A first course in linear algebra

430 LTSM Linear Transformation Scalar Multiplication.. 176Section MM SLEMM Systems of Linear Equations as Matrix Multiplication.. 423MLTCV Matrix of a Linear Transformation, Column Vecto

Trang 1

A First Course in Linear Algebra

Trang 3

A First Course in Linear Algebra

by Robert A Beezer Department of Mathematics and Computer Science

University of Puget Sound

Waldron EditionVersion 2.00 ContentJuly 16, 2008

Trang 4

Santa Clara in 1978, a M.S in Statistics from the University of Illinois at Urbana-Champaign in 1982 and a Ph.D.

in Mathematics from the University of Illinois at Urbana-Champaign in 1984 He teaches calculus, linear algebraand abstract algebra regularly, while his research interests include the applications of linear algebra to graph theory.His professional website is athttp://buzzard.ups.edu

Edition

Waldron Edition, July 16, 2008

Based on Version 2.00

Front Cover

“The Summer Chair” c

Gentium Font, SIL Open Font License

Back Cover

Author photo by Ross Mulhausen

Computer Modern Font, Public Domain

Publisher

Robert A Beezer

Department of Mathematics and Computer Science

University of Puget Sound

Docu-“GNU Free Documentation License”

The most recent version of this work can always be found athttp://linear.ups.edu

Trang 5

To my wife, Pat.

Trang 7

Part C Core

WILA What is Linear Algebra? 3

LA “Linear” + “Algebra” 3

AA An Application 4

READ Reading Questions 7

EXC Exercises 8

SOL Solutions 9

SSLE Solving Systems of Linear Equations 11

SLE Systems of Linear Equations 11

PSS Possibilities for Solution Sets 12

ESEO Equivalent Systems and Equation Operations 13

EXC Exercises 18

SOL Solutions 20

RREF Reduced Row-Echelon Form 23

MVNSE Matrix and Vector Notation for Systems of Equations 23

RO Row Operations 26

EXC Exercises 37

SOL Solutions 41

TSS Types of Solution Sets 47

CS Consistent Systems 47

FV Free Variables 51

EXC Exercises 54

SOL Solutions 55

vii

Trang 8

HSE Homogeneous Systems of Equations 57

SHS Solutions of Homogeneous Systems 57

NSM Null Space of a Matrix 59

EXC Exercises 62

SOL Solutions 64

NM Nonsingular Matrices 67

NSNM Null Space of a Nonsingular Matrix 69

EXC Exercises 72

SOL Solutions 74

SLE Systems of Linear Equations 77

Chapter V Vectors 79 VO Vector Operations 79

VEASM Vector Equality, Addition, Scalar Multiplication 79

VSP Vector Space Properties 81

EXC Exercises 84

SOL Solutions 85

LC Linear Combinations 87

VFSS Vector Form of Solution Sets 91

PSHS Particular Solutions, Homogeneous Solutions 100

EXC Exercises 103

SOL Solutions 105

SS Spanning Sets 107

SSV Span of a Set of Vectors 107

SSNS Spanning Sets of Null Spaces 112

EXC Exercises 116

SOL Solutions 118

LI Linear Independence 123

LISV Linearly Independent Sets of Vectors 123

LINM Linear Independence and Nonsingular Matrices 127

NSSLI Null Spaces, Spans, Linear Independence 128

EXC Exercises 132

SOL Solutions 135

LDS Linear Dependence and Spans 141

LDSS Linearly Dependent Sets and Spans 141

COV Casting Out Vectors 143

EXC Exercises 149

SOL Solutions 150

O Orthogonality 153

CAV Complex Arithmetic and Vectors 153

IP Inner products 154

N Norm 156

OV Orthogonal Vectors 158

GSP Gram-Schmidt Procedure 160

EXC Exercises 164

SOL Solutions 165

V Vectors 167

Trang 9

CONTENTS ix

MO Matrix Operations 169

MEASM Matrix Equality, Addition, Scalar Multiplication 169

TSM Transposes and Symmetric Matrices 171

MCC Matrices and Complex Conjugation 173

AM Adjoint of a Matrix 175

EXC Exercises 177

SOL Solutions 179

MM Matrix Multiplication 181

MVP Matrix-Vector Product 181

MMEE Matrix Multiplication, Entry-by-Entry 185

PMM Properties of Matrix Multiplication 186

HM Hermitian Matrices 190

EXC Exercises 192

SOL Solutions 193

MISLE Matrix Inverses and Systems of Linear Equations 195

IM Inverse of a Matrix 196

CIM Computing the Inverse of a Matrix 197

PMI Properties of Matrix Inverses 201

EXC Exercises 204

SOL Solutions 206

MINM Matrix Inverses and Nonsingular Matrices 209

NMI Nonsingular Matrices are Invertible 209

UM Unitary Matrices 211

EXC Exercises 216

SOL Solutions 217

CRS Column and Row Spaces 219

CSSE Column Spaces and Systems of Equations 219

CSSOC Column Space Spanned by Original Columns 221

CSNM Column Space of a Nonsingular Matrix 223

RSM Row Space of a Matrix 225

EXC Exercises 231

SOL Solutions 235

FS Four Subsets 239

LNS Left Null Space 239

CRS Computing Column Spaces 240

EEF Extended echelon form 242

FS Four Subsets 244

EXC Exercises 253

SOL Solutions 255

M Matrices 259

Chapter VS Vector Spaces 261 VS Vector Spaces 261

VS Vector Spaces 261

EVS Examples of Vector Spaces 262

RD Recycling Definitions 269

EXC Exercises 270

SOL Solutions 271

S Subspaces 273

Trang 10

TS Testing Subspaces 274

TSS The Span of a Set 277

SC Subspace Constructions 282

EXC Exercises 283

SOL Solutions 284

LISS Linear Independence and Spanning Sets 287

SS Spanning Sets 291

VR Vector Representation 294

EXC Exercises 297

SOL Solutions 299

B Bases 303

BSCV Bases for Spans of Column Vectors 306

BNM Bases and Nonsingular Matrices 307

OBC Orthonormal Bases and Coordinates 308

EXC Exercises 313

SOL Solutions 314

D Dimension 317

DVS Dimension of Vector Spaces 320

RNM Rank and Nullity of a Matrix 322

RNNM Rank and Nullity of a Nonsingular Matrix 323

EXC Exercises 325

SOL Solutions 326

PD Properties of Dimension 329

GT Goldilocks’ Theorem 329

RT Ranks and Transposes 332

DFS Dimension of Four Subspaces 333

DS Direct Sums 334

EXC Exercises 339

SOL Solutions 340

VS Vector Spaces 341

Chapter D Determinants 343 DM Determinant of a Matrix 343

EM Elementary Matrices 343

DD Definition of the Determinant 347

CD Computing Determinants 348

EXC Exercises 352

SOL Solutions 353

PDM Properties of Determinants of Matrices 355

DRO Determinants and Row Operations 355

DROEM Determinants, Row Operations, Elementary Matrices 359

DNMMM Determinants, Nonsingular Matrices, Matrix Multiplication 360

EXC Exercises 363

SOL Solutions 364

D Determinants 365

Chapter E Eigenvalues 367 EE Eigenvalues and Eigenvectors 367

EEM Eigenvalues and Eigenvectors of a Matrix 367

Trang 11

CONTENTS xi

PM Polynomials and Matrices 369

EEE Existence of Eigenvalues and Eigenvectors 370

CEE Computing Eigenvalues and Eigenvectors 373

ECEE Examples of Computing Eigenvalues and Eigenvectors 376

EXC Exercises 383

SOL Solutions 384

PEE Properties of Eigenvalues and Eigenvectors 387

ME Multiplicities of Eigenvalues 391

EHM Eigenvalues of Hermitian Matrices 394

EXC Exercises 396

SOL Solutions 397

SD Similarity and Diagonalization 399

SM Similar Matrices 399

PSM Properties of Similar Matrices 400

D Diagonalization 402

FS Fibonacci Sequences 408

EXC Exercises 411

SOL Solutions 412

E Eigenvalues 415

Chapter LT Linear Transformations 417 LT Linear Transformations 417

LT Linear Transformations 417

LTC Linear Transformation Cartoons 421

MLT Matrices and Linear Transformations 421

LTLC Linear Transformations and Linear Combinations 425

PI Pre-Images 428

NLTFO New Linear Transformations From Old 430

EXC Exercises 434

SOL Solutions 436

ILT Injective Linear Transformations 439

EILT Examples of Injective Linear Transformations 439

KLT Kernel of a Linear Transformation 442

ILTLI Injective Linear Transformations and Linear Independence 446

ILTD Injective Linear Transformations and Dimension 447

CILT Composition of Injective Linear Transformations 447

EXC Exercises 449

SOL Solutions 450

SLT Surjective Linear Transformations 453

ESLT Examples of Surjective Linear Transformations 453

RLT Range of a Linear Transformation 456

SSSLT Spanning Sets and Surjective Linear Transformations 460

SLTD Surjective Linear Transformations and Dimension 462

CSLT Composition of Surjective Linear Transformations 462

EXC Exercises 464

SOL Solutions 466

IVLT Invertible Linear Transformations 469

IV Invertibility 472

SI Structure and Isomorphism 475

RNLT Rank and Nullity of a Linear Transformation 476

SLELT Systems of Linear Equations and Linear Transformations 479

Trang 12

EXC Exercises 481

SOL Solutions 483

LT Linear Transformations 487

Chapter R Representations 489 VR Vector Representations 489

CVS Characterization of Vector Spaces 493

CP Coordinatization Principle 494

EXC Exercises 498

SOL Solutions 499

MR Matrix Representations 501

NRFO New Representations from Old 506

PMR Properties of Matrix Representations 510

EXC Exercises 518

SOL Solutions 520

CB Change of Basis 529

EELT Eigenvalues and Eigenvectors of Linear Transformations 529

CBM Change-of-Basis Matrix 530

MRS Matrix Representations and Similarity 535

CELT Computing Eigenvectors of Linear Transformations 540

EXC Exercises 548

SOL Solutions 549

OD Orthonormal Diagonalization 553

TM Triangular Matrices 553

UTMR Upper Triangular Matrix Representation 554

NM Normal Matrices 557

OD Orthonormal Diagonalization 558

NLT Nilpotent Linear Transformations 561

PNLT Properties of Nilpotent Linear Transformations 565

CFNLT Canonical Form for Nilpotent Linear Transformations 569

IS Invariant Subspaces 577

GEE Generalized Eigenvectors and Eigenspaces 580

RLT Restrictions of Linear Transformations 583

JCF Jordan Canonical Form 593

GESD Generalized Eigenspace Decomposition 593

JCF Jordan Canonical Form 598

CHT Cayley-Hamilton Theorem 609

R Representations 611

Appendix CN Computation Notes 613 MMA Mathematica 613

ME.MMA Matrix Entry 613

RR.MMA Row Reduce 613

LS.MMA Linear Solve 613

VLC.MMA Vector Linear Combinations 614

NS.MMA Null Space 614

VFSS.MMA Vector Form of Solution Set 615

GSP.MMA Gram-Schmidt Procedure 616

TM.MMA Transpose of a Matrix 616

MM.MMA Matrix Multiplication 616

MI.MMA Matrix Inverse 617

SAGE SAGE: Open Source Mathematics Software 617

R.SAGE Rings 617

Trang 13

CONTENTS xiii

ME.SAGE Matrix Entry 617

MI.SAGE Matrix Inverse 618

TM.SAGE Transpose of a Matrix 618

E.SAGE Eigenspaces 618

Appendix P Preliminaries 621 CNO Complex Number Operations 621

CNA Arithmetic with complex numbers 621

CCN Conjugates of Complex Numbers 623

MCN Modulus of a Complex Number 624

SET Sets 625

SC Set Cardinality 626

SO Set Operations 626

PT Proof Techniques 629

D Definitions 629

T Theorems 629

L Language 630

GS Getting Started 631

C Constructive Proofs 631

E Equivalences 631

N Negation 632

CP Contrapositives 632

CV Converses 632

CD Contradiction 633

U Uniqueness 633

ME Multiple Equivalences 634

PI Proving Identities 634

DC Decompositions 634

I Induction 635

P Practice 636

LC Lemmas and Corollaries 636

Appendix A Archetypes 639 A 643

B 647

C 651

D 654

E 657

F 660

G 665

H 669

I 673

J 677

K 681

L 684

M 687

N 689

O 691

P 693

Q 695

R 698

S 701

T 703

U 705

V 707

W 709

X 711

Trang 14

1 APPLICABILITY AND DEFINITIONS 715

2 VERBATIM COPYING 716

3 COPYING IN QUANTITY 716

4 MODIFICATIONS 717

5 COMBINING DOCUMENTS 718

6 COLLECTIONS OF DOCUMENTS 718

7 AGGREGATION WITH INDEPENDENT WORKS 718

8 TRANSLATION 718

9 TERMINATION 718

10 FUTURE REVISIONS OF THIS LICENSE 719

ADDENDUM: How to use this License for your documents 719

Index

Trang 15

Beezer, David Belarmine Preparatory School, Tacoma

Beezer, Robert University of Puget Sound http://buzzard.ups.edu/

Bucht, Sara University of Puget Sound

Canfield, Steve University of Puget Sound

Hubert, Dupont Cr´eteil, France

Fellez, Sarah University of Puget Sound

Fickenscher, Eric University of Puget Sound

Jackson, Martin University of Puget Sound http://www.math.ups.edu/ ˜martinjHamrick, Mark St Louis University

Linenthal, Jacob University of Puget Sound

Million, Elizabeth University of Puget Sound

Osborne, Travis University of Puget Sound

Riegsecker, Joe Middlebury, Indiana joepye (at) pobox (dot) com

Phelps, Douglas University of Puget Sound

Shoemaker, Mark University of Puget Sound

Zimmer, Andy University of Puget Sound

xv

Trang 17

Section WILA

Section SSLE

SLE System of Linear Equations 11

ESYS Equivalent Systems 13

EO Equation Operations 13

Section RREF M Matrix 23

CV Column Vector 23

ZCV Zero Column Vector 24

CM Coefficient Matrix 24

VOC Vector of Constants 24

SOLV Solution Vector 24

MRLS Matrix Representation of a Linear System 25

AM Augmented Matrix 25

RO Row Operations 26

REM Row-Equivalent Matrices 26

RR Row-Reducing 36

Section TSS CS Consistent System 47

IDV Independent and Dependent Variables 49

Section HSE HS Homogeneous System 57

TSHSE Trivial Solution to Homogeneous Systems of Equations 57

NSM Null Space of a Matrix 59

Section NM SQM Square Matrix 67

NM Nonsingular Matrix 67

IM Identity Matrix 68

Section VO VSCV Vector Space of Column Vectors 79

CVE Column Vector Equality 79

CVA Column Vector Addition 80

CVSM Column Vector Scalar Multiplication 81

Section LC LCCV Linear Combination of Column Vectors 87

Section SS SSCV Span of a Set of Column Vectors 107

xvii

Trang 18

Section LI

RLDCV Relation of Linear Dependence for Column Vectors 123

LICV Linear Independence of Column Vectors 123

Section LDS Section O CCCV Complex Conjugate of a Column Vector 153

IP Inner Product 154

NV Norm of a Vector 156

OV Orthogonal Vectors 158

OSV Orthogonal Set of Vectors 158

SUV Standard Unit Vectors 158

ONS OrthoNormal Set 162

Section MO VSM Vector Space of m × n Matrices 169

ME Matrix Equality 169

MA Matrix Addition 169

MSM Matrix Scalar Multiplication 170

ZM Zero Matrix 171

TM Transpose of a Matrix 172

SYM Symmetric Matrix 172

CCM Complex Conjugate of a Matrix 173

A Adjoint 175

Section MM MVP Matrix-Vector Product 181

HM Hermitian Matrix 190

Section MISLE MI Matrix Inverse 196

Section MINM UM Unitary Matrices 211

Section CRS CSM Column Space of a Matrix 219

RSM Row Space of a Matrix 225

Section FS LNS Left Null Space 239

EEF Extended Echelon Form 242

Section VS VS Vector Space 261

Section S S Subspace 273

TS Trivial Subspaces 276

LC Linear Combination 277

SS Span of a Set 278

Section LISS RLD Relation of Linear Dependence 287

TSVS To Span a Vector Space 291

Trang 19

DEFINITIONS xix

Section B

B Basis 303

Section D D Dimension 317

NOM Nullity Of a Matrix 322

ROM Rank Of a Matrix 322

Section PD DS Direct Sum 334

Section DM ELEM Elementary Matrices 343

SM SubMatrix 347

DM Determinant of a Matrix 347

Section PDM Section EE EEM Eigenvalues and Eigenvectors of a Matrix 367

CP Characteristic Polynomial 373

EM Eigenspace of a Matrix 374

AME Algebraic Multiplicity of an Eigenvalue 376

GME Geometric Multiplicity of an Eigenvalue 376

Section PEE Section SD SIM Similar Matrices 399

DIM Diagonal Matrix 402

DZM Diagonalizable Matrix 402

Section LT LT Linear Transformation 417

PI Pre-Image 428

LTA Linear Transformation Addition 430

LTSM Linear Transformation Scalar Multiplication 431

LTC Linear Transformation Composition 432

Section ILT ILT Injective Linear Transformation 439

KLT Kernel of a Linear Transformation 442

Section SLT SLT Surjective Linear Transformation 453

RLT Range of a Linear Transformation 456

Section IVLT IDLT Identity Linear Transformation 469

IVS Isomorphic Vector Spaces 475

ROLT Rank Of a Linear Transformation 476

NOLT Nullity Of a Linear Transformation 476

Section VR VR Vector Representation 489

Section MR MR Matrix Representation 501

Trang 20

Section CB

EELT Eigenvalue and Eigenvector of a Linear Transformation 529

CBM Change-of-Basis Matrix 530

Section OD UTM Upper Triangular Matrix 553

LTM Lower Triangular Matrix 553

NRML Normal Matrix 557

Section NLT NLT Nilpotent Linear Transformation 561

JB Jordan Block 563

Section IS IS Invariant Subspace 577

GEV Generalized Eigenvector 580

GES Generalized Eigenspace 580

LTR Linear Transformation Restriction 584

IE Index of an Eigenvalue 588

Section JCF JCF Jordan Canonical Form 598

Section CNO CNE Complex Number Equality 622

CNA Complex Number Addition 622

CNM Complex Number Multiplication 622

CCN Conjugate of a Complex Number 623

MCN Modulus of a Complex Number 624

Section SET SET Set 625

SSET Subset 625

ES Empty Set 625

SE Set Equality 625

C Cardinality 626

SU Set Union 626

SI Set Intersection 627

SC Set Complement 627 Section PT

Trang 21

Section WILA

Section SSLE

EOPSS Equation Operations Preserve Solution Sets 13

Section RREF REMES Row-Equivalent Matrices represent Equivalent Systems 27

REMEF Row-Equivalent Matrix in Echelon Form 29

RREFU Reduced Row-Echelon Form is Unique 30

Section TSS RCLS Recognizing Consistency of a Linear System 50

ISRN Inconsistent Systems, r and n 50

CSRN Consistent Systems, r and n 51

FVCS Free Variables for Consistent Systems 51

PSSLS Possible Solution Sets for Linear Systems 52

CMVEI Consistent, More Variables than Equations, Infinite solutions 52

Section HSE HSC Homogeneous Systems are Consistent 57

HMVEI Homogeneous, More Variables than Equations, Infinite solutions 59

Section NM NMRRI Nonsingular Matrices Row Reduce to the Identity matrix 68

NMTNS Nonsingular Matrices have Trivial Null Spaces 70

NMUS Nonsingular Matrices and Unique Solutions 70

NME1 Nonsingular Matrix Equivalences, Round 1 70

Section VO VSPCV Vector Space Properties of Column Vectors 82

Section LC SLSLC Solutions to Linear Systems are Linear Combinations 90

VFSLS Vector Form of Solutions to Linear Systems 95

PSPHS Particular Solution Plus Homogeneous Solutions 100

Section SS SSNS Spanning Sets for Null Spaces 112

Section LI LIVHS Linearly Independent Vectors and Homogeneous Systems 125

LIVRN Linearly Independent Vectors, r and n 126

MVSLD More Vectors than Size implies Linear Dependence 127

NMLIC Nonsingular Matrices have Linearly Independent Columns 128

NME2 Nonsingular Matrix Equivalences, Round 2 128

BNS Basis for Null Spaces 129

xxi

Trang 22

Section LDS

DLDS Dependency in Linearly Dependent Sets 141

BS Basis of a Span 145Section O

CRVA Conjugation Respects Vector Addition 153CRSM Conjugation Respects Vector Scalar Multiplication 153IPVA Inner Product and Vector Addition 155IPSM Inner Product and Scalar Multiplication 155IPAC Inner Product is Anti-Commutative 156IPN Inner Products and Norms 157PIP Positive Inner Products 157OSLI Orthogonal Sets are Linearly Independent 159GSP Gram-Schmidt Procedure 160Section MO

VSPM Vector Space Properties of Matrices 170SMS Symmetric Matrices are Square 172TMA Transpose and Matrix Addition 173TMSM Transpose and Matrix Scalar Multiplication 173

TT Transpose of a Transpose 173CRMA Conjugation Respects Matrix Addition 174CRMSM Conjugation Respects Matrix Scalar Multiplication 174CCM Conjugate of the Conjugate of a Matrix 174MCT Matrix Conjugation and Transposes 175AMA Adjoint and Matrix Addition 175AMSM Adjoint and Matrix Scalar Multiplication 175

AA Adjoint of an Adjoint 176Section MM

SLEMM Systems of Linear Equations as Matrix Multiplication 181EMMVP Equal Matrices and Matrix-Vector Products 183EMP Entries of Matrix Products 185MMZM Matrix Multiplication and the Zero Matrix 186MMIM Matrix Multiplication and Identity Matrix 186MMDAA Matrix Multiplication Distributes Across Addition 187MMSMM Matrix Multiplication and Scalar Matrix Multiplication 187MMA Matrix Multiplication is Associative 187MMIP Matrix Multiplication and Inner Products 188MMCC Matrix Multiplication and Complex Conjugation 188MMT Matrix Multiplication and Transposes 189MMAD Matrix Multiplication and Adjoints 189AIP Adjoint and Inner Product 190HMIP Hermitian Matrices and Inner Products 190Section MISLE

TTMI Two-by-Two Matrix Inverse 197CINM Computing the Inverse of a Nonsingular Matrix 199MIU Matrix Inverse is Unique 201

SS Socks and Shoes 201MIMI Matrix Inverse of a Matrix Inverse 202MIT Matrix Inverse of a Transpose 202MISM Matrix Inverse of a Scalar Multiple 202Section MINM

NPNT Nonsingular Product has Nonsingular Terms 209OSIS One-Sided Inverse is Sufficient 210

NI Nonsingularity is Invertibility 210

Trang 23

THEOREMS xxiii

NME3 Nonsingular Matrix Equivalences, Round 3 211SNCM Solution with Nonsingular Coefficient Matrix 211UMI Unitary Matrices are Invertible 212CUMOS Columns of Unitary Matrices are Orthonormal Sets 212UMPIP Unitary Matrices Preserve Inner Products 213Section CRS

CSCS Column Spaces and Consistent Systems 220BCS Basis of the Column Space 222CSNM Column Space of a Nonsingular Matrix 224NME4 Nonsingular Matrix Equivalences, Round 4 224REMRS Row-Equivalent Matrices have equal Row Spaces 226BRS Basis for the Row Space 227CSRST Column Space, Row Space, Transpose 228Section FS

PEEF Properties of Extended Echelon Form 243

FS Four Subsets 244Section VS

ZVU Zero Vector is Unique 267AIU Additive Inverses are Unique 267ZSSM Zero Scalar in Scalar Multiplication 267ZVSM Zero Vector in Scalar Multiplication 267AISM Additive Inverses from Scalar Multiplication 268SMEZV Scalar Multiplication Equals the Zero Vector 268Section S

TSS Testing Subsets for Subspaces 274NSMS Null Space of a Matrix is a Subspace 276SSS Span of a Set is a Subspace 278CSMS Column Space of a Matrix is a Subspace 282RSMS Row Space of a Matrix is a Subspace 282LNSMS Left Null Space of a Matrix is a Subspace 282Section LISS

VRRB Vector Representation Relative to a Basis 294Section B

SUVB Standard Unit Vectors are a Basis 303CNMB Columns of Nonsingular Matrix are a Basis 307NME5 Nonsingular Matrix Equivalences, Round 5 308COB Coordinates and Orthonormal Bases 309UMCOB Unitary Matrices Convert Orthonormal Bases 311Section D

SSLD Spanning Sets and Linear Dependence 317BIS Bases have Identical Sizes 320DCM Dimension of Cm 320

DP Dimension of Pn 320

DM Dimension of Mmn 320CRN Computing Rank and Nullity 323RPNC Rank Plus Nullity is Columns 323RNNM Rank and Nullity of a Nonsingular Matrix 324NME6 Nonsingular Matrix Equivalences, Round 6 324Section PD

ELIS Extending Linearly Independent Sets 329

Trang 24

G Goldilocks 329PSSD Proper Subspaces have Smaller Dimension 332EDYES Equal Dimensions Yields Equal Subspaces 332RMRT Rank of a Matrix is the Rank of the Transpose 332DFS Dimensions of Four Subspaces 333DSFB Direct Sum From a Basis 335DSFOS Direct Sum From One Subspace 335DSZV Direct Sums and Zero Vectors 335DSZI Direct Sums and Zero Intersection 336DSLI Direct Sums and Linear Independence 336DSD Direct Sums and Dimension 337RDS Repeated Direct Sums 337Section DM

EMDRO Elementary Matrices Do Row Operations 344EMN Elementary Matrices are Nonsingular 346NMPEM Nonsingular Matrices are Products of Elementary Matrices 346DMST Determinant of Matrices of Size Two 348DER Determinant Expansion about Rows 348

DT Determinant of the Transpose 349DEC Determinant Expansion about Columns 350Section PDM

DZRC Determinant with Zero Row or Column 355DRCS Determinant for Row or Column Swap 355DRCM Determinant for Row or Column Multiples 356DERC Determinant with Equal Rows or Columns 356DRCMA Determinant for Row or Column Multiples and Addition 357DIM Determinant of the Identity Matrix 359DEM Determinants of Elementary Matrices 359DEMMM Determinants, Elementary Matrices, Matrix Multiplication 360SMZD Singular Matrices have Zero Determinants 361NME7 Nonsingular Matrix Equivalences, Round 7 361DRMM Determinant Respects Matrix Multiplication 362Section EE

EMHE Every Matrix Has an Eigenvalue 370EMRCP Eigenvalues of a Matrix are Roots of Characteristic Polynomials 374EMS Eigenspace for a Matrix is a Subspace 374EMNS Eigenspace of a Matrix is a Null Space 375Section PEE

EDELI Eigenvectors with Distinct Eigenvalues are Linearly Independent 387SMZE Singular Matrices have Zero Eigenvalues 387NME8 Nonsingular Matrix Equivalences, Round 8 388ESMM Eigenvalues of a Scalar Multiple of a Matrix 388EOMP Eigenvalues Of Matrix Powers 388EPM Eigenvalues of the Polynomial of a Matrix 389EIM Eigenvalues of the Inverse of a Matrix 390ETM Eigenvalues of the Transpose of a Matrix 390ERMCP Eigenvalues of Real Matrices come in Conjugate Pairs 391DCP Degree of the Characteristic Polynomial 391NEM Number of Eigenvalues of a Matrix 392

ME Multiplicities of an Eigenvalue 392MNEM Maximum Number of Eigenvalues of a Matrix 393HMRE Hermitian Matrices have Real Eigenvalues 394HMOE Hermitian Matrices have Orthogonal Eigenvectors 394

Trang 25

LTTZZ Linear Transformations Take Zero to Zero 420MBLT Matrices Build Linear Transformations 423MLTCV Matrix of a Linear Transformation, Column Vectors 424LTLC Linear Transformations and Linear Combinations 425LTDB Linear Transformation Defined on a Basis 426SLTLT Sum of Linear Transformations is a Linear Transformation 430MLTLT Multiple of a Linear Transformation is a Linear Transformation 431VSLT Vector Space of Linear Transformations 432CLTLT Composition of Linear Transformations is a Linear Transformation 432Section ILT

KLTS Kernel of a Linear Transformation is a Subspace 443KPI Kernel and Pre-Image 444KILT Kernel of an Injective Linear Transformation 445ILTLI Injective Linear Transformations and Linear Independence 446ILTB Injective Linear Transformations and Bases 446ILTD Injective Linear Transformations and Dimension 447CILTI Composition of Injective Linear Transformations is Injective 448Section SLT

RLTS Range of a Linear Transformation is a Subspace 457RSLT Range of a Surjective Linear Transformation 459SSRLT Spanning Set for Range of a Linear Transformation 460RPI Range and Pre-Image 461SLTB Surjective Linear Transformations and Bases 461SLTD Surjective Linear Transformations and Dimension 462CSLTS Composition of Surjective Linear Transformations is Surjective 462Section IVLT

ILTLT Inverse of a Linear Transformation is a Linear Transformation 471IILT Inverse of an Invertible Linear Transformation 471ILTIS Invertible Linear Transformations are Injective and Surjective 472CIVLT Composition of Invertible Linear Transformations 474ICLT Inverse of a Composition of Linear Transformations 474IVSED Isomorphic Vector Spaces have Equal Dimension 476ROSLT Rank Of a Surjective Linear Transformation 477NOILT Nullity Of an Injective Linear Transformation 477RPNDD Rank Plus Nullity is Domain Dimension 477Section VR

VRLT Vector Representation is a Linear Transformation 489VRI Vector Representation is Injective 492VRS Vector Representation is Surjective 493VRILT Vector Representation is an Invertible Linear Transformation 493CFDVS Characterization of Finite Dimensional Vector Spaces 493IFDVS Isomorphism of Finite Dimensional Vector Spaces 494CLI Coordinatization and Linear Independence 494CSS Coordinatization and Spanning Sets 495Section MR

Trang 26

FTMR Fundamental Theorem of Matrix Representation 503MRSLT Matrix Representation of a Sum of Linear Transformations 506MRMLT Matrix Representation of a Multiple of a Linear Transformation 506MRCLT Matrix Representation of a Composition of Linear Transformations 507KNSI Kernel and Null Space Isomorphism 510RCSI Range and Column Space Isomorphism 512IMR Invertible Matrix Representations 514IMILT Invertible Matrices, Invertible Linear Transformation 516NME9 Nonsingular Matrix Equivalences, Round 9 517Section CB

CB Change-of-Basis 530ICBM Inverse of Change-of-Basis Matrix 531MRCB Matrix Representation and Change of Basis 535SCB Similarity and Change of Basis 537EER Eigenvalues, Eigenvectors, Representations 539Section OD

PTMT Product of Triangular Matrices is Triangular 553ITMT Inverse of a Triangular Matrix is Triangular 554UTMR Upper Triangular Matrix Representation 554OBUTR Orthonormal Basis for Upper Triangular Representation 556

OD Orthonormal Diagonalization 558OBNM Orthonormal Bases and Normal Matrices 560Section NLT

NJB Nilpotent Jordan Blocks 565ENLT Eigenvalues of Nilpotent Linear Transformations 566DNLT Diagonalizable Nilpotent Linear Transformations 566KPLT Kernels of Powers of Linear Transformations 566KPNLT Kernels of Powers of Nilpotent Linear Transformations 567CFNLT Canonical Form for Nilpotent Linear Transformations 569Section IS

EIS Eigenspaces are Invariant Subspaces 578KPIS Kernels of Powers are Invariant Subspaces 579GESIS Generalized Eigenspace is an Invariant Subspace 580GEK Generalized Eigenspace as a Kernel 581RGEN Restriction to Generalized Eigenspace is Nilpotent 588MRRGE Matrix Representation of a Restriction to a Generalized Eigenspace 590Section JCF

GESD Generalized Eigenspace Decomposition 593DGES Dimension of Generalized Eigenspaces 598JCFLT Jordan Canonical Form for a Linear Transformation 599CHT Cayley-Hamilton Theorem 609Section CNO

PCNA Properties of Complex Number Arithmetic 622CCRA Complex Conjugation Respects Addition 623CCRM Complex Conjugation Respects Multiplication 623CCT Complex Conjugation Twice 623Section SET

Section PT

Trang 27

M A: Matrix 23

MC [A]ij: Matrix Components 23

CV v: Column Vector 23CVC [v]i: Column Vector Components 23ZCV 0: Zero Column Vector 24MRLS LS(A, b): Matrix Representation of a Linear System 25

AM [A | b]: Augmented Matrix 25

RO Ri↔ Rj, αRi, αRi+ Rj: Row Operations 26RREFA r, D, F : Reduced Row-Echelon Form Analysis 28NSM N (A): Null Space of a Matrix 59

IM Im: Identity Matrix 68VSCV Cm: Vector Space of Column Vectors 79CVE u = v: Column Vector Equality 80CVA u + v: Column Vector Addition 80CVSM αu: Column Vector Scalar Multiplication 81SSV hSi: Span of a Set of Vectors 107CCCV u: Complex Conjugate of a Column Vector 153

IP hu, vi: Inner Product 154

NV kvk: Norm of a Vector 156SUV ei: Standard Unit Vectors 158VSM Mmn: Vector Space of Matrices 169

ME A = B: Matrix Equality 169

MA A + B: Matrix Addition 169MSM αA: Matrix Scalar Multiplication 170

ZM O: Zero Matrix 171

TM At: Transpose of a Matrix 172CCM A: Complex Conjugate of a Matrix 173

A A∗: Adjoint 175MVP Au: Matrix-Vector Product 181

MI A−1: Matrix Inverse 196CSM C(A): Column Space of a Matrix 219RSM R(A): Row Space of a Matrix 225LNS L(A): Left Null Space 239

D dim (V ): Dimension 317NOM n (A): Nullity of a Matrix 322ROM r (A): Rank of a Matrix 322

DS V = U ⊕ W : Direct Sum 334ELEM Ei,j, Ei(α), Ei,j(α): Elementary Matrix 344

SM A (i|j): SubMatrix 347

DM det (A), |A|: Determinant of a Matrix 347AME αA(λ): Algebraic Multiplicity of an Eigenvalue 376GME γA(λ): Geometric Multiplicity of an Eigenvalue 376

LT T : U 7→ V : Linear Transformation 417KLT K(T ): Kernel of a Linear Transformation 442RLT R(T ): Range of a Linear Transformation 456ROLT r (T ): Rank of a Linear Transformation 476

xxvii

Trang 28

NOLT n (T ): Nullity of a Linear Transformation 476

VR ρB(w): Vector Representation 489

B,C: Matrix Representation 501

JB Jn(λ): Jordan Block 563GES GT(λ): Generalized Eigenspace 580LTR T |U: Linear Transformation Restriction 584

IE ιT(λ): Index of an Eigenvalue 588CNE α = β: Complex Number Equality 622CNA α + β: Complex Number Addition 622CNM αβ: Complex Number Multiplication 622CCN c: Conjugate of a Complex Number 623SETM x ∈ S: Set Membership 625SSET S ⊆ T : Subset 625

Trang 29

DTSLS Decision Tree for Solving Linear Systems 52CSRST Column Space and Row Space Techniques 251DLTA Definition of Linear Transformation, Additive 418DLTM Definition of Linear Transformation, Multiplicative 418GLT General Linear Transformation 421NILT Non-Injective Linear Transformation 440ILT Injective Linear Transformation 441FTMR Fundamental Theorem of Matrix Representations 504FTMRA Fundamental Theorem of Matrix Representations (Alternate) 504MRCLT Matrix Representation and Composition of Linear Transformations 510

xxix

Trang 31

US Three equations, one solution 15

IS Three equations, infinitely many solutions 16Section RREF

AM A matrix 23NSLE Notation for systems of linear equations 25AMAA Augmented matrix for Archetype A 26TREM Two row-equivalent matrices 26USR Three equations, one solution, reprised 27RREF A matrix in reduced row-echelon form 28NRREF A matrix not in reduced row-echelon form 28SAB Solutions for Archetype B 33SAA Solutions for Archetype A 34SAE Solutions for Archetype E 35Section TSS

RREFN Reduced row-echelon form notation 47ISSI Describing infinite solution sets, Archetype I 48FDV Free and dependent variables 49CFV Counting free variables 51OSGMD One solution gives many, Archetype D 52Section HSE

AHSAC Archetype C as a homogeneous system 57HUSAB Homogeneous, unique solution, Archetype B 57HISAA Homogeneous, infinite solutions, Archetype A 58HISAD Homogeneous, infinite solutions, Archetype D 58NSEAI Null space elements of Archetype I 59CNS1 Computing a null space, #1 60CNS2 Computing a null space, #2 60Section NM

S A singular matrix, Archetype A 67

NM A nonsingular matrix, Archetype B 67

IM An identity matrix 68SRR Singular matrix, row-reduced 68NSR Nonsingular matrix, row-reduced 69NSS Null space of a singular matrix 69NSNM Null space of a nonsingular matrix 69

xxxi

Trang 32

Section VO

VESE Vector equality for a system of equations 80

VA Addition of two vectors in C4 80CVSM Scalar multiplication in C5 81Section LC

TLC Two linear combinations in C6 87ABLC Archetype B as a linear combination 88AALC Archetype A as a linear combination 89VFSAD Vector form of solutions for Archetype D 91VFS Vector form of solutions 92VFSAI Vector form of solutions for Archetype I 97VFSAL Vector form of solutions for Archetype L 98PSHS Particular solutions, homogeneous solutions, Archetype D 100Section SS

ABS A basic span 107SCAA Span of the columns of Archetype A 109SCAB Span of the columns of Archetype B 110SSNS Spanning set of a null space 112NSDS Null space directly as a span 113SCAD Span of the columns of Archetype D 114Section LI

LDS Linearly dependent set in C5 123LIS Linearly independent set in C5 124LIHS Linearly independent, homogeneous system 125LDHS Linearly dependent, homogeneous system 125LDRN Linearly dependent, r < n 126LLDS Large linearly dependent set in C4 127LDCAA Linearly dependent columns in Archetype A 127LICAB Linearly independent columns in Archetype B 127LINSB Linear independence of null space basis 128NSLIL Null space spanned by linearly independent set, Archetype L 130Section LDS

RSC5 Reducing a span in C5 142COV Casting out vectors 143RSSC4 Reducing a span in C4 146RES Reworking elements of a span 147Section O

CSIP Computing some inner products 154CNSV Computing the norm of some vectors 156TOV Two orthogonal vectors 158SUVOS Standard Unit Vectors are an Orthogonal Set 158AOS An orthogonal set 159GSTV Gram-Schmidt of three vectors 161ONTV Orthonormal set, three vectors 162ONFV Orthonormal set, four vectors 162Section MO

MA Addition of two matrices in M23 170MSM Scalar multiplication in M32 170

TM Transpose of a 3 × 4 matrix 172SYM A symmetric 5 × 5 matrix 172CCM Complex conjugate of a matrix 174

Trang 33

EXAMPLES xxxiii

Section MM

MTV A matrix times a vector 181MNSLE Matrix notation for systems of linear equations 182MBC Money’s best cities 182PTM Product of two matrices 184MMNC Matrix multiplication is not commutative 184PTMEE Product of two matrices, entry-by-entry 185Section MISLE

SABMI Solutions to Archetype B with a matrix inverse 195MWIAA A matrix without an inverse, Archetype A 196

MI Matrix inverse 196CMI Computing a matrix inverse 198CMIAB Computing a matrix inverse, Archetype B 200Section MINM

UM3 Unitary matrix of size 3 212UPM Unitary permutation matrix 212OSMC Orthonormal set from matrix columns 213Section CRS

CSMCS Column space of a matrix and consistent systems 219MCSM Membership in the column space of a matrix 220CSTW Column space, two ways 221CSOCD Column space, original columns, Archetype D 222CSAA Column space of Archetype A 223CSAB Column space of Archetype B 224RSAI Row space of Archetype I 225RSREM Row spaces of two row-equivalent matrices 227IAS Improving a span 228CSROI Column space from row operations, Archetype I 229Section FS

LNS Left null space 239CSANS Column space as null space 240SEEF Submatrices of extended echelon form 242FS1 Four subsets, #1 248FS2 Four subsets, #2 248FSAG Four subsets, Archetype G 249Section VS

VSCV The vector space Cm 262VSM The vector space of matrices, Mmn 262VSP The vector space of polynomials, Pn 263VSIS The vector space of infinite sequences 264VSF The vector space of functions 264VSS The singleton vector space 264CVS The crazy vector space 265PCVS Properties for the Crazy Vector Space 268Section S

SC3 A subspace of C3 273SP4 A subspace of P4 275NSC2Z A non-subspace in C2, zero vector 276NSC2A A non-subspace in C2, additive closure 276NSC2S A non-subspace in C2, scalar multiplication closure 276RSNS Recasting a subspace as a null space 277

Trang 34

LCM A linear combination of matrices 278SSP Span of a set of polynomials 279SM32 A subspace of M32 280Section LISS

LIP4 Linear independence in P4 287LIM32 Linear independence in M32 288LIC Linearly independent set in the crazy vector space 290SSP4 Spanning set in P4 291SSM22 Spanning set in M22 292SSC Spanning set in the crazy vector space 293AVR A vector representation 294Section B

BP Bases for Pn 304

BM A basis for the vector space of matrices 304BSP4 A basis for a subspace of P4 304BSM22 A basis for a subspace of M22 305

BC Basis for the crazy vector space 305RSB Row space basis 306

RS Reducing a span 307CABAK Columns as Basis, Archetype K 308CROB4 Coordinatization relative to an orthonormal basis, C4 309CROB3 Coordinatization relative to an orthonormal basis, C3 310Section D

LDP4 Linearly dependent set in P4 319DSM22 Dimension of a subspace of M22 320DSP4 Dimension of a subspace of P4 321

DC Dimension of the crazy vector space 321VSPUD Vector space of polynomials with unbounded degree 322RNM Rank and nullity of a matrix 322RNSM Rank and nullity of a square matrix 323Section PD

BPR Bases for Pn, reprised 330BDM22 Basis by dimension in M22 330SVP4 Sets of vectors in P4 331RRTI Rank, rank of transpose, Archetype I 333SDS Simple direct sum 334Section DM

EMRO Elementary matrices and row operations 344

SS Some submatrices 347D33M Determinant of a 3 × 3 matrix 348TCSD Two computations, same determinant 350DUTM Determinant of an upper triangular matrix 351Section PDM

DRO Determinant by row operations 357ZNDAB Zero and nonzero determinant, Archetypes A and B 361Section EE

SEE Some eigenvalues and eigenvectors 367

PM Polynomial of a matrix 369CAEHW Computing an eigenvalue the hard way 371CPMS3 Characteristic polynomial of a matrix, size 3 373EMS3 Eigenvalues of a matrix, size 3 374

Trang 35

EXAMPLES xxxv

ESMS3 Eigenspaces of a matrix, size 3 375EMMS4 Eigenvalue multiplicities, matrix of size 4 376ESMS4 Eigenvalues, symmetric matrix of size 4 377HMEM5 High multiplicity eigenvalues, matrix of size 5 378CEMS6 Complex eigenvalues, matrix of size 6 378DEMS5 Distinct eigenvalues, matrix of size 5 380Section PEE

BDE Building desired eigenvalues 389Section SD

SMS5 Similar matrices of size 5 399SMS3 Similar matrices of size 3 399EENS Equal eigenvalues, not similar 401DAB Diagonalization of Archetype B 402DMS3 Diagonalizing a matrix of size 3 403NDMS4 A non-diagonalizable matrix of size 4 405DEHD Distinct eigenvalues, hence diagonalizable 406HPDM High power of a diagonalizable matrix 407FSCF Fibonacci sequence, closed form 408Section LT

ALT A linear transformation 418NLT Not a linear transformation 419LTPM Linear transformation, polynomials to matrices 419LTPP Linear transformation, polynomials to polynomials 420LTM Linear transformation from a matrix 421MFLT Matrix from a linear transformation 423MOLT Matrix of a linear transformation 424LTDB1 Linear transformation defined on a basis 427LTDB2 Linear transformation defined on a basis 427LTDB3 Linear transformation defined on a basis 428SPIAS Sample pre-images, Archetype S 428STLT Sum of two linear transformations 430SMLT Scalar multiple of a linear transformation 431CTLT Composition of two linear transformations 432Section ILT

NIAQ Not injective, Archetype Q 439IAR Injective, Archetype R 440IAV Injective, Archetype V 441NKAO Nontrivial kernel, Archetype O 442TKAP Trivial kernel, Archetype P 443NIAQR Not injective, Archetype Q, revisited 445NIAO Not injective, Archetype O 446IAP Injective, Archetype P 446NIDAU Not injective by dimension, Archetype U 447Section SLT

NSAQ Not surjective, Archetype Q 453SAR Surjective, Archetype R 454SAV Surjective, Archetype V 455RAO Range, Archetype O 456FRAN Full range, Archetype N 458NSAQR Not surjective, Archetype Q, revisited 459NSAO Not surjective, Archetype O 459SAN Surjective, Archetype N 460BRLT A basis for the range of a linear transformation 460

Trang 36

NSDAT Not surjective by dimension, Archetype T 462Section IVLT

AIVLT An invertible linear transformation 469ANILT A non-invertible linear transformation 470CIVLT Computing the Inverse of a Linear Transformations 473IVSAV Isomorphic vector spaces, Archetype V 475Section VR

VRC4 Vector representation in C4 490VRP2 Vector representations in P2 491TIVS Two isomorphic vector spaces 494CVSR Crazy vector space revealed 494ASC A subspace characterized 494MIVS Multiple isomorphic vector spaces 494CP2 Coordinatizing in P2 495CM32 Coordinatization in M32 496Section MR

OLTTR One linear transformation, three representations 501ALTMM A linear transformation as matrix multiplication 504MPMR Matrix product of matrix representations 507KVMR Kernel via matrix representation 511RVMR Range via matrix representation 513ILTVR Inverse of a linear transformation via a representation 515Section CB

ELTBM Eigenvectors of linear transformation between matrices 529ELTBP Eigenvectors of linear transformation between polynomials 529CBP Change of basis with polynomials 531CBCV Change of basis with column vectors 533MRCM Matrix representations and change-of-basis matrices 535MRBE Matrix representation with basis of eigenvectors 537ELTT Eigenvectors of a linear transformation, twice 540CELT Complex eigenvectors of a linear transformation 544Section OD

ANM A normal matrix 558Section NLT

NM64 Nilpotent matrix, size 6, index 4 561NM62 Nilpotent matrix, size 6, index 2 562JB4 Jordan block, size 4 563NJB5 Nilpotent Jordan block, size 5 563NM83 Nilpotent matrix, size 8, index 3 564KPNLT Kernels of powers of a nilpotent linear transformation 568CFNLT Canonical form for a nilpotent linear transformation 573Section IS

TIS Two invariant subspaces 577EIS Eigenspaces as invariant subspaces 578ISJB Invariant subspaces and Jordan blocks 579GE4 Generalized eigenspaces, dimension 4 domain 581GE6 Generalized eigenspaces, dimension 6 domain 582LTRGE Linear transformation restriction on generalized eigenspace 584ISMR4 Invariant subspaces, matrix representation, dimension 4 domain 586ISMR6 Invariant subspaces, matrix representation, dimension 6 domain 587GENR6 Generalized eigenspaces and nilpotent restrictions, dimension 6 domain 588

Trang 37

SETM Set membership 625SSET Subset 625

CS Cardinality and Size 626

SU Set union 627

SI Set intersection 627

SC Set complement 627Section PT

Trang 39

This textbook is designed to teach the university mathematics student the basics of linear algebra and the techniques

of formal mathematics There are no prerequisites other than ordinary algebra, but it is probably best used by

a student who has the “mathematical maturity” of a sophomore or junior The text has two goals: to teach thefundamental concepts and techniques of matrix algebra and abstract vector spaces, and to teach the techniquesassociated with understanding the definitions and theorems forming a coherent area of mathematics So there is

an emphasis on worked examples of nontrivial size and on proving theorems carefully

This book is copyrighted This means that governments have granted the author a monopoly — the exclusiveright to control the making of copies and derivative works for many years (too many years in some cases) Italso gives others limited rights, generally referred to as “fair use,” such as the right to quote sections in a reviewwithout seeking permission However, the author licenses this book to anyone under the terms of the GNU FreeDocumentation License (GFDL), which gives you more rights than most copyrights (see Appendix GFDL [715]).Loosely speaking, you may make as many copies as you like at no cost, and you may distribute these unmodifiedcopies if you please You may modify the book for your own use The catch is that if you make modificationsand you distribute the modified version, or make use of portions in excess of fair use in another work, then youmust also license the new work with the GFDL So the book has lots of inherent freedom, and no one is allowed

to distribute a derivative work that restricts these freedoms (See the license itself in the appendix for the exactdetails of the additional rights you have been given.)

Notice that initially most people are struck by the notion that this book is free (the French would say gratuit,

at no cost) And it is However, it is more important that the book has freedom (the French would say libert´e,liberty) It will never go “out of print” nor will there ever be trivial updates designed only to frustrate the usedbook market Those considering teaching a course with this book can examine it thoroughly in advance Addingnew exercises or new sections has been purposely made very easy, and the hope is that others will contribute thesemodifications back for incorporation into the book, for the benefit of all

Depending on how you received your copy, you may want to check for the latest version (and other news) at

http://linear.ups.edu/

Topics The first half of this text (through Chapter M [169]) is basically a course in matrix algebra, thoughthe foundation of some more advanced ideas is also being formed in these early sections Vectors are presentedexclusively as column vectors (since we also have the typographic freedom to avoid writing a column vector inline

as the transpose of a row vector), and linear combinations are presented very early Spans, null spaces, columnspaces and row spaces are also presented early, simply as sets, saving most of their vector space properties for later,

so they are familiar objects before being scrutinized carefully

You cannot do everything early, so in particular matrix multiplication comes later than usual However, with

a definition built on linear combinations of column vectors, it should seem more natural than the more frequentdefinition using dot products of rows with columns And this delay emphasizes that linear algebra is built uponvector addition and scalar multiplication Of course, matrix inverses must wait for matrix multiplication, but thisdoes not prevent nonsingular matrices from occurring sooner Vector space properties are hinted at when vectorand matrix operations are first defined, but the notion of a vector space is saved for a more axiomatic treatmentlater (Chapter VS [261]) Once bases and dimension have been explored in the context of vector spaces, lineartransformations and their matrix representations follow The goal of the book is to go as far as Jordan canonicalform in the Core (Part C [3]), with less central topics collected in the Topics (Part T [??]) A third part containscontributed applications (Part A [??]), with notation and theorems integrated with the earlier two parts

Linear algebra is an ideal subject for the novice mathematics student to learn how to develop a topic precisely,with all the rigor mathematics requires Unfortunately, much of this rigor seems to have escaped the standardcalculus curriculum, so for many university students this is their first exposure to careful definitions and theorems,and the expectation that they fully understand them, to say nothing of the expectation that they become proficient

in formulating their own proofs We have tried to make this text as helpful as possible with this transition Every

xxxix

Trang 40

definition is stated carefully, set apart from the text Likewise, every theorem is carefully stated, and almost everyone has a complete proof Theorems usually have just one conclusion, so they can be referenced precisely later.Definitions and theorems are cataloged in order of their appearance in the front of the book (Definitions [xvii],Theorems [xxi]), and alphabetical order in the index at the back Along the way, there are discussions of somemore important ideas relating to formulating proofs (Proof Techniques [??]), which is part advice and part logic.Origin and History This book is the result of the confluence of several related events and trends.

• At the University of Puget Sound we teach a one-semester, post-calculus linear algebra course to studentsmajoring in mathematics, computer science, physics, chemistry and economics Between January 1986 andJune 2002, I taught this course seventeen times For the Spring 2003 semester, I elected to convert mycourse notes to an electronic form so that it would be easier to incorporate the inevitable and nearly-constantrevisions Central to my new notes was a collection of stock examples that would be used repeatedly toillustrate new concepts (These would become the Archetypes, Appendix A [639].) It was only a short leap

to then decide to distribute copies of these notes and examples to the students in the two sections of thiscourse As the semester wore on, the notes began to look less like notes and more like a textbook

• I used the notes again in the Fall 2003 semester for a single section of the course Simultaneously, thetextbook I was using came out in a fifth edition A new chapter was added toward the start of the book,and a few additional exercises were added in other chapters This demanded the annoyance of reworking mynotes and list of suggested exercises to conform with the changed numbering of the chapters and exercises Ihad an almost identical experience with the third course I was teaching that semester I also learned that inthe next academic year I would be teaching a course where my textbook of choice had gone out of print Ifelt there had to be a better alternative to having the organization of my courses buffeted by the economics

of traditional textbook publishing

• I had used TEX and the Internet for many years, so there was little to stand in the way of typesetting, tributing and “marketing” a free book With recreational and professional interests in software development,

dis-I had long been fascinated by the open-source software movement, as exemplified by the success of GNU andLinux, though public-domain TEX might also deserve mention Obviously, this book is an attempt to carryover that model of creative endeavor to textbook publishing

• As a sabbatical project during the Spring 2004 semester, I embarked on the current project of creating afreely-distributable linear algebra textbook (Notice the implied financial support of the University of PugetSound to this project.) Most of the material was written from scratch since changes in notation and approachmade much of my notes of little use By August 2004 I had written half the material necessary for our Math

232 course The remaining half was written during the Fall 2004 semester as I taught another two sections

in the construction of simple models A desire to show that even in mathematics one could have funled to an exhibition of the results and attracted considerable attention throughout the school Sincethen the Sherborne collection has grown, ideas have come from many sources, and widespread interesthas been shown It seems therefore desirable to give permanent form to the lessons of experience sothat others can benefit by them and be encouraged to undertake similar work

How To Use This Book Chapters, Theorems, etc are not numbered in this book, but are instead referenced

by acronyms This means that Theorem XYZ will always be Theorem XYZ, no matter if new sections are added,

or if an individual decides to remove certain other sections Within sections, the subsections are acronyms thatbegin with the acronym of the section So Subsection XYZ.AB is the subsection AB in Section XYZ Acronymsare unique within their type, so for example there is just one Definition B [303], but there is also a Section B[303] At first, all the letters flying around may be confusing, but with time, you will begin to recognize the moreimportant ones on sight Furthermore, there are lists of theorems, examples, etc in the front of the book, and anindex that contains every acronym If you are reading this in an electronic version (PDF or XML), you will seethat all of the cross-references are hyperlinks, allowing you to click to a definition or example, and then use the

Tiêu đề	A First Course In Linear Algebra
Tác giả	Robert A. Beezer
Trường học	University of Puget Sound
Chuyên ngành	Mathematics
Thể loại	Essay
Năm xuất bản	2008
Thành phố	Tacoma

Định dạng
Số trang	794
Dung lượng	6,42 MB