Algebra and Geometry

340 Pages·2005·4.38 MB·English

by  Alan F. Beardon

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ALGEBRAANDGEOMETRY
Thistextgivesabasicintroductionandaunifiedapproachtoalgebraandgeometry.It
coverstheideasofcomplexnumbers,scalarandvectorproducts,determinants,linear
algebra,grouptheory,permutationgroups,symmetrygroupsandvariousaspectsof
geometryincludinggroupsofisometries,rotationsandsphericalgeometry.The
emphasisisalwaysontheinteractionbetweenthesetopics,andeachoneisconstantly
illustratedbyusingittodescribeanddiscusstheothers.Manyoftheideasare
developedgraduallythroughoutthebook.Forexample,thedefinitionofagroupis
giveninChapter1sothatitcanbeusedinadiscussionofthearithmeticofrealand
complexnumbers;however,manyofthepropertiesofgroupsaregivenlater,andata
timewhentheimportanceoftheconcepthasbecomeclear.Thetextisdividedinto
shortsections,withexercisesattheendofeachone.

ALGEBRA AND GEOMETRY
ALAN F. BEARDON
  
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Cambridge University Press
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ToDylan,Harry,FionnandFenella

Contents
Preface pagexi
1 Groupsandpermutations 1
1.1 Introduction 1
1.2 Groups 2
1.3 Permutationsofafiniteset 6
1.4 Thesignofapermutation 11
1.5 Permutationsofanarbitraryset 15
2 Therealnumbers 22
2.1 Theintegers 22
2.2 Therealnumbers 26
2.3 Fields 27
2.4 Modulararithmetic 28
3 Thecomplexplane 31
3.1 Complexnumbers 31
3.2 Polarcoordinates 36
3.3 Linesandcircles 40
3.4 Isometriesoftheplane 41
3.5 Rootsofunity 44
3.6 Cubicandquarticequations 46
3.7 TheFundamentalTheoremofAlgebra 48
4 Vectorsinthree-dimensionalspace 52
4.1 Vectors 52
4.2 Thescalarproduct 55
4.3 Thevectorproduct 57
4.4 Thescalartripleproduct 60
vii
viii Contents
4.5 Thevectortripleproduct 62
4.6 Orientationanddeterminants 63
4.7 Applicationstogeometry 68
4.8 Vectorequations 72
5 Sphericalgeometry 74
5.1 Sphericaldistance 74
5.2 Sphericaltrigonometry 75
5.3 Areaonthesphere 77
5.4 Euler’sformula 79
5.5 Regularpolyhedra 83
5.6 Generalpolyhedra 85
6 Quaternionsandisometries 89
6.1 IsometriesofEuclideanspace 89
6.2 Quaternions 95
6.3 Reflectionsandrotations 99
7 Vectorspaces 102
7.1 Vectorspaces 102
7.2 Dimension 106
7.3 Subspaces 111
7.4 Thedirectsumoftwosubspaces 115
7.5 Lineardifferenceequations 118
7.6 Thevectorspaceofpolynomials 120
7.7 Lineartransformations 124
7.8 Thekernelofalineartransformation 127
7.9 Isomorphisms 130
7.10 Thespaceoflinearmaps 132
8 Linearequations 135
8.1 Hyperplanes 135
8.2 Homogeneouslinearequations 136
8.3 Rowrankandcolumnrank 139
8.4 Inhomogeneouslinearequations 141
8.5 Determinantsandlinearequations 143
8.6 Determinants 144
9 Matrices 149
9.1 Thevectorspaceofmatrices 149
9.2 Amatrixasalineartransformation 154
9.3 Thematrixofalineartransformation 158