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  • Linear Algebra

    David Cherney, Tom Denton and Andrew Waldron

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    Edited by Katrina Glaeser, Rohit Thomas and Travis ScrimshawFirst Edition. Davis California, 2013.

    This work is licensed under a

    Creative Commons Attribution-NonCommercial-

    ShareAlike 3.0 Unported License.

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    http://creativecommons.org/licenses/by-nc-sa/3.0/deed.en_UShttp://creativecommons.org/licenses/by-nc-sa/3.0/deed.en_UShttp://creativecommons.org/licenses/by-nc-sa/3.0/deed.en_US

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    Preface

    This book grew out of a series of twenty five lecture notes for a sophomorelinear algebra class taught at the University of California, Davis. The audi-ence was primarily engineering students and students of pure sciences, someof whom may go on to major in mathematics. It was motivated by the lack ofa book that taught students basic structures of linear algebra without overdo-ing mathematical rigor or becoming a mindless exercise in crunching recipesat the cost of fundamental understanding. In particular we wanted a bookthat was suitable for all students, not just math majors, that focussed onconcepts and developing the ability to think in terms of abstract structuresin order to address the dizzying array of seemingly disparate applicationsthat can all actually be addressed with linear algebra methods.

    In addition we had practical concerns. We wanted to offer students aonline version of the book for free, both because we felt it our academicduty to do so, but also because we could seamlessly link an online book toa myriad of other resourcesin particular WeBWorK exercises and videos.We also wanted to make the LaTeX source available to other instructorsso they could easily customize the material to fit their own needs. Finally,we wanted to restructure the way the course was taught, by getting thestudents to direct most of their effort at more difficult problems where theyhad to think through concepts, present well-thought out logical argumentsand learn to turn word problems into ones where the usual array of linearalgebra recipes could take over.

    How to Use the Book

    At the end of each chapter there is a set of review questions. Our studentsfound these very difficult, mostly because they did not know where to begin,rather than needing a clever trick. We designed them this way to ensure thatstudents grappled with basic concepts. Our main aim was for students tomaster these problems, so that we could ask similar high caliber problemson midterm and final examinations. This meant that we did have to directresources to grading some of these problems. For this we used two tricks.First we asked students to hand in more problems than we could grade, andthen secretly selected a subset for grading. Second, because there are morereview questions than what an individual student could handle, we split theclass into groups of three or four and assigned the remaining problems to them

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    for grading. Teamwork is a skill our students will need in the workplace; alsoit really enhanced their enjoyment of mathematics.

    Learning math is like learning to play a violinmany technical exercisesare necessary before you can really make music! Therefore, each chapter hasa set of dedicated WeBWorK skills problems where students can test thatthey have mastered basic linear algebra skills. The beauty of WeBWorK isthat students get instant feedback and problems can be randomized, whichmeans that although students are working on the same types of problem,they cannot simply tell each other the answer. Instead, we encourage themto explain to one another how to do the WeBWorK exercises. Our experienceis that this way, students can mostly figure out how to do the WeBWorKproblems among themselves, freeing up discussion groups and office hours forweightier issues. Finally, we really wanted our students to carefully read thebook. Therefore, each chapter has several very simple WeBWorK readingproblems. These appear as links at strategic places. They are very simpleproblems that can answered rapidly if a student has read the preceding text.

    The Material

    We believe the entire book can be taught in twenty five fifty minute lecturesto a sophomore audience that has been exposed to a one year calculus course.Vector calculus is useful, but not necessary preparation for this book, whichattempts to be self-contained. Key concepts are presented multiple times,throughout the book, often first in a more intuitive setting, and then againin a definition, theorem, proof style later on. We do not aim for studentsto become agile mathematical proof writers, but we do expect them to beable to show and explain why key results hold. We also often use the reviewexercises to let students discover key results for themselves; before they arepresented again in detail later in the book.

    Linear algebra courses run the risk of becoming a conglomeration of learn-by-rote recipes involving arrays filled with numbers. In the modern computerera, understanding these recipes, why they work, and what they are for ismore important than ever. Therefore, we believe it is crucial to change thestudents approach to mathematics right from the beginning of the course.Instead of them asking us what do I do here?, we want them to ask whywould I do that? This means that students need to start to think in termsof abstract structures. In particular, they need to rapidly become conversantin sets and functionsthe first WeBWorK set will help them brush up these

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    skills.There is no best order to teach a linear algebra course. The book has

    been written such that instructors can reorder the chapters (using the La-TeX source) in any (reasonable) order and still have a consistent text. Wehammer the notions of abstract vectors and linear transformations hard andearly, while at the same time giving students the basic matrix skills neces-sary to perform computations. Gaussian elimination is followed directly byan exploration chapter on the simplex algorithm to open students mindsto problems beyond standard linear systems ones. Vectors in Rn and generalvector spaces are presented back to back so that students are not strandedwith the idea that vectors are just ordered lists of numbers. To this end, wealso labor the notion of all functions from a set to the real numbers. In thesame vein linear transformations and matrices are presented hand in hand.Once students see that a linear map is specified by its action on a limited setof inputs, they can already understand what a basis is. All the while studentsare studying linear systems and their solution sets, so after matrices deter-minants are introduced. This material can proceed rapidly since elementarymatrices were already introduced with Gaussian elimination. Only then is acareful discussion of spans, linear independence and dimension given to readystudents for a thorough treatment of eigenvectors and diagonalization. Thedimension formula therefore appears quite late, since we prefer not to elevaterote computations of column and row spaces to a pedestal. The book endswith applicationsleast squares and singular values. These are a fun way toend any lecture course. It would also be quite easy to spend any extra timeon systems of differential equations and simple Fourier transform problems.

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    One possible distribution of twenty five fifty minute lectures might be:

    Chapter LecturesWhat is Linear Algebra? 1SystemsofLinearEquations 3The Simplex Method 1Vectors in Space, n-Vectors 1Vector Spaces 1Linear Transformations 1Matrices 3Determinants 2Subspaces and Spanning Sets 1Linear Independence 1Basis and Dimension 1Eigenvalues and Eigenvectors 2Diagonalization 1Orthonormal Bases and Complements 2Diagonalizing Symmetric Matrices 1Kernel, Range, Nullity, Rank 1LeastSquaresandSingularValues 1

    Creating this book has taken the labor of many people. Special thanks aredue to Katrina Glaeser and Travis Scrimshaw for shooting many of the videosand LaTeXing their scripts. Rohit Thomas wrote many of the WeBWorKproblems. Bruno Nachtergaele and Anne Schilling provided inspiration forcreating a free resource for all students of linear algebra. Dan Comins helpedwith technical aspects. A University of California online pilot grant helpedfund the graduate students who worked on the project. Most of all we thankour students who found many errors in the book and taught us how to teachthis material!

    Finally, we admit the books many shortcomings: clumsy writing, lowquality artwork and low-tech video material. We welcome anybody whowishes to contribute new materialWeBWorK problems, videos, picturesto make this resource a better one and are glad to hear of any typographicalerrors, mathematical fallacies, or simply ideas how to improve the book.

    David, Tom and Andrew

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  • Contents

    1 What is Linear Algebra? 131.1 Vectors? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131.2 Linear Transformations . . . . . . . . . . . . . . . . . . . . . . 151.3 What is a Matrix? . . . . . . . . . . . . . . . . . . . . . . . . 191.4 Review Problems . . . . . . . . . . . . . . . . . . . . . . . . . 23

    2 Systems of Linear Equations 292.1 Gaussian Elimination . . . . . . . . . . . . . . . . . . . . . . . 29

    2.1.1 Augmented Matrix Notation . . . . . . . . . . . . . . . 292.1.2 Equivalence and the Act of Solving . . . . . . . . . . . 322.1.3 Reduced Row Echelon Form . . . . . . . . . . . . . . . 322.1.4 Solutions and RREF . . . . . . . . . . . . . . . . . . . 36

    2.2 Review Problems . . . . . . . . . . . . . . . . . . . . . . . . . 382.3 Elementary Row Operations . . . . . . . . . . . . . . . . . . . 42

    2.3.1 EROs and Matrices . . . . . . . . . . . . . . . . . . . . 422.3.2 Recording EROs in (M |I ) . . . . . . . . . . . . . . . . 442.3.3 The Three Elementary Mat

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