Linear Algebra Problem Book

348 Pages·2013·1.97 MB·English

LINEAR ALGEBRA
PROBLEM BOOK
Paul R. Halmos
THE
DOLCIANI MATHEMATICAL EXPOSITIONS
Publishedby
THEMATHEMATICAL ASSOCIATIONOFAMERICA
CommitteeonPublications
JAMESW. DANIEL,Chairman
DolcianiMathematicalExpositionsEditorialBoard
BRUCEP.PALKA, Editor
CHRISTINEW. AYOUB
IRLC. BIVENS
BRUCEA. REZNICK
The DolcianiMathematicalExpositions
NUMBERSIXTEEN
LINEAR ALGEBRA
PROBLEM BOOK
PAUL R. HALMOS
PublishedandDistributedby
THEMATHEMATICAL ASSOCIATIONOFAMERICA
(cid:13)c1995by
TheMathematicalAssociationofAmerica(Incorporated)
LibraryofCongressCatalogCardNumber94-79588
PrintISBN978-0-88385-322-1
ElectronicISBN978-1-61444-212-7
PrintedintheUnitedStatesofAmerica
Currentprinting(lastdigit):
10987654
The DOLCIANI MATHEMATICAL EXPOSITIONS series of the Mathematical
Association of America was established through a generous gift to the Association
from Mary P. Dolciani, Professor of MathematicsatHunter College of the City Uni-
versity of New York. Inmakingthe gift, Professor Dolciani, herself anexceptionally
talented and successful expositor of mathematics, had the purpose of furthering the
ideal of excellencein mathematicalexposition.
TheAssociation,foritspart,wasdelightedtoacceptthegraciousgestureinitiating
the revolving fund for this series from one who has served the Association with
distinction, both asamember of the Committee on Publicationsandasamember of
the Board of Governors. It was with genuine pleasure that the Board chose to name
the seriesin herhonor.
Thebooksintheseriesareselectedfortheirlucidexpositorystyleandstimulating
mathematical content. Typically, they contain an ample supply of exercises, many
with accompanyingsolutions. Theyare intendedto besufficiently elementaryfor the
undergraduateandeventhemathematicallyinclinedhigh-schoolstudenttounderstand
andenjoy,butalsotobeinteresting andsometimeschallengingtothemoreadvanced
mathematician.
1. MathematicalGems,RossHonsberger
2. MathematicalGemsII, RossHonsberger
3. MathematicalMorsels,RossHonsberger
4. MathematicalPlums, RossHonsberger (ed.)
5. GreatMomentsinMathematics(Before1650),HowardEves
6. MaximaandMinima withoutCalculus,IvanNiven
7. GreatMomentsinMathematics(After1650),Howard Eves
8. MapColoring,Polyhedra,andtheFour-ColorProblem,DavidBarnette
9. MathematicalGemsIII, RossHonsberger
10. MoreMathematicalMorsels,RossHonsberger
11. OldandNewUnsolvedProblemsinPlaneGeometryandNumberTheory,Victor
Klee andStanWagon
12. ProblemsforMathematicians,YoungandOld, PaulR. Halmos
13. ExcursionsinCalculus:AnInterplayoftheContinuousandtheDiscrete,Robert
M.Young
14. The WohascumCounty Problem Book, George T. Gilbert, Mark I. Krusemeyer,
LorenC. Larson
15. Lion Hunting and Other Mathematical Pursuits: A Collection of Mathematics,
Verse, and Stories by Ralph P. Boas, Jr., Gerald L. Alexanderson and Dale H.
Mugler (eds.)
16. LinearAlgebraProblemBook,PaulR. Halmos
17. FromErdo}stoKiev:Problemsof OlympiadCaliber,RossHonsberger
18. WhichWayDidtheBicycleGo?...andOtherIntriguingMathematicalMysteries,
JosephD.E. Konhauser,DanVelleman,andStanWagon
19. InPo(cid:19)lya’sFootsteps:MiscellaneousProblemsandEssays,RossHonsberger
20. Diophantusand Diophantine Equations, I. G. Bashmakova(Updated by Joseph
Silvermanandtranslated byAbe Shenitzer)
21. LogicasAlgebra,PaulHalmosandStevenGivant
22. Euler:TheMasterof UsAll, William Dunham
23. TheBeginningsandEvolutionofAlgebra,I.G.BashmakovaandG.S.Smirnova
(Translated byAbeShenitzer)
24. MathematicalChestnutsfrom AroundtheWorld,RossHonsberger
25. Counting on Frameworks:Mathematics to Aid the Design of Rigid Structures,
JackE. Graver
26. MathematicalDiamonds,RossHonsberger
27. Proofs that Really Count: The Art of Combinatorial Proof, Arthur T. Benjamin
andJennifer J.Quinn
28. MathematicalDelights, RossHonsberger
29. Conics,Keith Kendig
30. Hesiod’sAnvil:fallingandspinningthroughheavenandearth,AndrewJ.Simo-
son
31. A GardenofIntegrals,FrankE.Burk
32. A Guideto ComplexVariables(MAA Guides#1),Steven G. Krantz
33. SinkorFloat?ThoughtProblemsin MathandPhysics,Keith Kendig
34. Biscuitsof NumberTheory,Arthur T. BenjaminandEzraBrown
35. UncommonMathematicalExcursions:PolynomiaandRelatedRealms,DanKalman
36. When Lessis More: Visualizing BasicInequalities, Claudi Alsina andRoger B.
Nelsen
37. A Guideto AdvancedRealAnalysis(MAA Guides#2),Gerald B.Folland
38. A Guideto RealVariables(MAA Guides#3), Steven G.Krantz
39. Voltaire’s Riddle: Microme(cid:19)gas and the measure of all things, Andrew J.
Simoson
40. A Guideto Topology,(MAA Guides#4), StevenG. Krantz
41. A Guideto ElementaryNumberTheory,(MAA Guides#5), UnderwoodDudley
42. CharmingProofs:AJourneyintoElegantMathematics,ClaudiAlsinaandRoger
B. Nelsen
43. MathematicsandSports,editedbyJosephA. Gallian
44. A Guideto AdvancedLinearAlgebra,(MAA Guides#6),Steven H. Weintraub
45. Iconsof Mathematics:AnExplorationofTwentyKeyImages,ClaudiAlsinaand
Roger B.Nelsen
46. A Guideto PlaneAlgebraicCurves,(MAA Guides#7),Keith Kendig
47. NewHorizonsinGeometry,Tom M. Apostol andMamikonA. Mnatsakanian
48. A Guideto Groups,Rings,andFields,(MAA Guides#8),FernandoQ. Gouve^a
49. A Guideto FunctionalAnalysis,(MAA Guides#9),Steven G. Krantz
MathematicalAssociationof America
P.O. Box 91112
Washington,DC20090-1112
1-800-331-1MAA FAX: [phone]
PREFACE
Is it fun to solve problems, and is solving problems about somethinga good
way to learn something?The answers seem to be yes, providedtheproblems
are neithertoo hard nor too easy.
Thebook is addressed to students(and teachers) of undergraduatelinear
algebra|itmightsupplementbutnot(I hope)replacemyoldFinite-Dimen-
sionalVectorSpaces.Itlargelyfollowsthatoldbookinorganizationandlevel
and order|but only \largely"|the principle is often violated. This is not a
step-by-step textbook|the problems vary back and forth between subjects,
theyvaryback and forthfromeasy tohardand back again. Thelocationof a
problem is not always a hint to what methods might be appropriate to solve
it or how hard it is.
Words like \hard" and \easy" are subjective of course. I tried to make
some of the problems accessible to any interested grade school student, and
at the same time to insert some that might stump even a professional expert
(at least for a minute or two). Correspondingly, the statements of the prob-
lems, and the introductions that precede and the solutions that follow them
sometimes laboriously explain elementary concepts, and, at other times as-
sume that you are at home with the language and attitude of mathematics
at the research level. Example: sometimes I assume that you know nothing,
and carefullyexplaintheassociative law, butat othertimesI assume thatthe
word \topology", while it may not refer to something that you are an expert
in, refers to something that you have heard about.
The solutionsare intrinsicparts of the exposition.You are urged to look
at the solution of each problem even if you can solve the problem without
doing so|the solution sometimes contains comments that couldn’t be made
inthestatementof theproblem,or even inthehint,withoutgiving too much
of the show away.
vii
viii LINEARALGEBRAPROBLEMBOOK
I hope youwillenjoy tryingto solve theproblems, I hope youwilllearn
something by doing so, and I hope you will have fun.
CONTENTS
Preface .............................................................vi
Chapter 1. Scalars ...................................................1
1. Double addition
2. Half doubleaddition
3. Exponentiation
4. Complex numbers
5. Affine transformations
6. Matrix multiplication
7. Modular multiplication
8. Small operations
9. Identityelements
10. Complex inverses
11. Affine inverses
12. Matrix inverses
13. Abelian groups
14. Groups
15. Independentgroup axioms
16. Fields
17. Additionand multiplicationin fields
18. Distributivefailure
19. Finitefields
Chapter 2. Vectors ..................................................17
20. Vector spaces
21. Examples
22. Linear combinations
23. Subspaces
24. Unions of subspaces
ix