementary Linear Algebra with Applications NINTH EDITION. See the start of each optional section for dependency information. Preface Acknowledgments 1 Systems of Linear Equations 1. This book is designed for use in a standard linear algebra course for an applied audience, usually populated by sophomores and juniors.

While the majority of students in this type of course are majoring in engineering, some also come from the sciences, economics, and other disciplines. To accommodate a broad audience, applications covering a variety of topics are included. Although this book is targeted toward an applied audience, full development of the theoretical side of linear algebra is included, so this textbook also can be used as an introductory course for mathematics majors. I have designed this book so that instructors can teach from it at a conceptual level that is appropriate for their individual course. There is a collection of core topics that appear in virtually all linear algebra texts, and these are included in this text.

In particular, the core topics recommended by the Linear Algebra Curriculum Study Group are covered here. The organization here reflects this trend, with the chapters approximately alternating between computational and conceptual topics. New to this Edition Practice Problems have been added to the end of each section, with full solutions available in the back of the book. This allows students to check their understanding of the material as they progress through the chapter and also can be used by instructors as extra examples. Supplementary Exercises have been added to the end of each chapter to provide instructors and students with an even greater variety of exercises. The section on Change of Basis has been moved from Chapter 6 to Chapter 4. Reviewers suggested that the Change of Basis material would be useful earlier, so it is now introduced closer to the section on Basis and Dimension. Examples have been revised and added throughout the book.

The subsection on the Shortcut Method has been moved up in section 5. In Chapters 1 and 6, the sections on Approximation Methods have been moved to the end of the chapter for a more streamlined presentation. In Chapters 4 and 9, the notation for coordinate vectors has been updated. Text Features Early Presentation of Key Concepts. Traditional linear algebra texts initially focus on computational topics, then treat more conceptual subjects soon after introducing abstract vector spaces. As a result, at the point abstract vector spaces are introduced, students face two simultaneous challenges: a A change in mode of thinking, from largely mechanical and computational solving systems of equations, performing matrix arithmetic to wrestling with conceptual topics span, linear independence.

b A change in context from the familiar and concrete Rn to abstract vector spaces. Many students cannot effectively meet both these challenges at the same time. The organization of topics in this book is designed to address this significant problem. In Linear Algebra with Applications, we first address challenge a. Conceptual topics are explored early and often, blended in with topics that are more computational. This spreads out the impact of conceptual topics, giving students more time to digest them. The first six chapters are presented solely in the context of Euclidean space, which is relatively familiar to students. This defers challenge b and also allows for a treatment of eigenvalues and eigenvectors that comes earlier than in other texts. Challenge b is taken up in Chapter 7, where abstract vector spaces are introduced.

Here, many of the conceptual topics explored in the context of Euclidean space are revisited in this more general setting. Definitions and theorems presented are similar to those given earlier with explicit references to reinforce connections , so students have less trouble grasping them and can focus more attention on the new concept of an abstract vector space. From a mathematical standpoint, there is a certain amount of redundancy in this book. Quite a lot of the material in Chapters 7, 9, and 10, where the majority of the development of abstract vector spaces resides, has close analogs in earlier chapters.

This is a deliberate part of the book design, to give students a second pass through key ideas to reinforce understanding and promote success. Topics Introduced and Motivated Through Applications. To provide understanding of why a topic is of interest, when it makes sense I use applications to introduce and motivate new topics, definitions, and concepts. In particular, many sections open with an application. Applications are also distributed in other places, including the exercises. In a few instances, entire sections are devoted to applications. Extensive Exercise Sets. When will textbook authors learn that the most important consideration is the exercises? Linear Algebra with Applications, Second Edition, contains over exercises, covering a wide range of types computational to conceptual to proofs and difficulty levels.

The Second Edition now includes Supplementary Exercises at the end of each chapter, providing an even greater variety of exercises from which to choose. Ample Instructional Examples. For many students, a primary use of a mathematics text is to learn by studying examples. Besides those examples used to introduce new topics, this text contains a large number of additional representative examples. Perhaps the number one complaint from students about mathematics texts is that there are not enough examples. I have tried to address that in this text.

The Second Edition also includes Practice Problems at the end of each section, which can be used as extra examples. Full solutions to all of the Practice Problems are included in the back of the book, allowing students to check their understanding of the material as they progress through the chapter. Support for Theory and Proofs. Many students in a first linear algebra course are not usually math majors, and many have limited experience with proofs. Proofs of most theorems are supplied in this book, but it is possible for a course instructor to vary the level of emphasis given to proofs through choice of lecture topics and homework exercises. Throughout the book, the goal of proofs is to help students understand why a statement is true. Thus, proofs are presented in different ways. Sometimes a theorem might be proved for a special case, when it is clear that no additional understanding results from presenting the more general case especially if the general case is more notationally messy.

If a proof is difficult and will not help students understand why the theorem is true, then it might be given at the end of the section or omitted entirely. If it provides a source of motivation for the theorem, the proof might come before the statement of the theorem. I also have written an appendix containing an overview of how to read and write proofs to assist those with limited experience. See the text website at www. Most linear algebra texts handle theorems and proofs in similar ways, although there is some variety in the level of rigor. However, it seems that often there is not enough concern for whether or not the proof is conveying why the theorem is true, with the goal instead being to keep the proof as short as possible.

Sometimes it is worth taking a bit of extra time to give a complete explanation. For example, in Section 1. However, it is also possible that many students will not know why the system has no solutions, so a brief explanation is included. Organization of Material Roughly speaking, the chapters alternate between computational and conceptual topics. This is deliberate, in order to spread out the challenge of the conceptual topics and to give students more time to digest them. The material in Chapters 1—6 and 8 is exclusively in the context of Euclidean space and includes the core topics recommended by the Linear Algebra Curriculum Study Group.

Chapters 7, 9, and 10 cover topics in the context of abstract vector space, and Chapter 11 contains a collection of optional topics that can be included at the end of a course. Systems of Linear Equations 1. Iterative solutions to systems are also treated. The chapter includes a section containing in-depth descriptions of several applications of linear systems. By the end of this chapter, students should be proficient in using augmented matrices and row operations to find the set of solutions to a linear system. Euclidean Space 2. This chapter is devoted to introducing vectors and the important concepts of span and linear independence, all in the concrete context of Rn. These topics appear early so that students have more time to absorb these important concepts. Matrices 3. This is used to motivate the definition of matrix multiplication, which is covered in the next section along with other matrix arithmetic.

This is followed by a section on computing the inverse of a matrix, motivated by finding the inverse of a linear transformation. Matrix factorizations, arguably related to numerical methods, provide an alternate way of organizing computations. The chapter closes with Markov chains, a topic not typically covered until after discussing eigenvalues and eigenvectors. But this subject easily can be covered earlier, and as there are a number of interesting applications of Markov chains, they are included here. Subspaces 4. The first section provides the definition of subspace along with examples. The second section develops the notion of basis and dimension for subspaces in Rn, and the third section thoroughly treats row and column spaces. The last section is optional and covers change of basis in Euclidean space. By the end of this chapter, students will have been exposed to many of the central conceptual topics typically covered in a linear algebra course.

These are revisited and eventually generalized throughout the remainder of the book. Determinants 5. This topic has moved around in texts in recent years. For some time, the trend was to reduce the emphasis on determinants, but lately they have made something of a comeback. This chapter is relatively short and is introduced at this point in the text to support the introduction of eigenvalues and eigenvectors in the next chapter. Those who want only enough of determinants for eigenvalues can cover only Section 5. Eigenvalues and Eigenvectors 6. Diagonalization is presented here and is revisited for symmetric matrices in Chapter 8. The remaining sections are optional and can be covered as needed. Vector Spaces 7. This relatively late introduction allows students time to internalize key concepts such as span, linear independence, and subspaces before being presented with the challenge of abstract vector spaces.

To further smooth this transition, definitions and theorems in this chapter typically include specific references to analogs in earlier chapters to reinforce connections. Since most proofs are similar to those given in Euclidean space, many are left as homework exercises. Making the parallels between Euclidean space and abstract vector spaces very explicit helps students more easily assimilate this material. The order of Chapter 7 and Chapter 8 can be reversed, so if time is limited, Chapter 8 can be covered immediately after Chapter 6. However, if both Chapters 7 and 8 are going to be covered, it is recommended that Chapter 7 be covered first so that this abstract material is not appearing at the end of the course. Orthogonality 8. Chapter 7 is placed before Chapter 8 to allow for an introduction to abstract vector spaces that does not come at the end of the term, and to a degree preserves the chapter alternation between computational and conceptual.

However, the two chapters are interchangeable if necessary. Linear Transformations 9. As in Chapter 7, there is some deliberate redundancy between the material in Chapter 9 and that presented in earlier chapters. Explicit references to earlier analogous definitions and theorems are provided to reinforce connections and improve understanding. Inner Product Spaces The content is somewhat parallel to the first two sections of Chapter 8, with explicit analogs noted. The first section defines the inner product and inner product spaces and gives numerous examples of each. The second section generalizes the notion of projection and the Gram—Schmidt process to inner product spaces, and the last section provides applications of inner products. For the most part, Chapter 10 is independent of Chapter 9 except for a small number of exercise references to linear transformations , so Chapter 10 can be covered without covering Chapter 9.

These can be inserted at the end of a course as desired. Course Coverage Most schools teach linear algebra as a semester-long course that meets 3 hours per week. This usually does not allow enough time to cover everything in this book, so decisions about coverage are required. The dependencies among chapters are fairly straightforward. The first six chapters are designed to be covered in order, although there are some optional sections flagged in the table of contents that can be skipped. The order of Chapter 7 and Chapter 8 can be interchanged. The order of Chapter 9 and Chapter 10 can be interchanged except for a small number of exercises in Chapter 10 that use linear transformations. Chapter 9 assumes Chapter 7, and Chapter 10 assumes Chapter 7 and Chapter 8. Below are a few options for course coverage. Note that some sections or even subsections can be omitted to fine-tune the course to local needs.

Modest Pace: Chapters 1—8. This course covers all key concepts in the context of Euclidean space and provides an introduction to abstract vector spaces. Intermediate Pace: Chapters 1—9 or Chapters 1—8 and This includes everything from the Modest Pace course and either linear transformations on abstract vector spaces Chapter 9 or inner product spaces Chapter Brisk Pace: Chapters 1— This will include everything from the Modest Pace course, as well as both linear transformations on abstract vector spaces and inner product spaces. This is roughly the syllabus we follow here at the University of Virginia, although we omit a few optional sections and we give exams in the evening, which makes available more lecture time. A detailed list of sections that we cover is available on request from the author. Chapter Transitions Each chapter opens with a photo of an example of green engineering, including dams, wind turbines, solar panels, and more. Although linear algebra has applications in many fields, engineering is perhaps one of the most visible.

With our society becoming more and more environmentally conscious, these photos demonstrate ways in which engineering is used in attempts to provide environmentally friendly benefits. Although the mathematics behind these examples is not discussed, we hope the chapter opener photos and captions serve as a small reminder to students of the importance and relevance of linear algebra to their everyday lives. WebAssign lets you easily create assignments, grade homework, and give your students instant feedback. Along with flexible features, class and question-level analytics are available for instructors and students. org W. Freeman offers algorithmically generated questions with full solutions through this free open source online homework system developed at the University of Rochester.

The hassle-free solution created for educators by educators. For more information, visit www. Thank you to Melanie Fulton for accuracy reviewing the Test Bank, Mark Hughes for accuracy reviewing the Solutions, and Paul Lorzsak for accuracy reviewing the Practice Quizzes and iclicker Questions. A large number of experienced mathematics faculty were generous in sharing their thoughts as this Second Edition was developed. Their input contributed in countless ways to the improvement of this book. I gratefully acknowledge the comments and suggestions from the following individuals: Trefor Bazett, University of Toronto Yuri Berest, Cornell University Papiya Bhattacharjee, Penn State Erie, The Behrend College Darin Brezeale, University of Texas at Arlington Terry Bridgman, Colorado School of Mines Robert G.

Brown, Old Dominion University Dietrich Burbulla, University of Toronto Robert Byerly, Texas Tech University Thomas R. Cameron, Washington State University Jen-Mei Chang, California State University, Long Beach Edward Curtis, University of Washington Stefaan De Winter, Michigan Technological University Kosmas Diveris, St. Attaway, Boston University Chris Bernhardt, Fairfield University Eddie Boyd Jr. Bradley, University of Louisville Fernando Burgos, University of South Florida Nancy Childress, Arizona State University Peter Cholak, University of Notre Dame Matthew T. Clay, University of Arkansas Adam Coffman, Indiana University-Purdue University Fort Wayne Ray E.

Collings, Georgia Perimeter College Ben W. Daniel, University of Texas at Austin Gregory Daubenmire, Las Positas College Donald Davis, Lehigh University Tristan Denley, Austin Peay State University Yssa DeWoody, Texas Tech University Caren Diefenderfer, Hollins University Javid Dizgam, University of Illinois Urbana-Champaign Neil Malcolm Donaldson, University of California, Irvine Alina Raluca Dumitru, University of North Florida Della Duncan-Schnell, California State University, Fresno Alexander Dynin, The Ohio State University Daniel J. Endres, University of Central Oklahoma Alex Feingold, SUNY Binghamton John Fink, Kalamazoo College Timothy J. Flaherty, Carnegie Mellon University Bill Fleissner, The University of Kansas Chris Francisco, Oklahoma State University Natalie P. Frank, Vassar College Chris Frenzen, Naval Postgraduate School Anda Gadidov, Kennesaw State University Scott Glasgow, Brigham Young University Jay Gopalakrishnan, Portland State University Anton Gorodetski, University of California-Irvine John Goulet, Worcester Polytechnic Institute William Hager, University of Florida Patricia Hale, California State Polytechnic University, Pomona Chungsim Han, Baldwin Wallace College James Hartsman, Colorado School of Mines Willy Hereman, Colorado School of Mines Konrad J.

Heuvers, Michigan Technological University Allen Hibbard, Central College Rudy Horne, Morehouse College Kevin James, Clemson University Naihuan Jing, North Carolina State University Raymond L. Johnson, Rice University Thomas W. Judson, Stephen F. Austin State University Steven Kahan, CUNY Queens College Jennifer D. Key, Clemson University In-Jae Kim, Minnesota State University Alan Koch, Agnes Scott College Joseph D. Lakey, New Mexico State University Namyong Lee, Minnesota State University Luen-Chau Li, Penn State University Lucy Lifschitz, University of Oklahoma Roger Lipsett, Brandeis University Xinfeng Liu, University of South Carolina Satyagopal Mandal, The University of Kansas Aldo J.

Manfroi, University of Illinois at Urbana-Champaign Douglas Bradley Meade, University of South Carolina Valentin Milanov, Fayetteville State University Mona Mocanasu, Northwestern University Carrie Muir, University of Colorado at Boulder Shashikant Mulay, The University of Tennessee Bruno Nachtergaele, University of California, Davis Ralph Oberste-Vorth, Marshall University Timothy E. Olson, University of Florida Boon Wee Ong, Penn State Erie, The Behrend College Seth F. Oppenheimer, Mississippi State University Bonsu M. Osei, Eastern Connecticut State University Allison Pacelli, Williams College Richard O.

Pellerin, Northern Virginia Community College Jack Porter, The University of Kansas Chuanxi Qian, Mississippi State University David Richter, Western Michigan University John Rossi, Virginia Tech Matthew Saltzman, Clemson University Alicia Sevilla, Moravian College Alexander Shibakov, Tennessee Tech University Rick L. Smith, University of Florida Katherine F. Stevenson, California State University, Northridge Allan Struthers, Michigan Technological University Alexey Sukhinin, University of New Mexico Gnana Bhaskar Tenali, Florida Institute of Technology Magdalena Toda, Texas Tech University Mark Tomforde, University of Houston Douglas Torrance, University of Idaho Scott Wilson, CUNY Queens College Amy Yielding, Eastern Oregon University Jeong-Mi Yoon, University of Houston-Downtown John Zerger, Catawba College Jianqiang Zhao, Eckerd College A large round of thanks is due to all of the people at Macmillan Learning for their assistance and guidance during this project.

These include Nikki Miller, Jorge Amaral, Jodi Isman, and Victoria Garvey. I particularly want to thank Terri Ward, who has been with this project since the beginning and displayed a remarkable level of patience with my consistently inconsistent progress. I gratefully acknowledge the support of the University of Virginia, where I class tested portions of this book. I also thank Simon Fraser University and the IRMACS Centre, for their warm hospitality during my sabbatical visit while I completed the final draft of the first edition. Last but certainly not least, I thank my family for their ongoing and unconditional support as I wrote this book. CHAPTER 1 Systems of Linear Equations Linear algebra has applications in science, computer science, social science, business, and other fields.

Engineering is filled with visible applications of linear algebra, with examples all around us of infrastructure that affects our everyday lives. For example, Glen Canyon Dam, located in northern Arizona, was built to provide flood control, store water to be used during droughts, and produce hydroelectric power for several surrounding states. The construction of Glen Canyon Dam was an impressive feat of engineering, but dams also can be controversial. Even though dams can be used to create environmentally friendly hydroelectric power, they are also criticized for harmful ecological and environmental consequences. Throughout the rest of the chapter openers, we will take a look at other examples of engineering marvels that seek to provide environmentally friendly benefits. T here are endless applications of linear algebra in the sciences, social sciences, and business, and many are included throughout this book.

Chapter 1 begins our tour of linear algebra in territory that may be familiar, systems of linear equations. In Section 1. Section 1. The following example is a good place to start. Although not complicated, it contains the essential elements of other applications and also serves as a gateway to our treatment of more general systems of linear equations. Example 1 A discount movie theater sells tickets for an afternoon matinee. By the same reasoning, the revenue from adult tickets is 5y. The pair x, y that satisfies 1 and 2 will lie on both lines, so must be at the intersection of the lines. We cannot precisely identify x and y from the graph, so we use algebraic methods.

We conclude that tickets were sold for 39 children and 61 adults. Substituting into the equations is not part of finding the solution, but it is a way to check that the solution is correct. Systems of Linear Equations Linear Equation The equations in the preceding problem are examples of linear equations. When the equation has two variables, the graph of the solution set is a line. In three variables, the graph of a solution set is a plane. See Figure 2 for an example. The set of two linear equations in 3 is an example of a system of linear equations. Solution for Linear System, Solution Set for a Linear System The system 7 has m equations with n unknowns. It is possible for m to be greater than, equal to, or less than n, and we will encounter all three cases. The collection of all solutions to a linear system is called the solution set for the system.

In Example 1, there was exactly one solution to the linear system. This is not always the case. So, it must be that our original assumption that there are values of x1 and x2 that satisfy 8 is false, and therefore we can conclude that the system has no solutions. See the Preface for the Web address. The graphs of the two equations in Example 2 are parallel lines see Figure 3. Since the lines do not have any points in common, there cannot be values that satisfy both equations, matching what we found algebraically. Consistent Linear System, Inconsistent Linear System If a linear system has at least one solution, then we say that it is consistent.

If not as in Example 2 , then it is inconsistent. This tells us that the relationship between x1 and x2 is the same in both equations. Therefore there are infinitely many solutions. This is known as the general solution because it gives all solutions to the system of equations. We note that 10 is not the only way to describe the solutions. Figure 4 shows the graphs of the two equations in 9. It looks like something is missing, but there is only one line because the two equations have the same graph. Since the graphs coincide, they have infinitely many points in common, which agrees with our algebraic conclusion that there are infinitely many solutions to 9.

Table 1 Percentage of Glycol Required to Prevent Freezing Minimum Temp. The system works by circulating a mixture of water and propylene glycol through rooftop solar panels to absorb heat, and then through a heat exchanger to heat household water Figure 5. The glycol is included in the mixture to prevent freezing during cold weather. Table 1 shows the percentage of glycol required for various minimum temperatures. How much of each type of solution is required? Figure 5 Schematic of a solar hot water system. Source: Information from U. Thus the total amount of glycol in the system must be 0. We will get 0. This leads to a second equation 0. Therefore a combination of Finding Solutions: Triangular Systems Here we begin developing a method for finding the solutions for a linear system.

For the remainder of this section we concentrate on special types of systems, and will take on general systems in the next section. Consider the two systems below. Although not obvious, these systems have exactly the same solution set. Looking at the system, we see that the easiest place to start is at the bottom. Now we know the values of x3 and x4. In a system of linear equations, a variable that appears as the first term in at least one equation is called a leading variable. Thus in Example 5 each of x1, x2, x3, and x4 is a leading variable. A key reason why the system in Example 5 is easy to solve is that every variable is a leading variable in exactly one equation. This feature is useful because as we back substitute from the bottom equation upward, at each step we are working with an equation that has only one unknown variable. Triangular Form, Triangular System The system in Example 5 is said to be in triangular form, with the name suggested by the triangular shape of the left side of the system.

It is straightforward to verify that triangular systems have the following properties. b There are the same number of equations as variables. c There is exactly one solution. The velocity and acceleration are negative because the ball is moving downward. What is the height of the bridge and when does the ball hit the water? Our model ignores forces other than gravity. Solution We need to find the values of a, b, and c in order to answer these questions. Figure 7 Golden Gate Bridge. John Holt Finding Solutions: Echelon Systems In the next example, we consider a linear system where each variable is a leading variable for at most one equation.

Although this system is not quite triangular, it is close enough that the solutions can be found using back substitution. The middle equation has x2 as the leading variable, but we do not yet have a value for x3. Note that each distinct choice for s1 gives a new solution, so the system has infinitely many solutions. All triangular systems are in echelon form. For example, the system 15 is in echelon form but the system 13 is not, because x4 is the leading variable of two equations. b There are no solutions, exactly one solution, or infinitely many solutions. Free Variable For a system in echelon form, any variable that is not a leading variable is called a free variable.

For instance, x3 is a free variable in Example 7. For a system in echelon form, the total number of variables is equal to the number of leading variables plus the number of free variables. To find the general solution to a system in echelon form, we use the following two-step procedure. Set each free variable equal to a free parameter. Back substitute to solve for the leading variables. Since the last equation has no solutions, the entire system has no solutions. Example 11 looks similar to Example 10, but the set of solutions is very different. Otherwise, there are two possibilities: Figure 8 Graphs of equations in Examples 1—3.

If the system has no free variables, then there is exactly one solution. If the system has at least one free variable, then the general solution has free parameters and there are infinitely many solutions. Number of Solutions: Geometry In Examples 1—3 we saw that a linear system can have a single solution, no solutions, or infinitely many solutions. Figure 8 shows that our examples illustrate all possibilities for two lines: They can intersect in exactly one point, they can be parallel and have no points in common, or they can coincide and have infinitely many points of intersection. Thus it follows that a system of two linear equations with two variables can have zero, one, or infinitely many solutions. Now consider systems of linear equations with three variables. Recall that the graph of the solutions for each equation is a plane. To explore the solutions such a system can have, you can experiment by using a few pieces of cardboard to represent planes.

A theorem is a mathematical statement that has been rigorously proved to be true. As we progress through this book, theorems will serve to organize our expanding body of linear algebra knowledge. Starting with two pieces, you will quickly discover that the only two possibilities for the number of points of intersection is either none or infinitely many. See Figure 9. This geometric observation is equivalent to the algebraic statement that a system of two linear equations in three variables has either no solutions or infinitely many solutions. Now try three pieces of cardboard. More configurations are possible, with some shown in Figure This time, we see that the number of points of intersection can be zero, one, or infinitely many. Note that this also held for a pair of lines.

In fact, this turns out to be true in general, not only for planes but also for solution sets in higher dimensions. The equivalent statement for systems of linear equations is contained in Theorem 1. THEOREM 1. We will prove this theorem at the end of the next section. Figure 9 Graphs of systems of two equations with three variables. Figure 10 Graphs of systems of three equations with three variables. Practice problems can also be used as additional examples. PRACTICE PROBLEMS Do the practice problems and then check your answers against the full solutions in the back of the book. Find the solutions to each linear system. a A triangular system can have infinitely many solutions. b An echelon system with five equations all with variables and eight variables must have three free variables. c A linear system with two variables and three equations cannot have one solution. d Every variable in an echelon system is either a free variable or a free parameter.

Suppose that an echelon system has four equations all with variables and nine variables. How many leading variables are there? How many free variables are there? How many free parameters are in the solution? How many solutions are there? A play is sold out in a theater with a capacity of How many of each type of ticket was sold? You may want to use a calculator for the computations in this problem. Consider a pile of nickels, dimes, and quarters. If there are a total of 31 coins, how many of each type of coin are there? EXERCISES In each exercise set, problems marked with are designed to be solved using a programmable calculator or computer algebra system. HINT: All parameters of a solution must cancel completely when substituted into each equation. If so, identify the leading variables and the free variables. If not, explain why not. Write the system in echelon form, and then find the set of solutions. Find value s of k so that the linear system is consistent.

Find values of h and k so that the linear system has no solutions. For each, assume that all equations have variables. A linear system is in echelon form. If there are four free variables and five leading variables, how many variables are there? Suppose that a linear system with five equations and eight unknowns is in echelon form. Suppose that a linear system with seven equations and thirteen unknowns is in echelon form. There are a total of nine variables, of which four are free variables. How many equations does the system have? FIND AN EXAMPLE Exercises 43— Find an example that meets the given specifications. A linear system with three equations and three variables that has exactly one solution. A linear system with three equations and three variables that has infinitely many solutions. A linear system with four equations and three variables that has infinitely many solutions.

A linear system with three equations and four variables that has no solutions. Come up with an application that has a solution found by solving an echelon linear system. Then solve the system to find the solution. TRUE OR FALSE Exercises 51— Determine if the statement is true or false, and justify your answer. a A linear system with three equations and two variables must be inconsistent. b A linear system with three equations and five variables must be consistent. a There is only one way to express the general solution for a linear system. b A triangular system always has exactly one solution. a All triangular systems are in echelon form. b All systems in echelon form are also triangular systems.

a A system in echelon form can be inconsistent. b A system in echelon form can have more equations than variables. a If a triangular system has integer coefficients including the constant terms , then the solution consists of rational numbers. b A system in echelon form can have more variables than equations. a If a general solution has free parameters, then there must be infinitely many solutions. Assume both equations have variables. A total of people attend the premiere of a new movie. A plane holds passengers. How many of each type of ticket were sold? Referring to Example 4, suppose that the minimum outside temperature is 10°F. In this case, how much of each type of solution is required? How much should be invested in each type of bond? A gallon bathtub is to be filled with water that is exactly ° F. The hot water supply is ° F and the cold water supply is 60° F. When mixed, the temperature will be a weighted average based on the amount of each water source in the mix.

How much of each should be used to fill the tub as specified? The hot water supply is ° F and the cold water supply is 70° F. Pure water freezes at 32° F and 0° C, and boils at ° F and ° C. Use this information to find a and b. For tax and accounting purposes, corporations depreciate the value of equipment each year. This problem requires 8 nickels, 8 quarters, and a sheet of 8. The goal is to estimate the diameter of each type of coin as follows: Using trial and error, find a combination of nickels and quarters that, when placed side by side, extend the height long side of the paper.

Then do the same along the width short side of the paper. Use the information obtained to write two linear equations involving the unknown diameters of each type of coin, then solve the resulting system to find the diameter for each type of coin. Bixby Creek Bridge. Suppose that a bag of concrete is projected downward from the bridge deck at an initial rate of 5 meters per second. After 3 seconds, the bag is Use the model in Example 6 to find a formula for H t , the height at time t. Exercises 71— Use computational assistance to find the set of solutions to the linear system.

In this section, we develop a method for converting any linear system into a system in echelon form, so that we can apply back substitution. To get us started, consider the following projectile motion problem. Suppose that a cannon sits on a hill and fires a ball across a flat field below. Figure 1 shows the elevation of the ball at three separate places. This system is not in echelon form, so back substitution is not easy to use here. We will return to this system shortly. The primary goal of this section is to develop a systematic procedure for transforming any linear system into a system that is in echelon form. The key feature of our transformation procedure is that it produces a new linear system that is in echelon form hence solvable using back substitution and has exactly the same set of solutions as the original system.

Two linear systems are said to be equivalent if they have the same set of solutions. Elementary Operations Elementary Operations We can transform a linear system using a sequence of elementary operations. Each operation produces a new system that is equivalent to the old one, so the solution set is unchanged. There are three types of elementary operations. Interchange the position of two equations. This amounts to nothing more than rewriting the system of equations. For example, we exchange the places of the first and second equations in the following system. Multiply an equation by a nonzero constant. Add a multiple of one equation to another. For this operation, we multiply one of the equations by a constant and then add it to another equation, replacing the latter with the result. It is similar to the method used in the first three examples of Section 1. Note that this is exactly what happened here, with the lower left coefficient becoming zero, transforming the system closer to echelon form.

This illustrates a single step of our basic strategy for transforming any linear system into a system that is in echelon form. Our goal is to transform the system to echelon form, so we want to eliminate the x1 terms in the second and third equations. This will leave x1, as the leading variable in only the top equation. Going forward we identify coefficients using the notation for a generic system of equations introduced in Section 1. We need to transform a21 and a31 to 0. We do this in two parts. Next, we focus on the x2 coefficients. Since our goal is to reach echelon form, we do not care about the coefficient on x2 in the top equation, so we concentrate on the second and third equations. Here we need to transform a32 to 0. To check our solution, we plug these values into the original system. Thanks to the previous step, we need only add the second equation to the third to transform a32 to 0. Figure 2 shows a graph of the model together with the known points.

Figure 2 Cannonball data and the graph of the model. The Augmented Matrix Matrix Augmented Matrix When manipulating systems of equations, the coefficients change but the variables do not. We can simplify notation by transferring the coefficients to a matrix, which for the moment we can think of as a rectangular table of numbers. When a matrix contains all the coefficients of a linear system, including the constant terms on the right side of each equation, it is called an augmented matrix. Elementary Row Operations The three elementary operations that we performed on equations can be translated into equivalent elementary row operations for matrices. Interchange two rows. Multiply a row by a nonzero constant.

Replace a row with the sum of that row and the scalar multiple of another row. Equivalent Matrices Borrowing from the terminology for systems of equations, we say that two matrices are equivalent if one can be obtained from the other through a sequence of elementary row operations. Hence equivalent augmented matrices correspond to equivalent linear systems. Zero Row, Zero Column When discussing matrices, the rows are numbered from top to bottom, and the columns are numbered from left to right. A zero row is a row consisting entirely of zeros, and a nonzero row contains at least one nonzero entry. The terms zero column and nonzero column are similarly defined. In the examples that follow, we transfer the system of equations to an augmented matrix, but our goal is the same as before, to find an equivalent system in echelon form. We focus on the first column of the matrix, which contains the coefficients of x1.

Although this step is not required, exchanging Row 1 and Row 2 will move a 1 into the upper left position and avoid the early introduction of fractions. To transform the system to echelon form, we need to introduce zeros in the first column below Row 1. This requires two operations. With the first column complete, we move down to the second row and to the right to the second column. We now extract the transformed system of equations from the matrix. We can substitute into the original system to verify our solution. The resulting matrix is said to be in echelon form or row echelon form and will have the properties given in Definition 1.

In the definition, the leading term of a row is the leftmost nonzero term in that row, and a row of all zeros has no leading term. b Any zero rows are at the bottom of the matrix. Echelon form is the counterpart to echelon systems from Section 1. Note that the first condition in the definition implies that a matrix in echelon form will have zeros filling out the column below each of the leading terms. The entries in the pivot positions for the matrices in 2 are shown in boldface. The pivot columns are the columns that contain pivot positions, and a pivot is a nonzero number in a pivot position. In what follows, it will be handy to have a general augmented matrix when referring to entries in specific positions. We adopt a notation similar to that for a general system of equations given in 7 of Section 1. It is named in honor of German mathematician Carl Friedrich Gauss, who independently discovered the method and introduced it to the West in the nineteenth century.

Next, we need zeroes down the first column below the pivot position. This makes a good pivot, so we move to the elimination step. Down the remainder of the second column, we need only introduce a zero at a32 by using the operation shown in the margin. However, there are nonzero terms down the first column, so we interchange Row 1 and Row 3 to place a 1 in the pivot position. Next, we need zeros down the first column below the pivot position. This will undo the zeros in the first column. Since all the entries below a22 are also zero, interchanging with lower rows will not put a nonzero term in the a22 position. Thus a22 cannot be a pivot position, so we move to the right to the third column to determine if a23 is a suitable pivot position. From the Row 2 pivot position, we move down one row and to the right one column to a This entry is 0, as is the entry below, so interchanging rows will not yield an acceptable pivot.

As we did before, we move one column to the right. We introduce a zero in the a45 position by using the operation shown in the margin. Since Row 4 is the only remaining row and consists entirely of zeros, it has no pivot position. The matrix is now in echelon form, so no additional row operations are required. Gaussian elimination can be applied to any matrix to find an equivalent matrix that is in echelon form. If matrix A is equivalent to matrix B that is in echelon form, we say that B is an echelon form of A.

Different sequences of row operations can produce different echelon forms of the same starting matrix, but all echelon forms of a given matrix will have the same pivot positions. We introduce zeros down the first column with the row operations shown in the margin. Thus this system has no solutions, and so is inconsistent. The preceding example illustrates a general principle. When applying row operations to an augmented matrix, if at any point in the process the matrix has a row of the form […0 c] 3 where c is nonzero, then stop. The system is inconsistent.

Gauss, and Wilhelm Jordan — , a German engineer who popularized this method for finding solutions to linear systems in his book on geodesy the science of measuring earth shapes. After extracting the linear system from this matrix, we back substituted and simplified to find the general solution. We can make it easier to find the general solution by performing additional row operations on the matrix. Specifically, we do the following: 1. Multiply each nonzero row by the reciprocal of the pivot so that we end up with a 1 as the leading term in each nonzero row.

Use row operations to introduce zeros in the entries above each pivot position. Picking up with our matrix, we see that the first and third rows already have a 1 in the pivot position. When implementing Gaussian elimination, we worked from left to right. To put zeros above pivot positions, we work from right to left, starting with the rightmost pivot, which in this case appears in the fifth column. Two row operations are required to introduce zeros above this pivot. One row operation is required to introduce a zero in the a13 position. Changing the order can result in a circular sequence of operations that lead to endless misery.

Note that when the system is expressed in this form, the leading variables appear only in the equation that they lead. Thus during back substitution we need only plug in free parameters and then subtract to solve for the leading variables, simplifying the process considerably. The matrix on the right in 4 is said to be in reduced echelon form. b All pivot positions contain a 1. c The only nonzero term in a pivot column is in the pivot position. The combination of the forward and backward phases is referred to as Gauss—Jordan elimination. Although a given matrix can be equivalent to many different echelon form matrices, the same is not true of reduced echelon form matrices. The proof of Theorem 1. Next, we implement the backward phase to transform the matrix to reduced echelon form. If there are additional solutions, they are called nontrivial solutions.

We determine if there are nontrivial solutions in the usual way, using elimination methods. To find the other solutions, we load the system into an augmented matrix and transform to reduced echelon form. Proof of Theorem 1. Recall the statement of the theorem. Proof We can take any linear system, form the augmented matrix, use Gaussian elimination to reduce to echelon form, and extract the transformed system. In this case, the system has no solutions. If a does not occur, then one of b or c must: b The transformed system has no free variables and hence exactly one solution. c The transformed system has one or more free variables and hence infinitely many solutions.

Homogeneous linear systems are even simpler. Since all such systems have the trivial solution, a cannot happen. Therefore a homogeneous linear system has either a unique solution or infinitely many solutions. Which is better? For a system of n equations with n unknowns, Gaussian elimination requires approximately 23n3 flops i. Back substitution is slightly more complicated for Gaussian elimination than for Gauss—Jordan, but overall Gaussian elimination is more efficient and is the method that is usually implemented in computer software. When performing row operations by hand, partial pivoting tends to introduce fractions and leads to messy calculations, so we avoided the topic. However, it is discussed Section 1. Counting flops gives a measure of algorithm efficiency. Transform the following matrices to echelon form. Transform the following matrices to reduced row echelon form. Convert the system to an augmented matrix, transform to echelon form, then back substitute to find the solutions to the system.

Convert the system to an augmented matrix, transform to reduced row echelon form, then back substitute to find the solutions to the system. Determine if each statement is true or false, and justify your answer. a A matrix that has more columns than rows cannot be transformed to reduced row echelon form. b A matrix that has more rows than columns and is in echelon form must have a row of zeros. c Every elementary row operation is reversible. d A linear system with free variables cannot have one solution. Suppose that a matrix with four rows and nine columns is in echelon form. a If the matrix has no row of all zeros, then how many pivot positions are there? b What is the minimum number of zeros in the bottom row? c What is the minimum number of zeros in the matrix?

d If this is the augmented matrix for a linear system, then what is the minimum number of free variables? Exercises 1—4: Convert the augmented matrix to the equivalent linear system. Identify the row operation.

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For some time, the trend was to reduce the emphasis on determinants, but lately they have made something of a comeback. For every dollar of goods A sells, A requires 20 cents of goods from B. The system given in Exercise 9. Digital back Digital Offerings Achieve LaunchPad E-books iOLab iClicker Inclusive Access LMS Integration Curriculum Solutions Lab Solutions Training and Demos First Day of Class. Proof We can take any linear system, form the augmented matrix, use Gaussian elimination to reduce to echelon form, and extract the transformed system.

To find the general solution to a system in echelon form, we use the following two-step procedure. Thus we need 4 days of production at facility A, 2 days of production at facility B,