Solution Manual For Linear Algebra Plus Mymathlab Getting Started Kit For Linear Algebra And Its Applications, 4th Edition

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MATLAB®MANUALJEREMYR.CASETaylor UniversityJANEDAYSan Jose State UniversityLINEARALGEBRAANDITSAPPLICATIONSFOURTHEDITIONDavid C. LayUniversity of Maryland

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iiiContents1Getting Started...................................................1Prepare .............................................................................................1Obtain an Educational License ........................................................1Order Student MATLAB and Study Guides Early ..........................1Install Laydata4 Toolbox Before Classes Start................................1Obtain Data for Case Studies and Application Projects ..................2Prepare Student Computer Lab Instructions ...................................22Planning the Course ..........................................2Allow Time for Planning and Adjusting Plans ................................2Consider Purposes for Computer Assignments ...............................3Decide Emphasis on Computer Work..............................................4Design Computer Assignments .......................................................4Anticipate Commons Difficulties ....................................................5Consider Classroom Demonstrations...............................................5Decide How You Will Test Students...............................................6Be Creative.......................................................................................73Using Software for Demonstrations .................84Downloading M-Files from the Web................95Computer Projects...........................................10General Information.......................................................................10Partners ..........................................................................................10Notes about the Individual Projects ...............................................106Overview of the Case Studies/Applications ... 14Case Studies ...................................................................................14Application Projects.......................................................................147References.........................................................16Sample Computer Projects

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Getting Started11. Getting StartedPREPAREThis manual assumes that you will be using MATLAB, but there are other excellent software packages orcalculators you could use. Maple and Mathematica are examples of other mathematical software packagesconducive to linear algebra. Calculators with matrix capabilities such as those made by Hewlett Packard and TexasInstruments are other possibilities. Each of the technologies listed here has a manual to accompany Lay’s book andis available from Addison-Wesley. The exercises in Lay’s text are written so that any appropriate software orcalculator may be used.This manual assumes that in addition to MATLAB you will use Laydata4 Toolbox, which is a collection of M-filesthat can be downloaded from MyMathLab or accessed throughpearsonhighered.com/lay. These M-files will needto be made accessible to your students. It is also a good idea to use theStudy Guide,a supplement to the text, toaccompany this manual. The text, theStudy Guide,and Laydata4 Toolbox are highly coordinated to work together.For the students, theStudy Guideis the primary support for the use of technology during the semester. In addition tothe MATLAB boxes at the ends of many sections, the Appendix, “Getting Started with MATLAB,” provides theinitial information students may need to use MATLAB effectively. Furthermore, there are appendices for othertechnologies such as Maple, Mathematica, and several graphing calculators. For the instructor, your job will bemuch easier if you and your students have theStudy Guidewith its wealth of information.If you have not used MATLAB extensively before, spend some time learning the basic operations before you beginclass. MATLAB and linear algebra work very well together, and you can do some interesting things fairly quickly.You might work through the first project, “Getting Started with MATLAB,” and then try some other projects.Exercises designated by an [M] in Lay’s text are designed to be worked with the aid of technology, and you mighttry some of these problems. Read the MATLAB boxes in theStudy Guideto see how MATLAB could be used onhomework problems. One of theStudy Guide’s features is that it gradually introduces MATLAB commands as theyare needed. Alternatively, you could become introduced to MATLAB by working through a tutorial in a book or onthe Web.As you become more familiar with MATLAB’s capabilities, attempt more involved problems such as the casestudies and application projects available from the course website. These are usually found at the beginning of mostchapters, and an icon in the text references the course website for these resources.OBTAIN AN EDUCATIONAL LICENSEUsually the most cost effective way to provide MATLAB to students is for your school to buy an educational sitelicense. You can use any version of MATLAB for Lay’s [M] exercises and for most of the projects.ORDER STUDENT MATLAB AND STUDY GUIDES EARLYAsk your bookstore to stock Lay’sStudy Guidewith the textbook. Our institution has an educational site license forMATLAB, but if yours does not, request that theStudent Edition of MATLAB[10] be made available as well. Thelatest edition of Student MATLAB is usually what they will get. Some students may have access to earlier editions,and those will work fine. Student MATLAB costs about $100 and includes a goodUser's Guide. It is identical toprofessional MATLAB except in a few ways that rarely affect students' use.INSTALL LAYDATA4 TOOLBOX BEFORE CLASSES STARTThe M-files in Laydata4 Toolbox are not part of commercial MATLAB, so you must install them on the computersyour students will use. If students plan to use MATLAB on their personal computers, they must also install the M-files. See Section 4 below and Section 15 of the preliminary Computer Project “Getting Started With MATLAB.” Ifthere is an earlier Toolbox from Lay’s text such as Laydata Toolbox, you will need to delete this Toolbox so thatstudents download the correct data.

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2Planning the CourseYou should check the status of your software and M-files before each term begins. One year, our institution changedoperating systems between semesters. What had worked one month previously no longer worked, and I spent thefirst week scrambling to correct it. It got the class off on the wrong foot, and it took longer for them to feelcomfortable with the technology. It taught me to never take for granted what will work as computers are upgraded.The files in Laydata4 Toolbox provide the data for about 850 exercises in Lay's text, as well as the data for theindividual projects printed at the back of this manual. Having the data for all numerical exercises readily availablesaves the tedium of typing it in and ensures that students work with the intended numbers. Laydata4 Toolbox alsocontains some special MATLAB functions that enhance the teaching of linear algebra from Lay’s text. Thesefunctions are described in Section 3 below, as well as in theStudy Guideand in the projects as they are needed.OBTAIN DATA FOR CASE STUDIES AND APPLICATION PROJECTSIn addition to the hard copy projects at the end of this manual, there are Case Studies and Application Projectsavailable from the Web. The Case Studies expand topics introduced at the beginning of each chapter in Lay’stextbook and use real-world data. The Application Projects either extend existing topics in the text or introduce newapplications. The Data files for the Case Studies and Application Projects are contained in text files on the Web atpearsonhighered.com/lay. If you decide to assign one or more of these projects that has accompanying data, theneither you or your students must download the appropriate files from the Web and add them to the MATLAB path.See Section 4 below.PREPARE STUDENT COMPUTER LAB INSTRUCTIONSOn the first day of classes, students need information about how you plan to use MATLAB in the course and howthey can access the program and appropriate data. List theStudy Guideas the “lab manual” for the course. With theStudy Guidein hand,students rarely will need more documentation for the course other than MATLAB’shelpcommand and your local computer procedures. You can prepare a sheet to hand out, or put the information on a webpage, or do both. Here are some facts that students may need:•Location of campus computer lab facilities.•Hours and days when the labs are available for student use.•How to obtain and use computer log-on names and passwords.•How to start MATLAB in the lab, print output, and save the work.•Where to get help.•How to get a personal copy of MATLAB and data for the course.2. Planning the CourseALLOW TIME FOR PLANNING AND ADJUSTING PLANSIt would be very good to have some release time the first time you try using a significant number of computerexercises. However, some institutions like mine cannot always provide such release time. Pressed for time, I foundthe projects in this manual to be very helpful the first time I taught linear algebra.Starting out or making major changes in a course takes a lot of effort, and computers introduce another dimension forwhat can go wrong. I started slowly by using computer exercises as an “add on” to the traditional course. I think thisis not an unwise way to begin. As you understand the technology and the students better, you can change the styleand topics of your course. Students will have various interests and questions, and you should allow yourself someflexibility in modifying your course.

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Planning the Course3CONSIDER PURPOSES FOR COMPUTER ASSIGNMENTSAs you consider your students' interests and begin to appreciate the potential of computer exercises, decide whatpurposes are most appropriate for your class. Here are some possible ones:1.To teach applications2.To reinforce understanding of concepts and theory3.To think and problem solve4.To explore and conjecture5.To develop some computational wisdom6.To learn something new7.To reduce tedious hand calculation8.To practice routine calculations9.To write programs to solve problems, learn algorithms, etc.10. To introduce MATLAB for later coursesThe first four purposes listed are the most important to me, but all of these reasons have merit and are addressed invarious projects.Applications provide motivation as to why one should learn the material. For many of my students, they seem morelikely to forget material that they do not find interesting or that they cannot conceive of applying it in the future.You can expose your students to a variety of applications using the case studies in the book. Such examples arenatural topics for computer exercises because applications are more interesting when the data are not trivial.The second goal is very important. People tend to misuse theory when they do not understand it, so you might stressthe mastery of the big ideas such as linear independence, span, basis, dimension, eigenvalues and orthogonality. Ibelieve students should also understand why the major theorems are true. However, many of my students do notappreciate the abstract concepts as much as I do, and it helps if they connect the concepts to ideas they alreadyknow. Practicing the ideas in concrete calculations and applications also solidifies the ideas. Furthermore, if they canbe convinced that the theorems and ideas are reasonable and useful, they are more motivated. Computer exercisesare an attractive device towards this end. It gets them to grapple with some theory, and some of the projectsexplicitly address abstract ideas. All projects ask students to explain what they have seen, and most points should beplaced on those questions.I consider the third and fourth purposes part of the broader scheme of mathematics. I will then add some questions tothe projects asking students to extend the results. These are sometimes difficult to grade particularly if you have alarge class. Hence, while I did not modify the projects to include these questions in this manual, I encourage you toadapt the projects toward your goals.One of the original authors of this manual, Jane Day, stressed the development of computational wisdom. Numericalissues and peculiarities come up in several of the projects, including "Reduced Echelon Form andref" and"Roundoff Error in Matrix Computations." While not undermining the faith in good software, she wanted herstudents to learn to be wise and cautious. To her students she emphasized the following, in this order:1.Professionally written matrix software gives good answers to most problems, but there is almostalways some error.2.The matrix algorithms that work well on computers are more sophisticated than those presented inbasic linear algebra texts. So people should employ professionally written software in their jobs.3.Some problems are inherently difficult to solve accurately even with the best algorithms. So usersshould never ignore warnings from professional software.For students who want to know more, there are several well-written contemporary introductions to numerical linearalgebra. George Forsythe's paper [4] is a classic. It is remarkable how clearly he articulated the inevitable pitfalls offloating point matrix computations so early.

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4Planning the CourseDECIDE EMPHASIS ON COMPUTER WORKComputer projects are good vehicles for introducing simple applications, which are important for the majority of ourstudents. One important reason for having linear algebra students work with professional software like MATLAB isthat they need to know such software exists. In the workplace, they will be using professional software that will befar faster and more accurate than code based on algorithms alone. At the same time, I feel it is important for them tolearn the basic algorithms of linear algebra because those enhance understanding of concepts.Some instructors organize the computer work around weekly lists of five or more [M] exercises from the text. Otherinstructors include one or more [M] exercises in nearly every night’s homework. Here the point is to encourage dailyuse of MATLAB for most of the numerical exercises, not just those marked with [M]. Remember that Laydata4Toolbox provides data for most exercises in the text so the time entering data on the computer is minimal.One obvious factor in determining how much to emphasize computer work is the availability of a computer lab.Currently, my institution does not have the space available for a weekly lab so most of the computer work is doneoutside of class. I believe I would change my approach if the students could use the computer during class timerather than just me using the computer at the front of the room.How much you will emphasize computer assignments, applications, and theory depends on your personal situation.Your teaching style, your course objectives, and the specific needs at your school are all factors that must be givenconsideration. I suggest you look at book [2] which greatly helped me to become aware of issues I had notpreviously considered and to navigate these issues for myself.DESIGN COMPUTER ASSIGNMENTSThe first time you use computer assignments, you should probably proceed with some caution. Evaluate theeffectiveness and difficulty of each assignment before making another. Various details can require more attentionthan you might expect, especially at the beginning of the semester. For instance, you may find that access to thecomputer labs is inadequate, or a student who buys software has trouble installing it. Equipment has a way ofbreaking down when you need it most.It would be wise to assume "If it can go wrong, it will" and then bepleasantly surprised if things go smoothly. On the other hand, technology is so ubiquitous that I am finding fewerand fewer problems each year. (It could also be that I know now by experience which IT person to contact.)It is important to give an easy computer assignment early and collect it to get students started and to help youevaluate how they react to computer use. I suggest you ask them to work through "Getting Started" during the firstfew days. It will be good for them to know what topics are discussed there even if they don't understand all thematrix operations at first. You could then demonstrate the functionsreplace,swapandscale, and assign "PracticeRow Reduction." After they succeed with that project, assign a few [M] exercises from the first sections of the text.Work each computer problem yourself before assigning it. You will then know how this experience will fit withyour classroom lessons, what students should watch out for, how much time to allow, and how much otherhomework is reasonable to assign. Most of the projects are straightforward – students have to read some, and spendsome time doing the calculations, but hard thinking is required only occasionally. Emphasize that the questions thatask forinterpretation of what they calculateare the ones that really matter, and that's where most of the points are.In recent semesters, I have used 10 core projects, and then assigned a few more specific to the students’ needs andinterests. For example, the education majors were assigned the Cryptography project while those interested inbusiness were given the project “An Economy with an Open Sector.” Others require about 14 projects and letstudents select 2 or 3 more for extra credit, which seems a reasonable way to address the variety of student interests.It is a good idea to discuss each project briefly before assigning it and again when handing the papers back to besure everyone got the point. A very real danger in computer projects is that students push the buttons and miss theobvious points in understanding. The projects here are written like lab forms and from my experience are pretty easyto grade. Student graders, if available, can be used to grade the projects to save you time.

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Planning the Course5I encourage students to work in groups. I am almost convinced that students do better with technology when theyhave to figure it out for themselves rather than when it is explained to them. Furthermore, there are the MATLABboxes and the MATLAB Appendix in theStudy Guidefor them to use as a resource. Since it is unlikely thatsomeone will always be there to provide an explanation as newer technologies and upgrades are introduced, figuringout technology on your own seems to be an important skill to develop. Working in groups eases this struggle, but Istill am available for consultation—particularly for the shy student who has some aversion to computers.Sometimes students use MATLAB to check their answers to the textbook problems, but most of the time mystudents just use their calculators. Since one of the emphases in my class is theory, I collect homework that requiresexplanations or proof rather than routine problems. However, the routine problems and computer projects help buildthat understanding so that the theory is not developed in a “vacuum.”ANTICIPATE COMMON DIFFICULTIESMost students today are very computer literate. However there are still a few techno-phobes in every class. Theirproblems are primarily not with MATLAB, but rather with the interface between it and the outside environment,such as saving, editing and printing files. These projects were designed to minimize these issues because theseactivities do not enhance linear algebra much. Students could even record a few results by hand and print a graphoccasionally. The projects are much easier to grade than they used to be when students used thediarycommand allthe time and turned in huge stacks of paper.If, for example, you prefer to have your students turn in their projects electronically either by email or by a webcourse management system, you will have to think through how you want your students to submit their solutions.Editing documents created through thediarycommand is much easier today due to the interactive capabilities ofmost computers. Still, your students will need guidelines so that the problems that arise are related to the materialand not the word processing.CONSIDER CLASSROOM DEMONSTRATIONSMost schools have permanent or portable demonstration units in the classroom consisting of a computer and amultimedia projector. The portable ones have the inconvenience of setting them up and down before class, but theydo work. For linear algebra purposes, a color display is desirable because of the demonstrations related to graphics.While it is very difficult at our institution to reserve a computer lab on a regular basis, I can use such a projectionunit with a computer and a camera on a calculator for classroom demonstrations. This has benefited my classes somuch that I resent being assigned a classroom with just a chalkboard for my other courses. If I understand theeducational research correctly, students learn much more by seeing the material in a variety of contexts. I sometimesgive students “control” of the computer to break up the routine in class. They often can calculate faster than me, andthe class occasionally comes up with new insights since the method of using MATLAB is slightly different.One way classroom demonstrations can be used is to check on student learning. For example, if the students appearto understand a particular problem and its expected numerical results, students can witness an instant calculation tosee if that corresponds to their understanding. The class can also make conjectures, and those claims can be put tothe test rather quickly using technology. An aid to this end is the collection of special MATLAB functions thatcomes with Laydata4 Toolbox. See theStudy Guideand Section 3 on page 8 below for further descriptions of thespecial functions.If the computer is used appropriately, the material becomes more convincing and richer. The likelihood of gettingbogged down in the arithmetic increases the chances that students miss the larger, fuller picture. Using MATLABallows me to focus on the concepts and ideas that I want to stress. For example, I can ask what should be calculatedto verify that the answer is correct, and then use MATLAB to do that calculation. This process can be a much betteruse of class time for helping students grasp the material rather than working out a tedious solution by hand. Thisprocess ofanalysis,prediction,computer solution, andverificationis how professional scientists use computers, andthese are important skills for students to practice.

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6Planning the CourseDECIDE HOW YOU WILL TEST STUDENTSHow students should be assessed on exams is always a critical issue for an instructor. One major consideration iswhether to allow MATLAB on exams. If MATLAB is unavailable, you will need to decide if graphing calculatorswill be permitted. You obviously will have to consider the profiles of your students and the test conditions.My approach is that students should primarily be tested on conceptual ideas rather than computations. There areother settings, such as homework, where computational skills can be assessed. I allow TI-84 and TI-89 graphingcalculators on tests because the use of MATLAB for me is not practical. I still try to minimize the need for acalculator unless I give an untimed test or I link the data directly to their calculators. (I do not allow other hand helddevices although connectivity to the “outside world” will increasingly become an interesting and challenging issue.)As far as the type of questions to include on an exam, I like test questions that can be quickly done if one takes theappropriate perspective and thinks about the issues involved. I also value questions that require the synthesis of avariety of ideas and topics. When working on your exams, consider the text website which has many sample testsand review sheets for three different types of courses. You might choose to tell your students which types ofquestions on those exams are similar to the ones you sometimes create.Consider including on exams a question or two based on projects that the students have completed. Such questionscan be effective in reinforcing the objectives of the assignment and in determining which members of the groupactually participated in the project. The following sample questions are examples where computers were used duringinstruction but not on the test.1. The following matrices are row equivalent:123212082456A−−⎡⎤⎢⎥=−−⎢⎥⎢⎥−−⎣⎦,120800120000R−⎡⎤⎢⎥=−⎢⎥⎢⎥⎣⎦.Write the general solution toA=x0. Write a particular solution toA=x0. Consider the matrix transformationA→xx; is it 1-1? onto? Explain answers. Find a basis for the column space ofAand a basis for the null space ofA.2. There is a real33×matrixAfor which the general solution of some systemA=xbis3314210x⎡ ⎤⎡⎤⎢ ⎥⎢⎥=+−⎢ ⎥⎢⎥⎢ ⎥⎢⎥⎣ ⎦⎣⎦x.What is the general solution ofA=x0?3. A certain population of owls feeds almost exclusively on wood rats. Letting( )o kand( )r kdenote thenumber in each population in yeark, a biologist estimates that(1).5 ( ).05 ( )o ko kr k+=+and(1).9 ( )50 ( )r ko kr k+= −+.Write the matrix that describes the interaction of these two populations from yearktoyeark+1.Assume the pattern described will continue in the future. Don't calculate, but instead answer in words:(a)What would you calculate, and how would you interpret the results, to find out the number ofindividuals in each population five years from now?(b)How could you use eigenvalues and eigenvectors to describe the long term behavior of the owl and ratpopulations? Include any equations you need to discuss, and say what all your symbols mean.4. Consider the vectorsv1= (1,1,1,1),v2= (2,1,0,-3), andv3= (-1,2,0,0).(a)Which pairs of these vectors are orthogonal to each other? Show work.(b)Write the formulas for additional calculations which could be done to get an orthonornal basis for thesubspace spanned by {v1,v2,v3}. If you have time, complete the calculations for extra credit.

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Planning the Course75. LetAbe thenn×matrix in which each entry is 1. Justify your answers to the following questions. For (a)-(c), think, don’t calculate! A very little calculation will be needed for (d).(a)There are two distinct eigenvalues ofA. What are they?(b)What is dim(Nul(A))?(c)What is the characteristic polynomial ofA?(d)What is a basis for each eigenspace ofA?6. SupposeAis an invertible matrix, andxis an eigenvector ofA.(a) Which of the following matrices also havexas an eigenvector? Circle the ones that do:1A−TA2A3A3AA+(b) Choose one of those you circled in part (a) and justify it. For instance, if you circled2A, you mustshow that ifxis an eigenvector ofAthen it is also an eigenvector of2A.7. LetAbe annn×matrix. In each part below,circletheone best possibleexpression to complete thesentence truthfully. Only that one best choice will be counted correct:(a) SupposeAhas four distinct eigenvalues. ThenA(will) (will not)(could but doesn’t have to) havefour independent eigenvectors.(b) IfAhas zero as an eigenvalue, the eigenspace of zero (will) (will not) (could but doesn’t have to)equal Nul(A).(c) Suppose2Ais thenn×zero matrix. ThenA(will) (will not) (could but doesn’t have to) be the zeromatrix.BE CREATIVEThe very existence of powerful and accessible matrix utilities raises questions about what topics to emphasize, whatskills students need to learn, and what style of teaching is best. These issues are not easily resolved. Many have beeninfluenced by the constructivist theory that students learn best when making connections to what they already knowor what they want to know. As mentioned before, I have used certain projects for certain majors and had themexpand on those ideas by having them develop an education lesson plan or by writing a report after doing someindependent reading. Each year I try to use more group work in my courses so that students feel more involved inthe course. I encourage my students to study and to work on problems together, and I think working together lowersthe frustration level with regards to computers.I encourage you to discuss what is happening in other departments and disciplines at your institution. Also make useof instructors at other schools. My discussions with other colleagues have allowed me to understand better whatother faculty are trying to accomplish and to be exposed to the educational issues and curriculum developments inother disciplines. Linear algebra has many uses, and you might gain some interesting and motivating problems fromothers at your school or from your community as a service-learning project. At the same time, communicate whatyou are doing to allow them to appreciate your subject matter.

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8Planning the Course3. Using Software for DemonstrationsMost of the M-files in Laydata4 Toolbox simply contain data for exercises and projects, and this can be helpful fordemonstrations as discussed on page 5. In addition, there are a few files that are called the “special functions,”which do particular kinds of calculations or graphing. They were written for various exercises and projects but theycan also be effective for occasional demonstrations. I encourage you to experiment with them early so you’ll knowabout them when an occasion arises where they might be helpful. Here is a list of these special functions with a fewcomments:Special FunctionDescriptionreplace, swap, scaleSingle row operationsgauss, bgaussSweep out specified columnsnulbasisProduces a basis for the null spaceprojOrthogonal projection of vector onto a subspacegsPerforms Gram-Schmidt algorithmqrbasicBasic QR method for calculating eigenvaluesqrshiftQR method with shifts and deflationThe five functions below produce simple but effective graphics.seesum, seeprod, seecomVisualize vector arithmeticdrawpolyDraw polygonssingvecSearch visually for singular vectorsThe functionrandomintallows you to specify size and rank, and is very useful for generating quick, clearexamples. The command randomint(5,4,2) will create a54×matrix of rank 2.randomintCreate random integer matricesThe simple way to find out how to use any MATLAB function is withhelp. For example, at the MATLAB prompt,typehelp replaceorhelp seesum.Although all these special functions are very nice, at some point emphasize that they were developed for educationalpurposes and should not be used for professional applications. If such a need arises, they should use professionallywritten software that employs the most sophisticated and efficient algorithms known. For example, in MATLAB oneshould use MATLAB's backslash to solve linear systems, notref. The latter does not check for condition number,and its algorithm is not the most efficient. Usually itproduces accurate answers, but when a matrix is nearlysingular,refcan return the wrong reduced form, and the user will not be warned. See the project “Roundoff Error inMatrix Calculations” for more details. Similarly, one should use MATLAB'sqrfunction, not the special functiongsin any professional setting where an orthonormal basis is needed. The functiongswas written to help students learnthe Gram-Schmidt algorithm, and it works fine on a small set of vectors.

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Downloading M-files94. Downloading M-files from the WebTo get Laydata4 Toolbox files, go towww.pearsonhighered.com/layFollow the on-screen directions to obtain the version of Lay’s files you want.If you have used a Toolbox from an earlier edition of the book, you will want to delete the folder. In earlier editionsof the text, the Toolbox was named Laydata and will likely be found in the main MATLAB folder.Once you have the files, you will want to decompress them and make them accessible to the working path so thatMATLAB knows where to find them. To avoid having to type a possibly complicated path to the correct folder,create an empty folder namelaydata4inside the main MATLAB folder. It is a good idea to use lowercase letterssince MATLAB is case sensitive. For example, on a PC the file should be created inc:\matlab\(or whatever yourworking MATLAB path is). With the folder already named, navigation is easier as you move through the directorytree and decompress the downloaded files into the appropriate folder.If your access to MATLAB is through a network, ask your network administrator to install Laydata4 Toolbox andany other scripts or M-files you need so that they are accessible.If Laydata4 Toolbox or other M-files are saved someplace else, you can use the Set Path feature in MATLAB.•In MATLAB 6 and 7, selectSet Pathfrom theFilemenu. Click onAdd with Subfoldersand find thefolder name of the Laydata4 subdirectory. Highlight the folder and click on OK. Finally, click onSaveandClose.•If you are running MATLAB on a network, you should ask the system administrator to store Laydata4Toolbox to MATLAB’s path.For more details, see Section 15 of the preliminary Computer Project, “Getting Started with MATLAB” and theMATLAB appendix in theStudyGuide.

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10Computer Projects5. Computer ProjectsGENERAL INFORMATIONThe projects are intended to enrich and expand the material in Lay's text. They are independent of each other. Eachone begins by stating a purpose, the prerequisite sections from text, and the MATLAB functions used. On the linelisting the MATLAB functions, the commands inherent to MATLAB are listed first and are followed by asemicolon. The functions and data files from Laydata4 Toolbox follow the semicolon. You may copy and use theprojects as written or adapt them. Most projects should require 1-2 hours, and a few may take longer. They do notrequire very hard thinking (except for an occasional extra credit question, and the project "Subspaces"). TheMATLAB commands are given, so the time required depends mostly on how much independent reading studentsmust do. Their work will go faster if you lecture a little on the material before they begin a project, but any of thesecan be "read and do" assignments.I allow about a week for each project. You may want to be lenient with your deadlines as equipment misbehaves,networks go down, and students have more difficulty with a project than you expected.PARTNERSI encourage but do not require students to do their computer work with a partner. This helps the students workthrough some of the computer issues together and cuts down on computer frustration. It also reduces my grading.Most of my classes do well in pairing up, but some students need help finding a partner even after the first couple ofweeks.Before assigning the lab it is a good idea to present some ground rules. For example, both people should work on thelab and understand the solutions. Their signature on their paper indicates that both did the work and that both agreeto the work submitted. I suggest you follow up the computer assignment after it has been submitted to confirm thesignatures and to discuss briefly the objectives. One colleague of mine picks one group to present their solution tothe rest of the class.NOTES ABOUT THE INDIVIDUAL PROJECTSHere are a few comments about each project. The symbolRmeans the project is especially recommended becauseof its value. I usually grade each project out of 10 points and am generally more lenient with my grading than onother homework.Getting Started With MATLAB. This is long and it is not necessary to do it all, but it can be helpful fornovices and for reference.RPractice Row Reduction(5). This is easy and could be assigned soon after students have learned to do rowoperations by hand. They will practice doing them with Lay's functionsreplace,scale, andswap.Exchange Economy and Homogeneous Systems(5).Students seem to like economic models.Assignthis one immediately after covering homogeneous systems in Section 1.6, and perhaps letting them read aboutLeontief models by themselves. The last two questions provoke them to think about row operations abstractly. Forthe extra credit question, consider giving one point if a student works out a symbolic example with three rows andcolumns say, but no points if they give only numerical examples. A nice general argument could be worth 2-3points.

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Computer Projects11Reduced Echelon Form andref(5). This is easy and shows students how to row-reduce a matrix usinggaussandref. They also see that roundoff error can causerefto produce the wrong answer, by experimentingwith different values for the tolerance inref. Although the commandrefcan be introduced as early as Section1.2, the text emphasizes echelon form rather than reduced echelon form until Section 1.5. TheStudy Guideusesgauss,swap, andscalefor row operations until Section 4.3 (and 2.9).Rank and Linear Independence(5). This project can be used to introduce rank earlier than the text does.The last question here is an "explore and conjecture" type.(Section 1.7 is the prerequisite section.)RVisualizing Linear Transformations of the Plane(5). This looks long but much of it goes quickly. It usesdrawpolyfor some graphics, and helps students start thinking about a matrix as a transformation early.Moststudents seem to have very little geometric intuition and need all the practice they can get to develop some.(Section1.9)Population Migration(5). Students like this, especially the plotting. Before assigning this one and goingover the city-suburb example in the text, you might ask your students what will happen if this pattern of migrationpersists. Will everyone move to the suburbs? Have them calculatekxfor some large values ofkand report back atthe next class.(Section 1.10)RElementary Analysis of the Spotted Owl Population(5).This one does not need as much introduction,but students should read the simple example at the start of Chapter 5. Mention that they will do a naive analysis ofthe population’s long term behavior in this project and later will use eigenvalues and vectors to analyze it moredeeply in “Using Eigenvalues to Study Spotted Owls.”(Section 1.10)Lower Triangular Matrices(5).This is a simple but nice exploration of matrix multiplication.Studentswill discover that the product of (unit) lower triangular matrices is (unit) lower triangular, and write proofs.(Section2.1)The Adjacency Matrix of a Graph(10).This is a more sophisticated look at matrix multiplication.Students examine very carefully how an entry of2Ais calculated.They must also create a definition, which is anew kind of exercise for most of them.Before assigning this project, I recommend you discuss graphs a little andexplain "contact levelk."The answers for question 4(b) are "All but W8" and "All."A good answer to 4(d) would be "Dangerousmeans highest level one contact, and W6, W4 and W1 are most dangerous." Occasionally an observant student willsee that W6 is in level two contact with everyone else and is the only worker like that.Consider giving 2 extracredit points for that answer, and report the insight to the class when handing the papers back. It never fails to causea stir, and it motivates the others to pay more attention to details. Students' definitions in 4(d) are often vague, like"high contact level" and perhaps take off 1-2 points but try to help them say what they really wanted the definition tobe, based on their explanations at the end.(Section 2.1)RCryptography(5-10).Manystudentslikethis one---particularlytheeducation majors.TheyuseMATLAB’s remainder functionremto do some arithmetic modulo 26. In the extra credit question they calculateby hand a matrix inverse modulo 26. This project was originally created by a student, Sanja Petrovic.(Section 2.2)Using Backslash to SolveAx=b(5). The purpose is to see why the backslash operator is preferable tosolving matrix equations whenAis invertible. Students compare solutions of equations involving the ill-conditionedHilbert matrices using the backslash command, the matrix inverse commandinv, and therefcommand.(Section2.3)

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12Computer ProjectsRRoundoff Error in Matrix Calculations(5).Students use backslash,refandinvto solve 8 linearsystems three different ways. They see that the different algorithms give somewhat different answers in every caseand very different answers for the poorly conditioned systems. They see definitions of floating point notation,residual vector, condition, and Hilbert matrix; learn to watch for warnings; and usenorm. There are briefdiscussions of condition number and the algorithms used in the three methods. The illustrated computationalrealities are important since many students will do matrix calculations in scientific applications.(Section 2.3)Partitioned Matrices(5).This is an important topic, as partitioned matrices are used frequently inapplications. Question 2 is designed to reinforce the fact that[]⎥⎦⎤⎢⎣⎡YXBAequalsAXBY+, notXAYB+.Invariably, one or two people make that mistake. What leads them astray is the definition ofAxearlier in the textwhere the scalarsixare written on the left of the columns. To try to forestall this common error with partitionedmatrices, when definingAx, point out that theix’s are scalars so they look more natural on the left of a vector, buttechnically a scalar could be written on the left or right. However, when we learn to multiply two matrices, we’ll seethat the order makes a big difference, and one has to be especially careful when multiplying partitioned matrices.(Section 2.4)Schur Complement(5). This is a nice application of partitioning. Students learn three ways to calculate aSchur complement, including using row operations. The extra credit question is challenging.(Section 2.4)LU Factorization(5).This explains how the LU factorization algorithm in Section 2.5 differs fromMATLAB'slufunction, and provides practice using both.(Section 2.5)An Economy With An Open Sector(5).Students will verify Theorem 11 in Section 2.6 and thenexperiment to see that the result can fail if they change just one entry of the consumption matrixCenough. Theywill probably discover that increasing11cto about .95 will cause the solution of1C=+xxdto have some negativeentries. They should give something like the following explanation: if the Chemicals sector consumes .95 of its ownoutput then it is not surprising that the economy cannot meet the demands from other sectors, and this is what thenonsense solution says.Matrix Inverses and Infinite Series(5). This explores the meaning of Theorem 13 in Section 2.6.Students will experiment with the series2ISS+++…finding matricesSfor which it does and does not seem toconverge. They could do this project any time after Section 2.2 as a "read and do" assignment.HomogeneousCoordinatesforComputerGraphics(5).Thisusesdrawpoly.Homogeneouscoordinates for2\and how they can be manipulated with33×matrices are novel ideas to most students.Computer science majors especially like this.(Section 2.7)Subspaces(20). Span is a hard concept for many students, and this has proved to be the most challengingproject. You can pair up the students and assign each group a different pair of matrices,AandB.Each matrixAis54×,Bis55×, and both have rank 4. There are 25 pairs of integer matrices in the filesubmats. The first 17pairs do have the same column space, and the last 8 pairs do not. If you prefer to generate your own matrices, hereare commands create a pair that do have the same 4-dimensional column spaces:A = randomint(5,5,4),B = A*randomint(5,4,4)If you wantAandBthat do not have the same column spaces, try this instead:A = randomint(5,5,4),B = randomint(5,4,4)(The commandrandomint(m,n,k)yields anmn×matrix with rankk.)Students quickly reduceAandBand see they have the same rank. They don't feel very comfortable withsets, but after they discuss the problem with me and each other for a while, usually they figure out that they can rowreduce[A B]and then explain in words why the result shows that each column ofBis in ColA, hence ColBis asubset of ColA. An elementary way to finish would be to reduce [B A] to see that ColAis also a subset of ColB,and then conclude that the two sets must be the same, but few students seem to think of doing that. Instead they

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Computer Projects13explain that ColBis inside ColAand has the same dimension so by Theorem 15, a basis for ColBmust span ColA.This project could be assigned after Section 4.6 (or Section 2.9 if you cover that instead).Markov Chains and Long-Range Predictions(5-10 points, depending on how much help is given aheadof time). This is fun and good motivation for eigenvalues and eigenvectors. If you ask everyone to do this project,you should give a brief introduction to Markov processes.(Section 4.9)RReal and Complex Eigenvalues(5). This is easy and you could assign it instead of lecturing on complexeigenvalues. Students calculate some complex eigenvalues by hand and then create some examples of their own tobe sure they really look at the matrix entries. Then they learn how to use MATLAB’seigfunction to findeigenvalues and eigenvectors.(Section 5.5)RUsing Eigenvalues to Study Spotted Owls(10).Before assigning this, either lecture on complexeigenvalues or assign the previous project. This is a long project but students like it. It is a lovely application ofeigenvalues and diagonalizability, and includes some plotting. Most of the theory from Sections 5.1-5.3 is applied.Students useeigand experiment to find the critical value oft, the survival rate for juvenile→subadult ("critical"means the minimum value oftwhich makes the dominant eigenvalue at least 1).The extra credit question asks users to verify theoretically what they have seen experimentally.It ischallenging but many hints are given.The idea for this question is due to Andre Weideman and used with hispermission. Techniques from calculus can be used to show that the stage matrix will always have its dominanteigenvalue real and positive and be diagonalizable, and then one can derive a formula for the critical value oft.(Sections 5.5 and 5.6)QR Factorization(5). This is not hard and should be of interest to many students, since QR factorizationsare widely used in practice. Students will learn how MATLAB'sqrfunction differs from the QR factorizationdeveloped in the text and also will verify the connection beween QR factorization and the Gram Schmidt Process.(Section 6.4)The QR Method for Calculating Eigenvalues(10). Some students will be very interested in this project.Here they use theqrfunction to experiment with the basic QR algorithm for eigenvalues and with a shift-deflateversion of it. This type of iterative process is used in modern software for calculating eigenvalues. It never fails toamaze that the processes usually work! Convergence is discussed briefly and a reference given for moreinformation. Two functions in Laydata4 Toolbox,qrbasicandqrshift, assist with the calculations.(Sections5.2 and 6.4)RLeast-Squares Solutions and Curve Fitting(5).This is easy and can be done early in Chapter 6 tomotivate the ideas. The text algorithm is used to calculate coefficients for the least squares line, quadratic and cubiccurves;normis used to calculate least squares error, andplotis used to graph the data and the curves.An endnote describes how to usepolyvalandpolyfit, which are MATLAB functions that can do most of the workfor you.In question 2(b), most students guess3()veland say the error is smaller for the cubic than for the line orquadratic, and that is acceptable to some including me.However, the correct answer is that drag depends on2()vel. Consider giving an extra credit point to anyone who sees that this is sensible guess since the quadratic curve is adramatically better fit to the data than the line, but the cubic gives very little improvement. These things are evidentboth from the graph and the errors. In fact, the theoretical formula is2()dragc velAρ=whereρis air density,cisthe coefficient of drag, which varies with the angle of attack, andAis the surface area of the wing.

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