Table Of ContentLinear Algebra Gems
Assets for
Undergraduate Mathematics
(cid:13)c2002 by the Mathematical Association of America(Inc.)
ISBN: 0-88385-170-9
Library of Congress Catalog Card Number 2001097387
Printedin the UnitedStatesof America
Current Printing
10 9 8 7 6 5 4 3 2 1
Linear Algebra Gems
Assets for
Undergraduate Mathematics
Edited by
David Carlson
Charles R. Johnson
David C. Lay
A. Duane Porter
Published and Distributed by
The Mathematical Association of America
The MAA Notes Series, started in 1982,addresses a broad range of topics and themes of interest to allwho
are involved with undergraduate mathematics. The volumes in this series are readable, informative, and
useful, and help the mathematicalcommunity keep up with developments of importance to mathematics.
Editorial Board
Sr. Barbara E. Reynolds, Editor
Nancy Baxter Hastings, Associate Editor
Jack Bookman Annalisa Crannell Daniel John Curtin
Paul E. Fishback Nancy L. Hagelgans Richard D. Ja¨rvinen
Ellen J. Maycock Mary R. Parker
MAA Notes
11. Keys to Improved Instruction by Teaching Assistants and Part-Time Instructors, Committee on Teaching
AssistantsandPart-Time Instructors,BettyeAnneCase,Editor.
13. Reshaping College Mathematics, Committee on the UndergraduateProgram in Mathematics, Lynn A. Steen,
Editor.
14. Mathematical Writing,by DonaldE. Knuth, Tracy Larrabee,and PaulM. Roberts.
16. Using Writingto Teach Mathematics, AndrewSterrett,Editor.
17. Priming the Calculus Pump: Innovations and Resources, Committee on Calculus Reform and the First Two
Years,a subcomittee ofthe Committeeon the Undergraduate Program in Mathematics, Thomas W. Tucker,
Editor.
18. Models for Undergraduate Research in Mathematics, Lester Senechal,Editor.
19. Visualization in Teaching and Learning Mathematics, Committee on Computers in Mathematics Education,
SteveCunninghamandWalter S. Zimmermann,Editors.
20. The Laboratory Approach to Teaching Calculus,L. Carl Leinbachet al.,Editors.
21. Perspectives on Contemporary Statistics,David C. HoaglinandDavidS. Moore, Editors.
22. Heeding the Callfor Change: Suggestions for CurricularAction, LynnA. Steen,Editor.
24. Symbolic Computation inUndergraduate Mathematics Education, ZavenA. Karian,Editor.
25. TheConcept ofFunction: AspectsofEpistemology andPedagogy, GuershonHarelandEdDubinsky,Editors.
26. Statisticsfor the Twenty-First Century, Florenceand SheldonGordon,Editors.
27. Resources for Calculus Collection, Volume 1: Learning by Discovery: A Lab Manual for Calculus, Anita E.
Solow,Editor.
28. Resources for CalculusCollection,Volume 2: Calculus Problems for a NewCentury, Robert Fraga,Editor.
29. Resources for CalculusCollection,Volume 3: Applications of Calculus, PhilipStraffin,Editor.
30. Resources for Calculus Collection, Volume 4: Problems for Student Investigation, Michael B. Jackson and
JohnR. Ramsay,Editors.
31. Resources for CalculusCollection,Volume 5: Readings for Calculus,UnderwoodDudley,Editor.
32. Essays in Humanistic Mathematics, AlvinWhite,Editor.
33. Research Issues inUndergraduate Mathematics Learning: PreliminaryAnalyses and Results, James J. Kaput
andEd Dubinsky,Editors.
34. In Eves’ Circles,Joby Milo Anthony,Editor.
35. You’re the Professor, What Next? Ideas and Resources for Preparing College Teachers, The Committee on
Preparationfor College Teaching,BettyeAnneCase,Editor.
36. Preparing for a New Calculus: Conference Proceedings, AnitaE. Solow,Editor.
37. A Practical Guide to Cooperative Learning in Collegiate Mathematics, Nancy L. Hagelgans, Barbara E.
Reynolds,SDS,KeithSchwingendorf,DragaVidakovic,EdDubinsky,MazenShahin,G.JosephWimbish,Jr.
38. ModelsThatWork: CaseStudiesinEffectiveUndergraduate MathematicsPrograms,AlanC. Tucker,Editor.
39. Calculus: The Dynamics of Change, CUPM Subcommittee on Calculus Reform and the First Two Years, A.
WayneRoberts,Editor.
40. VitaMathematica: HistoricalResearch and Integration withTeaching, RonaldCalinger,Editor.
41. Geometry Turned On: Dynamic Software in Learning, Teaching, and Research, James R. King and Doris
Schattschneider,Editors.
42. Resources for TeachingLinear Algebra,DavidCarlson,CharlesR. Johnson,DavidC. Lay, A.DuanePorter,
AnnE. Watkins,William Watkins,Editors.
43. Student Assessment in Calculus: A Report of the NSF Working Group on Assessment in Calculus, Alan
Schoenfeld,Editor.
44. ReadingsinCooperativeLearningforUndergraduateMathematics,EdDubinsky,DavidMathews,andBarbara
E. Reynolds,Editors.
45. Confronting the Core Curriculum: Considering Change in the Undergraduate Mathematics Major, John A.
Dossey,Editor.
46. Women in Mathematics: Scaling the Heights, DeborahNolan,Editor.
47. Exemplary Programs in Introductory College Mathematics: Innovative Programs Using Technology, Susan
Lenker,Editor.
48. Writing inthe Teaching and Learning of Mathematics, JohnMeier andThomas Rishel.
49. Assessment Practices in Undergraduate Mathematics, BonnieGold,Editor.
50. RevolutionsinDifferentialEquations: ExploringODEswithModernTechnology,MichaelJ.Kallaher,Editor.
51. Using Historyto Teach Mathematics: An International Perspective, VictorJ. Katz,Editor.
52. Teaching Statistics: Resources forUndergraduate Instructors, Thomas L. Moore,Editor.
53. Geometry at Work: Papers in Applied Geometry, CatherineA. Gorini,Editor.
54. Teaching First: A Guide for NewMathematicians, Thomas W. Rishel.
55. CooperativeLearninginUndergraduateMathematics: IssuesThatMatterandStrategiesThatWork,Elizabeth
C. Rogers,BarbaraE. Reynolds,NeilA. Davidson,and AnthonyD. Thomas,Editors.
56. ChangingCalculus: AReportonEvaluationEffortsandNationalImpactfrom1988to1998,SusanL.Ganter.
57. Learning to Teach and Teaching to Learn Mathematics: Resources for Professional Development, Matthew
DelongandDale Winter.
58. Fractals,Graphics, and Mathematics Education, BenoitMandelbrotandMichael Frame,Editors.
59. LinearAlgebra Gems: Assetsfor Undergraduate Mathematics, DavidCarlson,CharlesR. Johnson,DavidC.
Lay,and A. DuanePorter,Editors.
MAAService Center
P.O. Box 91112
Washington, DC 20090-1112
800-331-1622 fax: 301-206-9789
Foreword
This volume, compiled by the editors on behalf of the Linear Algebra Curriculum Study
Group, is for instructors and students of linear algebra as well as all those interested in the
ideas of elementary linear algebra. We have noticed, through attendance at special sessions
organized at the Joint Annual Meetings and through talks given at other conferences and
universities,that thereis broad and sustained interestin the content of undergraduate linear
algebra courses.
Since this course became a centerpieceof the mathematics curriculum, beginning around
1960, new topics and new treatments have gradually reshaped it, with noticeably greater
evolution than in calculus courses. In addition, current courses are often taught by those
not trained in the subject or by those who learned linear algebra in a course rather different
from the present one. In this setting, it is not surprising that there is considerable interest
in the context and subtleties of ideas in the linear algebra course and in a perspective based
upon what lies just beyond. With this in mind, we have selected 74 items and an array of
problems, some previously published and some submitted in response to our request for such
items (College Mathematics Journal 23 (1992), 299–303). We hope that these will provide a
useful background and alternative techniques for instructors, sources of enrichment projects
and extended problems for teachers and students, impetus for further textbook evolution to
writers, and the enjoyment of discovery to others.
The Linear Algebra Curriculum Study Group (LACSG) began with a special session,
at the January 1990 Joint Annual Meetings, focusing upon the elementary linear algebra
course. This session was organized by Duane Porter, following upon an NSF-sponsored
Rocky Mountain Mathematics Consortium Lecture Series given by Charles Johnson at the
Universityof Wyoming; David Carlson and David Lay were panel members for that session.
With NSF encouragement and support, these four organized a five-day workshop held at
the College of William and Mary in August, 1990. The goal was to initiate substantial and
sustained national interest in improving the undergraduate linear algebra curriculum. The
workshop panel was broadly based, both geographically and with regard to the nature of
institutions represented. In addition, consultants from clientdisciplinesdescribed the role of
linear algebra in their areas and suggested ways in which the curriculum could be improved
from their perspective.
Preliminary versions of LACSG recommendations were completed at this workshop and
widelycirculatedforcomment. Afterreceivingmanycommentsand with thebenefitof much
discussion, a version was publishedin 1993 (College Mathematics Journal 24 (1993), 41–46).
This was followedby a companion volumeto this one in 1997 (Resources for Teaching Linear
vii
viii Linear Algebra Gems: Assets for Undergraduate Mathematics
Algebra, MAA Notes No. 42, Washington, D.C., 1997, 297 pages). Work of the LACSG has
continued with the organization of multiple special sessions at each of the Joint Annual
Meetings from 1990 through 1998. With sustained strong attendance at these sessions,
acknowledged influence on newer textbooks, discussion in letters to the AMS Notices, and
the ATLAST workshops, the general goal of the LACSG is being met.
Though a few items in this volume are several pages, we have generally chosen short,
crisp items that we feel contain an interesting idea. Previously published items have been
screenedfrom severalhundredwereviewedfrom thepast 50+ years,most fromtheAmerican
Mathematical Monthly and the College Mathematics Journal. New (previously unpublished)
items were selected from about 100 responses to our call for contributions. Generally, we
have chosen items that relate in some way to the first course, might evolve into the first
course, or are just beyond it. However, second courses are an important recommendation
of the LACSG, and some items are, perhaps, only appropriate at this level. Typically, we
have avoided items for which both the topic and treatment are well established in current
textbooks. For example, there has been dramatic improvement in the last few decades in
the use of row operations and reduced echelonform to elucidate or make calculations related
to basic issues in linear algebra. But, many of these have quickly found their way into
textbooks and become well established, so that we have not included discussion of them.
Also, because of the ATLAST volume, we have not concentrated upon items that emphasize
the use of technology in transmitting elementary ideas, though this is quite important. We
do not claim that each item is a “gem” in every respect, but something intrigued us about
each one. Thus, we feel that each has something to offer and, also, that every reader will
find something of interest.
Based upon what we found, the volume is organized into ten topical “parts.” The parts
and the items within each part are in no particular order, except that we have tried to
juxtapose itemsthat are closelyrelated. Many itemsdo relateto parts other than theone we
chose. The introduction to each part provides a bit of background or emphasizes important
issues about constituent items. Because of the number of items reprinted without editing,
we have not adopted a common notation. Each item should be regarded as a stand-alone
piece with its own notation or utilizing fairly standard notation.
Each item is attributed at the end of the item, typically in one of three ways. If it is a
reprinted item, the author, affiliation at the time, and original journal reference are given. If
it is an original contribution, the author and affiliation are given. In a few cases, the editors
chose to author a discussion they felt important, and such items are simply attributed to
“the Editors.”
We would liketo thank, first of all, the many colleagues and friends who contributed new
items to this volume, as well as the authors of items we have chosen to reprint here. We
also are grateful for the many more contributions we were unable to use due to limitations
of space or fit with the final themes of the volume. We give special thanks to the National
Science Foundation for its support of the activities of the Linear Algebra Curriculum Study
Group (USE 89-50086, USE 90-53422, USE 91-53284), including part of the preparation of
this volume, and for the support of other linear algebra education initiatives. The Mathe-
matical Association of America receives equal thanks, not only for the publication of this
volumeand the previous LACSG volume(Resources for Teaching Linear Algebra, 1997), but
The Editors: Foreword ix
also for its sponsorship of the well-attended LACSG special sessions at the January Joint
AnnualMeetings(along withAMS) since1990 and for the manypermissions to reprintitems
from MAA publications (the American Mathematical Monthly and the College Mathemat-
ics Journal) used in this volume. We thank the many attendees of these sessions for their
encouragement and interest, and we thank the Rocky Mountain Mathematics Consortium
and the University of Wyoming for hosting Charles Johnson’s initiating lecture series and
the College of William and Mary for hosting the 1990 founding workshop.
Finally, we are most grateful to our several colleagues who assisted with the preparation
of this volume: John Alsup, while a PhD. student at the University of Wyoming; Joshua
Porter, while an undergraduate student at the University of Wyoming; Chris Boner, while
an REU student at the College of William and Mary; Fuzhen Zhang, while a visitor at the
College of William and Mary; Jeanette Reisenburg for her many hours of excellenttyping in
Laramie; and the several reviewers for their encouragement and helpful suggestions.
Lastly, we are verypleased with the opportunity to prepare this volume. In spite of more
than 125 years of collective experience in teaching linear algebra and thinking about the
elementarycourse, we all learned a great deal from many hours of reading and then chatting
about the items. Indeed, weeach feelthat we learn new subtletieseach timewe undertake to
describe elementary linear algebra to a new cohort of students. We hope that all instructors
will find similar value in this volume and will continue to reflect upon and enhance their
understanding of ideas that underlie and relate to the course.
The Editors:
Charles R. Johnson, The College of William and Mary
David Carlson, San Diego State University
David C. Lay, University of Maryland
A. Duane Porter, University of Wyoming
