A First Course in Linear Algebra

(1)

(2)

by

Robert A. Beezer

Department of Mathematics and Computer Science

University of Puget Sound

(3)

Robert A. Beezer is a Professor of Mathematics at the University of Puget Sound, where he has been on the faculty since 1984. He received a B.S. in Mathematics (with an Emphasis in Computer Science) from the University of Santa Clara in 1978, a M.S. in Statistics from the University of Illinois at Champaign in 1982 and a Ph.D. in Mathematics from the University of Illinois at Urbana-Champaign in 1984. He teaches calculus, linear algebra and 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 Version 2.00. July 16, 2008.

Publisher Robert A. Beezer

Department of Mathematics and Computer Science University of Puget Sound

1500 North Warner

Tacoma, Washington 98416-1043 USA

c

2004 by Robert A. Beezer.

Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version 1.2 or any later version published by the Free Software Foundation; with no Invariant Sections, no Front-Cover Texts, and no Back-Cover Texts. A copy of the license is included in the appendix entitled “GNU Free Documentation License”.

(4)

(5)

Contributors

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/ ˜martinj Hamrick, 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

(16)

Definitions

Section WILA Section SSLE

SLE System of Linear Equations . . . 10

ESYS Equivalent Systems . . . 12

EO Equation Operations . . . 13

Section RREF M Matrix . . . 25

CV Column Vector . . . 26

ZCV Zero Column Vector . . . 26

CM Coefficient Matrix . . . 26

VOC Vector of Constants . . . 26

SOLV Solution Vector . . . 27

MRLS Matrix Representation of a Linear System . . . 27

AM Augmented Matrix . . . 28

RO Row Operations . . . 29

REM Row-Equivalent Matrices . . . 29

RR Row-Reducing . . . 41

Section TSS CS Consistent System . . . 52

IDV Independent and Dependent Variables . . . 54

Section HSE HS Homogeneous System . . . 65

TSHSE Trivial Solution to Homogeneous Systems of Equations . . . 65

NSM Null Space of a Matrix . . . 68

Section NM SQM Square Matrix . . . 75

NM Nonsingular Matrix . . . 75

IM Identity Matrix . . . 76

Section VO VSCV Vector Space of Column Vectors . . . 88

CVE Column Vector Equality . . . 89

(17)

DEFINITIONS xviii

CVSM Column Vector Scalar Multiplication . . . 90

Section LC LCCV Linear Combination of Column Vectors . . . 96

Section SS SSCV Span of a Set of Column Vectors . . . 119

Section LI RLDCV Relation of Linear Dependence for Column Vectors . . . 139

LICV Linear Independence of Column Vectors . . . 139

Section LDS Section O CCCV Complex Conjugate of a Column Vector . . . 175

IP Inner Product . . . 176

NV Norm of a Vector. . . 179

OV Orthogonal Vectors . . . 181

OSV Orthogonal Set of Vectors . . . 181

SUV Standard Unit Vectors . . . 181

ONS OrthoNormal Set. . . 186

Section MO VSM Vector Space of m_×n Matrices . . . 191

ME Matrix Equality . . . 191

MA Matrix Addition . . . 192

MSM Matrix Scalar Multiplication . . . 192

ZM Zero Matrix . . . 194

TM Transpose of a Matrix . . . 194

SYM Symmetric Matrix . . . 195

CCM Complex Conjugate of a Matrix . . . 196

A Adjoint . . . 198

Section MM MVP Matrix-Vector Product . . . 204

HM Hermitian Matrix . . . 216

Section MISLE MI Matrix Inverse . . . 224

Section MINM UM Unitary Matrices . . . 242

Section CRS CSM Column Space of a Matrix . . . 249

(18)

DEFINITIONS xix

Section FS

LNS Left Null Space . . . 271

EEF Extended Echelon Form . . . 275

Section VS VS Vector Space . . . 295

Section S S Subspace . . . 308

TS Trivial Subspaces . . . 312

LC Linear Combination . . . 313

SS Span of a Set . . . 314

Section LISS RLD Relation of Linear Dependence . . . 325

LI Linear Independence . . . 325

TSVS To Span a Vector Space . . . 330

Section B B Basis . . . 343

Section D D Dimension . . . 359

NOM Nullity Of a Matrix . . . 365

ROM Rank Of a Matrix . . . 365

Section PD DS Direct Sum . . . 379

Section DM ELEM Elementary Matrices . . . 389

SM SubMatrix . . . 394

DM Determinant of a Matrix . . . 394

Section PDM Section EE EEM Eigenvalues and Eigenvectors of a Matrix . . . 415

CP Characteristic Polynomial . . . 422

EM Eigenspace of a Matrix . . . 424

AME Algebraic Multiplicity of an Eigenvalue . . . 426

GME Geometric Multiplicity of an Eigenvalue . . . 426

Section PEE Section SD SIM Similar Matrices . . . 453

DIM Diagonal Matrix . . . 456

(19)

DEFINITIONS xx

Section LT

LT Linear Transformation . . . 473

PI Pre-Image . . . 486

LTA Linear Transformation Addition . . . 488

LTSM Linear Transformation Scalar Multiplication . . . 489

LTC Linear Transformation Composition . . . 491

Section ILT ILT Injective Linear Transformation. . . 499

KLT Kernel of a Linear Transformation . . . 503

Section SLT SLT Surjective Linear Transformation . . . 516

RLT Range of a Linear Transformation . . . 520

Section IVLT IDLT Identity Linear Transformation . . . 534

IVLT Invertible Linear Transformations . . . 534

IVS Isomorphic Vector Spaces . . . 541

ROLT Rank Of a Linear Transformation . . . 543

NOLT Nullity Of a Linear Transformation. . . 543

Section VR VR Vector Representation . . . 557

Section MR MR Matrix Representation . . . 570

Section CB EELT Eigenvalue and Eigenvector of a Linear Transformation . . . 603

CBM Change-of-Basis Matrix . . . 604

Section OD UTM Upper Triangular Matrix . . . 631

LTM Lower Triangular Matrix . . . 631

NRML Normal Matrix . . . 636

Section NLT NLT Nilpotent Linear Transformation . . . 641

JB Jordan Block . . . 643

Section IS IS Invariant Subspace . . . 659

GEV Generalized Eigenvector . . . 663

GES Generalized Eigenspace . . . 663

LTR Linear Transformation Restriction . . . 667

(20)

DEFINITIONS xxi

Section JCF

Section CNO CNE Complex Number Equality . . . 713

CNA Complex Number Addition . . . 713

CNM Complex Number Multiplication . . . 713

CCN Conjugate of a Complex Number . . . 714

MCN Modulus of a Complex Number . . . 715

Section SET SET Set . . . 716

SSET Subset . . . 716

ES Empty Set . . . 716

SE Set Equality . . . 717

C Cardinality . . . 717

SU Set Union . . . 718

SI Set Intersection . . . 718

SC Set Complement . . . 718

Section PT Section F F Field . . . 828

IMP Integers Modulo a Prime . . . 830

Section T T Trace . . . 837

Section HP HP Hadamard Product . . . 843

HID Hadamard Identity. . . 844

HI Hadamard Inverse . . . 844

Section VM VM Vandermonde Matrix . . . 849

Section PSM PSM Positive Semi-Definite Matrix . . . 853

Section ROD Section TD Section SVD SV Singular Values . . . 875

(21)

DEFINITIONS xxii

Section CF

(22)

Theorems

Section WILA Section SSLE

EOPSS Equation Operations Preserve Solution Sets . . . 13 Section RREF

REMES Row-Equivalent Matrices represent Equivalent Systems . . . 30

REMEF Row-Equivalent Matrix in Echelon Form. . . 32

RREFU Reduced Row-Echelon Form is Unique . . . 34 Section TSS

RCLS Recognizing Consistency of a Linear System . . . 55

ISRN Inconsistent Systems, r and n . . . 56

CSRN Consistent Systems, r and n . . . 56

FVCS Free Variables for Consistent Systems . . . 57

PSSLS Possible Solution Sets for Linear Systems . . . 58

CMVEI Consistent, More Variables than Equations, Infinite solutions . . . 58 Section HSE

HSC Homogeneous Systems are Consistent . . . 65

HMVEI Homogeneous, More Variables than Equations, Infinite solutions . . . 67 Section NM

NMRRI Nonsingular Matrices Row Reduce to the Identity matrix . . . 76

NMTNS Nonsingular Matrices have Trivial Null Spaces . . . 78

NMUS Nonsingular Matrices and Unique Solutions . . . 78

NME1 Nonsingular Matrix Equivalences, Round 1 . . . 79 Section VO

VSPCV Vector Space Properties of Column Vectors . . . 91 Section LC

SLSLC Solutions to Linear Systems are Linear Combinations . . . 99

VFSLS Vector Form of Solutions to Linear Systems . . . 106

PSPHS Particular Solution Plus Homogeneous Solutions . . . 112 Section SS

(23)

THEOREMS xxiv

Section LI

LIVHS Linearly Independent Vectors and Homogeneous Systems . . . 141

LIVRN Linearly Independent Vectors, r and n . . . 143

MVSLD More Vectors than Size implies Linear Dependence . . . 144

NMLIC Nonsingular Matrices have Linearly Independent Columns . . . 145

NME2 Nonsingular Matrix Equivalences, Round 2 . . . 145

BNS Basis for Null Spaces . . . 147 Section LDS

DLDS Dependency in Linearly Dependent Sets . . . 160

BS Basis of a Span . . . 165 Section O

CRVA Conjugation Respects Vector Addition . . . 175

CRSM Conjugation Respects Vector Scalar Multiplication . . . 176

IPVA Inner Product and Vector Addition. . . 177

IPSM Inner Product and Scalar Multiplication . . . 178

IPAC Inner Product is Anti-Commutative . . . 178

IPN Inner Products and Norms . . . 180

PIP Positive Inner Products . . . 180

OSLI Orthogonal Sets are Linearly Independent . . . 183

GSP Gram-Schmidt Procedure . . . 183 Section MO

VSPM Vector Space Properties of Matrices . . . 193

SMS Symmetric Matrices are Square . . . 195

TMA Transpose and Matrix Addition. . . 195

TMSM Transpose and Matrix Scalar Multiplication . . . 196

TT Transpose of a Transpose . . . 196

CRMA Conjugation Respects Matrix Addition . . . 197

CRMSM Conjugation Respects Matrix Scalar Multiplication . . . 197

CCM Conjugate of the Conjugate of a Matrix . . . 197

MCT Matrix Conjugation and Transposes . . . 198

AMA Adjoint and Matrix Addition . . . 198

AMSM Adjoint and Matrix Scalar Multiplication . . . 199

AA Adjoint of an Adjoint . . . 199 Section MM

SLEMM Systems of Linear Equations as Matrix Multiplication . . . 205

EMMVP Equal Matrices and Matrix-Vector Products . . . 207

EMP Entries of Matrix Products . . . 209

MMZM Matrix Multiplication and the Zero Matrix . . . 210

MMIM Matrix Multiplication and Identity Matrix . . . 211

MMDAA Matrix Multiplication Distributes Across Addition . . . 211

MMSMM Matrix Multiplication and Scalar Matrix Multiplication . . . 212

MMA Matrix Multiplication is Associative . . . 212

MMIP Matrix Multiplication and Inner Products . . . 213

(24)

THEOREMS xxv

MMT Matrix Multiplication and Transposes . . . 214

MMAD Matrix Multiplication and Adjoints . . . 214

AIP Adjoint and Inner Product . . . 215

HMIP Hermitian Matrices and Inner Products . . . 216 Section MISLE

TTMI Two-by-Two Matrix Inverse . . . 226

CINM Computing the Inverse of a Nonsingular Matrix . . . 229

MIU Matrix Inverse is Unique . . . 230

SS Socks and Shoes . . . 231

MIMI Matrix Inverse of a Matrix Inverse . . . 231

MIT Matrix Inverse of a Transpose. . . 232

MISM Matrix Inverse of a Scalar Multiple . . . 232 Section MINM

NPNT Nonsingular Product has Nonsingular Terms . . . 239

OSIS One-Sided Inverse is Sufficient . . . 240

NI Nonsingularity is Invertibility . . . 241

SNCM Solution with Nonsingular Coefficient Matrix . . . 242

UMI Unitary Matrices are Invertible . . . 243

CUMOS Columns of Unitary Matrices are Orthonormal Sets . . . 243

UMPIP Unitary Matrices Preserve Inner Products . . . 244 Section CRS

CSCS Column Spaces and Consistent Systems . . . 250

BCS Basis of the Column Space . . . 253

CSNM Column Space of a Nonsingular Matrix . . . 255

REMRS Row-Equivalent Matrices have equal Row Spaces . . . 258

BRS Basis for the Row Space . . . 259

CSRST Column Space, Row Space, Transpose . . . 261 Section FS

PEEF Properties of Extended Echelon Form . . . 276

FS Four Subsets . . . 278 Section VS

ZVU Zero Vector is Unique . . . 302

AIU Additive Inverses are Unique . . . 302

ZSSM Zero Scalar in Scalar Multiplication . . . 302

ZVSM Zero Vector in Scalar Multiplication . . . 303

AISM Additive Inverses from Scalar Multiplication. . . 303

SMEZV Scalar Multiplication Equals the Zero Vector . . . 304 Section S

TSS Testing Subsets for Subspaces. . . 309

(25)

THEOREMS xxvi

SSS Span of a Set is a Subspace . . . 314

CSMS Column Space of a Matrix is a Subspace . . . 319

RSMS Row Space of a Matrix is a Subspace . . . 319

LNSMS Left Null Space of a Matrix is a Subspace . . . 319 Section LISS

VRRB Vector Representation Relative to a Basis . . . 334 Section B

SUVB Standard Unit Vectors are a Basis . . . 343

CNMB Columns of Nonsingular Matrix are a Basis . . . 348

COB Coordinates and Orthonormal Bases . . . 350

UMCOB Unitary Matrices Convert Orthonormal Bases . . . 352 Section D

SSLD Spanning Sets and Linear Dependence . . . 359

BIS Bases have Identical Sizes . . . 362

DCM Dimension of Cm . . . 363

DP Dimension of Pn . . . 363 DM Dimension of Mmn . . . 363 CRN Computing Rank and Nullity . . . 366

RPNC Rank Plus Nullity is Columns. . . 366

RNNM Rank and Nullity of a Nonsingular Matrix . . . 367

NME6 Nonsingular Matrix Equivalences, Round 6 . . . 368 Section PD

ELIS Extending Linearly Independent Sets. . . 373

G Goldilocks . . . 374

PSSD Proper Subspaces have Smaller Dimension . . . 376

EDYES Equal Dimensions Yields Equal Subspaces . . . 377

RMRT Rank of a Matrix is the Rank of the Transpose . . . 377

DFS Dimensions of Four Subspaces . . . 378

DSFB Direct Sum From a Basis . . . 380

DSFOS Direct Sum From One Subspace . . . 381

DSZV Direct Sums and Zero Vectors . . . 381

DSZI Direct Sums and Zero Intersection . . . 382

DSLI Direct Sums and Linear Independence . . . 382

DSD Direct Sums and Dimension . . . 383

RDS Repeated Direct Sums . . . 383 Section DM

EMDRO Elementary Matrices Do Row Operations . . . 391

EMN Elementary Matrices are Nonsingular . . . 393

NMPEM Nonsingular Matrices are Products of Elementary Matrices . . . 393

DMST Determinant of Matrices of Size Two . . . 395

DER Determinant Expansion about Rows . . . 395

(26)

THEOREMS xxvii

DEC Determinant Expansion about Columns . . . 397 Section PDM

DZRC Determinant with Zero Row or Column . . . 402

DRCS Determinant for Row or Column Swap . . . 402

DRCM Determinant for Row or Column Multiples . . . 403

DERC Determinant with Equal Rows or Columns . . . 404

DRCMA Determinant for Row or Column Multiples and Addition . . . 404

DIM Determinant of the Identity Matrix. . . 407

DEM Determinants of Elementary Matrices . . . 407

DEMMM Determinants, Elementary Matrices, Matrix Multiplication . . . 408

SMZD Singular Matrices have Zero Determinants . . . 409

DRMM Determinant Respects Matrix Multiplication . . . 410 Section EE

EMHE Every Matrix Has an Eigenvalue . . . 419

EMRCP Eigenvalues of a Matrix are Roots of Characteristic Polynomials . . . 423

EMS Eigenspace for a Matrix is a Subspace . . . 424

EMNS Eigenspace of a Matrix is a Null Space . . . 424 Section PEE

EDELI Eigenvectors with Distinct Eigenvalues are Linearly Independent . . . 439

SMZE Singular Matrices have Zero Eigenvalues . . . 440

ESMM Eigenvalues of a Scalar Multiple of a Matrix . . . 441

EOMP Eigenvalues Of Matrix Powers . . . 441

EPM Eigenvalues of the Polynomial of a Matrix . . . 442

EIM Eigenvalues of the Inverse of a Matrix . . . 443

ETM Eigenvalues of the Transpose of a Matrix . . . 443

ERMCP Eigenvalues of Real Matrices come in Conjugate Pairs . . . 444

DCP Degree of the Characteristic Polynomial . . . 444

NEM Number of Eigenvalues of a Matrix . . . 445

ME Multiplicities of an Eigenvalue . . . 445

MNEM Maximum Number of Eigenvalues of a Matrix . . . 447

HMRE Hermitian Matrices have Real Eigenvalues . . . 448

HMOE Hermitian Matrices have Orthogonal Eigenvectors . . . 448 Section SD

SER Similarity is an Equivalence Relation . . . 454

SMEE Similar Matrices have Equal Eigenvalues . . . 455

DC Diagonalization Characterization . . . 457

DMFE Diagonalizable Matrices have Full Eigenspaces . . . 459

DED Distinct Eigenvalues implies Diagonalizable . . . 462 Section LT

LTTZZ Linear Transformations Take Zero to Zero . . . 477

(27)

THEOREMS xxviii

MLTCV Matrix of a Linear Transformation, Column Vectors . . . 481

LTLC Linear Transformations and Linear Combinations . . . 483

LTDB Linear Transformation Defined on a Basis . . . 483

SLTLT Sum of Linear Transformations is a Linear Transformation . . . 489

MLTLT Multiple of a Linear Transformation is a Linear Transformation . . . 490

VSLT Vector Space of Linear Transformations . . . 491

CLTLT Composition of Linear Transformations is a Linear Transformation . . . 491 Section ILT

KLTS Kernel of a Linear Transformation is a Subspace . . . 504

KPI Kernel and Pre-Image . . . 505

KILT Kernel of an Injective Linear Transformation . . . 506

ILTLI Injective Linear Transformations and Linear Independence . . . 508

ILTB Injective Linear Transformations and Bases . . . 508

ILTD Injective Linear Transformations and Dimension . . . 509

CILTI Composition of Injective Linear Transformations is Injective . . . 509 Section SLT

RLTS Range of a Linear Transformation is a Subspace . . . 521

RSLT Range of a Surjective Linear Transformation . . . 523

SSRLT Spanning Set for Range of a Linear Transformation . . . 524

RPI Range and Pre-Image . . . 526

SLTB Surjective Linear Transformations and Bases . . . 526

SLTD Surjective Linear Transformations and Dimension . . . 526

CSLTS Composition of Surjective Linear Transformations is Surjective . . . 527 Section IVLT

ILTLT Inverse of a Linear Transformation is a Linear Transformation . . . 537

IILT Inverse of an Invertible Linear Transformation . . . 537

ILTIS Invertible Linear Transformations are Injective and Surjective . . . 538

CIVLT Composition of Invertible Linear Transformations . . . 540

ICLT Inverse of a Composition of Linear Transformations . . . 540

IVSED Isomorphic Vector Spaces have Equal Dimension . . . 543

ROSLT Rank Of a Surjective Linear Transformation. . . 544

NOILT Nullity Of an Injective Linear Transformation . . . 544

RPNDD Rank Plus Nullity is Domain Dimension . . . 544 Section VR

VRLT Vector Representation is a Linear Transformation . . . 557

VRI Vector Representation is Injective . . . 561

VRS Vector Representation is Surjective . . . 562

VRILT Vector Representation is an Invertible Linear Transformation . . . 562

CFDVS Characterization of Finite Dimensional Vector Spaces . . . 562

IFDVS Isomorphism of Finite Dimensional Vector Spaces . . . 563

CLI Coordinatization and Linear Independence . . . 564

(28)

THEOREMS xxix

FTMR Fundamental Theorem of Matrix Representation . . . 573

MRSLT Matrix Representation of a Sum of Linear Transformations . . . 576

MRMLT Matrix Representation of a Multiple of a Linear Transformation . . . 577

MRCLT Matrix Representation of a Composition of Linear Transformations . . . 577

KNSI Kernel and Null Space Isomorphism . . . 581

RCSI Range and Column Space Isomorphism . . . 584

IMR Invertible Matrix Representations . . . 586

IMILT Invertible Matrices, Invertible Linear Transformation . . . 589

NME9 Nonsingular Matrix Equivalences, Round 9 . . . 589 Section CB

CB Change-of-Basis . . . 605

ICBM Inverse of Change-of-Basis Matrix . . . 605

MRCB Matrix Representation and Change of Basis . . . 610

SCB Similarity and Change of Basis . . . 613

EER Eigenvalues, Eigenvectors, Representations . . . 616 Section OD

PTMT Product of Triangular Matrices is Triangular . . . 631

ITMT Inverse of a Triangular Matrix is Triangular . . . 632

UTMR Upper Triangular Matrix Representation. . . 633

OBUTR Orthonormal Basis for Upper Triangular Representation . . . 635

OBNM Orthonormal Bases and Normal Matrices . . . 639 Section NLT

NJB Nilpotent Jordan Blocks . . . 646

ENLT Eigenvalues of Nilpotent Linear Transformations . . . 647

DNLT Diagonalizable Nilpotent Linear Transformations . . . 647

KPLT Kernels of Powers of Linear Transformations . . . 648

KPNLT Kernels of Powers of Nilpotent Linear Transformations . . . 649

CFNLT Canonical Form for Nilpotent Linear Transformations . . . 651 Section IS

EIS Eigenspaces are Invariant Subspaces . . . 661

KPIS Kernels of Powers are Invariant Subspaces . . . 662

GESIS Generalized Eigenspace is an Invariant Subspace . . . 663

GEK Generalized Eigenspace as a Kernel . . . 664

RGEN Restriction to Generalized Eigenspace is Nilpotent . . . 673

MRRGE Matrix Representation of a Restriction to a Generalized Eigenspace . . . 676 Section JCF

GESD Generalized Eigenspace Decomposition . . . 677

DGES Dimension of Generalized Eigenspaces . . . 683

JCFLT Jordan Canonical Form for a Linear Transformation . . . 684

(29)

THEOREMS xxx

PCNA Properties of Complex Number Arithmetic . . . 713

CCRA Complex Conjugation Respects Addition . . . 715

CCRM Complex Conjugation Respects Multiplication . . . 715

CCT Complex Conjugation Twice . . . 715 Section SET

Section PT Section F

FIMP Field of Integers Modulo a Prime . . . 830 Section T

TL Trace is Linear . . . 837

TSRM Trace is Symmetric with Respect to Multiplication . . . 838

TIST Trace is Invariant Under Similarity Transformations . . . 838

TSE Trace is the Sum of the Eigenvalues . . . 838 Section HP

HPC Hadamard Product is Commutative . . . 843

HPHID Hadamard Product with the Hadamard Identity . . . 844

HPHI Hadamard Product with Hadamard Inverses. . . 844

HPDAA Hadamard Product Distributes Across Addition . . . 845

HPSMM Hadamard Product and Scalar Matrix Multiplication . . . 845

DMHP Diagonalizable Matrices and the Hadamard Product . . . 846

DMMP Diagonal Matrices and Matrix Products . . . 846 Section VM

DVM Determinant of a Vandermonde Matrix . . . 849

NVM Nonsingular Vandermonde Matrix . . . 852 Section PSM

CPSM Creating Positive Semi-Definite Matrices. . . 853

EPSM Eigenvalues of Positive Semi-definite Matrices . . . 854 Section ROD

ROD Rank One Decomposition . . . 858 Section TD

TD Triangular Decomposition . . . 863

TDEE Triangular Decomposition, Entry by Entry . . . 867 Section SVD

EEMAP Eigenvalues and Eigenvectors of Matrix-Adjoint Product . . . 871

SVD Singular Value Decomposition . . . 875 Section SR

PSMSR Positive Semi-Definite Matrices and Square Roots . . . 877

EESR Eigenvalues and Eigenspaces of a Square Root . . . 878

(30)

THEOREMS xxxi

Section POD

PDM Polar Decomposition of a Matrix . . . 881 Section CF

IP Interpolating Polynomial . . . 884

(31)

Notation

M A: Matrix . . . 25

MC [A]_ij: Matrix Components . . . 25

CV v: Column Vector . . . 26

CVC [v]_i: Column Vector Components . . . 26

ZCV 0: Zero Column Vector . . . 26

MRLS LS(A, b): Matrix Representation of a Linear System . . . 27

AM [A| b]: Augmented Matrix . . . 28

RO Ri ↔Rj,αRi, αRi+Rj: Row Operations . . . 29 RREFA r,D, F: Reduced Row-Echelon Form Analysis . . . 31

NSM _N(A): Null Space of a Matrix. . . 68

IM Im: Identity Matrix . . . 76

VSCV Cm: Vector Space of Column Vectors . . . 88

CVE u=v: Column Vector Equality. . . 89

CVA u+v: Column Vector Addition. . . 90

CVSM αu: Column Vector Scalar Multiplication . . . 90

SSV _hS_i: Span of a Set of Vectors . . . 119

CCCV u: Complex Conjugate of a Column Vector . . . 175

IP hu, vi: Inner Product . . . 176

NV kvk: Norm of a Vector . . . 179

SUV ei: Standard Unit Vectors . . . 182 VSM Mmn: Vector Space of Matrices . . . 191

ME A=B: Matrix Equality . . . 191

MA A+B: Matrix Addition . . . 192

MSM αA: Matrix Scalar Multiplication . . . 192

ZM O: Zero Matrix . . . 194

TM At_{: Transpose of a Matrix} _{. . . .} ₁₉₄ CCM A: Complex Conjugate of a Matrix . . . 197

A A∗_{: Adjoint} _{. . . .} ₁₉₈

MVP Au: Matrix-Vector Product . . . 204

MI A−1_{: Matrix Inverse} _{. . . .} ₂₂₄

CSM C(A): Column Space of a Matrix . . . 249

RSM R(A): Row Space of a Matrix . . . 256

LNS _L(A): Left Null Space . . . 271

D dim (V): Dimension . . . 359

NOM n(A): Nullity of a Matrix . . . 365

ROM r(A): Rank of a Matrix . . . 365

DS V =U ⊕W: Direct Sum . . . 379

ELEM Ei,j,Ei(α),Ei,j(α): Elementary Matrix . . . 390

(32)

NOTATION xxxiii

SM A(i_|j): SubMatrix . . . 394

DM det (A),_|A_|: Determinant of a Matrix . . . 394

AME αA(λ): Algebraic Multiplicity of an Eigenvalue . . . 426 GME γA(λ): Geometric Multiplicity of an Eigenvalue . . . 426 LT T: U _7→V: Linear Transformation . . . 473

KLT K(T): Kernel of a Linear Transformation . . . 503

RLT _R(T): Range of a Linear Transformation . . . 520

ROLT r(T): Rank of a Linear Transformation . . . 543

NOLT n(T): Nullity of a Linear Transformation . . . 543

VR ρB(w): Vector Representation . . . 557 MR MT

B,C: Matrix Representation . . . 570 JB Jn(λ): Jordan Block . . . 643 GES GT (λ): Generalized Eigenspace . . . 663 LTR T_|U: Linear Transformation Restriction . . . 667 IE ιT(λ): Index of an Eigenvalue . . . 673 CNE α =β: Complex Number Equality . . . 713

CNA α+β: Complex Number Addition . . . 713

CNM αβ: Complex Number Multiplication . . . 713

CCN c: Conjugate of a Complex Number . . . 714

SETM x_∈S: Set Membership . . . 716

SSET S ⊆T: Subset . . . 716

ES ∅: Empty Set . . . 716

SE S =T: Set Equality . . . 717

C |S|: Cardinality. . . 717

SU S∪T: Set Union . . . 718

SI S_∩T: Set Intersection . . . 718

SC S: Set Complement . . . 719

T t(A): Trace . . . 837

HP A_◦B: Hadamard Product . . . 843

HID Jmn: Hadamard Identity . . . 844

HI Ab: Hadamard Inverse . . . 844

(33)

Diagrams

DTSLS Decision Tree for Solving Linear Systems . . . 58

CSRST Column Space and Row Space Techniques . . . 286

DLTA Definition of Linear Transformation, Additive . . . 474

DLTM Definition of Linear Transformation, Multiplicative . . . 474

GLT General Linear Transformation . . . 478

NILT Non-Injective Linear Transformation . . . 500

ILT Injective Linear Transformation. . . 502

FTMR Fundamental Theorem of Matrix Representations . . . 574

FTMRA Fundamental Theorem of Matrix Representations (Alternate) . . . 574

MRCLT Matrix Representation and Composition of Linear Transformations . . . 580

(34)

Examples

Section WILA

TMP Trail Mix Packaging . . . 3 Section SSLE

STNE Solving two (nonlinear) equations . . . 10

NSE Notation for a system of equations . . . 11

TTS Three typical systems . . . 11

US Three equations, one solution . . . 15

IS Three equations, infinitely many solutions . . . 16 Section RREF

AM A matrix . . . 25

NSLE Notation for systems of linear equations . . . 28

AMAA Augmented matrix for Archetype A . . . 28

TREM Two row-equivalent matrices . . . 29

USR Three equations, one solution, reprised . . . 30

RREF A matrix in reduced row-echelon form . . . 32

NRREF A matrix not in reduced row-echelon form . . . 32

SAB Solutions for Archetype B . . . 38

SAA Solutions for Archetype A . . . 39

SAE Solutions for Archetype E . . . 40 Section TSS

RREFN Reduced row-echelon form notation . . . 52

ISSI Describing infinite solution sets, Archetype I . . . 53

FDV Free and dependent variables . . . 54

CFV Counting free variables . . . 57

OSGMD One solution gives many, Archetype D . . . 58 Section HSE

AHSAC Archetype C as a homogeneous system . . . 65

HUSAB Homogeneous, unique solution, Archetype B. . . 66

HISAA Homogeneous, infinite solutions, Archetype A . . . 66

HISAD Homogeneous, infinite solutions, Archetype D . . . 66

NSEAI Null space elements of Archetype I . . . 68

CNS1 Computing a null space, #1 . . . 68

(35)

EXAMPLES xxxvi

Section NM

S A singular matrix, Archetype A . . . 75

NM A nonsingular matrix, Archetype B . . . 76

IM An identity matrix . . . 76

SRR Singular matrix, row-reduced . . . 77

NSR Nonsingular matrix, row-reduced . . . 77

NSS Null space of a singular matrix . . . 78

NSNM Null space of a nonsingular matrix . . . 78 Section VO

VESE Vector equality for a system of equations . . . 89

VA Addition of two vectors in C4 . . . 90

CVSM Scalar multiplication in C5 . . . 91 Section LC

TLC Two linear combinations in C6 . . . 96

ABLC Archetype B as a linear combination . . . 97

AALC Archetype A as a linear combination . . . 98

VFSAD Vector form of solutions for Archetype D . . . 101

VFS Vector form of solutions . . . 103

VFSAI Vector form of solutions for Archetype I . . . 108

VFSAL Vector form of solutions for Archetype L. . . 110

PSHS Particular solutions, homogeneous solutions, Archetype D . . . 112 Section SS

ABS A basic span . . . 119

SCAA Span of the columns of Archetype A . . . 121

SCAB Span of the columns of Archetype B . . . 123

SSNS Spanning set of a null space . . . 125

NSDS Null space directly as a span . . . 126

SCAD Span of the columns of Archetype D . . . 128 Section LI

LDS Linearly dependent set in C5 . . . 139

LIS Linearly independent set in C5 . . . 140

LIHS Linearly independent, homogeneous system . . . 141

LDHS Linearly dependent, homogeneous system . . . 142

LDRN Linearly dependent, r < n . . . 143

LLDS Large linearly dependent set in C4 . . . 143

LDCAA Linearly dependent columns in Archetype A . . . 144

LICAB Linearly independent columns in Archetype B . . . 145

LINSB Linear independence of null space basis . . . 146

NSLIL Null space spanned by linearly independent set, Archetype L . . . 148 Section LDS

RSC5 Reducing a span in C5 . . . 161

COV Casting out vectors . . . 162

(36)

EXAMPLES xxxvii

RES Reworking elements of a span . . . 168 Section O

CSIP Computing some inner products . . . 176

CNSV Computing the norm of some vectors. . . 179

TOV Two orthogonal vectors . . . 181

SUVOS Standard Unit Vectors are an Orthogonal Set . . . 182

AOS An orthogonal set . . . 182

GSTV Gram-Schmidt of three vectors . . . 185

ONTV Orthonormal set, three vectors . . . 186

ONFV Orthonormal set, four vectors . . . 186 Section MO

MA Addition of two matrices in M23 . . . 192

MSM Scalar multiplication in M32 . . . 192

TM Transpose of a 3×4 matrix . . . 194

SYM A symmetric 5×5 matrix . . . 195

CCM Complex conjugate of a matrix . . . 197 Section MM

MTV A matrix times a vector . . . 204

MNSLE Matrix notation for systems of linear equations . . . 205

MBC Money’s best cities . . . 205

PTM Product of two matrices . . . 208

MMNC Matrix multiplication is not commutative . . . 208

PTMEE Product of two matrices, entry-by-entry . . . 209 Section MISLE

SABMI Solutions to Archetype B with a matrix inverse . . . 223

MWIAA A matrix without an inverse, Archetype A . . . 224

MI Matrix inverse . . . 225

CMI Computing a matrix inverse . . . 227

CMIAB Computing a matrix inverse, Archetype B . . . 230 Section MINM

UM3 Unitary matrix of size 3 . . . 242

UPM Unitary permutation matrix . . . 243

OSMC Orthonormal set from matrix columns . . . 244 Section CRS

CSMCS Column space of a matrix and consistent systems . . . 249

MCSM Membership in the column space of a matrix . . . 250

CSTW Column space, two ways . . . 252

CSOCD Column space, original columns, Archetype D . . . 253

CSAA Column space of Archetype A . . . 254

CSAB Column space of Archetype B. . . 255

RSAI Row space of Archetype I . . . 257

(37)

EXAMPLES xxxviii

IAS Improving a span . . . 260

CSROI Column space from row operations, Archetype I . . . 261 Section FS

LNS Left null space . . . 271

CSANS Column space as null space . . . 272

SEEF Submatrices of extended echelon form . . . 275

FS1 Four subsets, #1 . . . 282

FS2 Four subsets, #2 . . . 283

FSAG Four subsets, Archetype G . . . 284 Section VS

VSCV The vector space Cm . . . 297

VSM The vector space of matrices, Mmn . . . 297 VSP The vector space of polynomials, Pn . . . 298 VSIS The vector space of infinite sequences . . . 298

VSF The vector space of functions . . . 299

VSS The singleton vector space . . . 299

CVS The crazy vector space . . . 300

PCVS Properties for the Crazy Vector Space . . . 304 Section S

SC3 A subspace of C3 . . . 308

SP4 A subspace of P4 . . . 310

NSC2Z A non-subspace in C2_{, zero vector} _{. . . .} ₃₁₁

NSC2A A non-subspace in C2, additive closure . . . 311

NSC2S A non-subspace in C2, scalar multiplication closure . . . 312

RSNS Recasting a subspace as a null space . . . 313

LCM A linear combination of matrices . . . 314

SSP Span of a set of polynomials . . . 315

SM32 A subspace of M32 . . . 316 Section LISS

LIP4 Linear independence in P4 . . . 325

LIM32 Linear independence in M32 . . . 327

LIC Linearly independent set in the crazy vector space . . . 329

SSP4 Spanning set in P4 . . . 330

SSM22 Spanning set in M22 . . . 332

SSC Spanning set in the crazy vector space . . . 332

AVR A vector representation . . . 334 Section B

BP Bases for Pn . . . 344 BM A basis for the vector space of matrices . . . 344

BSP4 A basis for a subspace of P4 . . . 344

BSM22 A basis for a subspace of M22 . . . 345

BC Basis for the crazy vector space . . . 346

(38)

EXAMPLES xxxix

RS Reducing a span . . . 347

CABAK Columns as Basis, Archetype K. . . 349

CROB4 Coordinatization relative to an orthonormal basis, C4 . . . 351

CROB3 Coordinatization relative to an orthonormal basis, C3 . . . 352 Section D

LDP4 Linearly dependent set in P4 . . . 362

DSM22 Dimension of a subspace of M22 . . . 363

DSP4 Dimension of a subspace of P4 . . . 364

DC Dimension of the crazy vector space . . . 364

VSPUD Vector space of polynomials with unbounded degree . . . 365

RNM Rank and nullity of a matrix . . . 365

RNSM Rank and nullity of a square matrix . . . 367 Section PD

BPR Bases for Pn, reprised . . . 375

BDM22 Basis by dimension in M22 . . . 375

SVP4 Sets of vectors in P4 . . . 376

RRTI Rank, rank of transpose, Archetype I . . . 378

SDS Simple direct sum . . . 380 Section DM

EMRO Elementary matrices and row operations . . . 390

SS Some submatrices . . . 394

D33M Determinant of a 3_×3 matrix . . . 394

TCSD Two computations, same determinant . . . 398

DUTM Determinant of an upper triangular matrix . . . 398 Section PDM

DRO Determinant by row operations . . . 405

ZNDAB Zero and nonzero determinant, Archetypes A and B . . . 409 Section EE

SEE Some eigenvalues and eigenvectors . . . 416

PM Polynomial of a matrix . . . 417

CAEHW Computing an eigenvalue the hard way . . . 420

CPMS3 Characteristic polynomial of a matrix, size 3 . . . 422

EMS3 Eigenvalues of a matrix, size 3 . . . 423

ESMS3 Eigenspaces of a matrix, size 3 . . . 425

EMMS4 Eigenvalue multiplicities, matrix of size 4 . . . 426

ESMS4 Eigenvalues, symmetric matrix of size 4 . . . 427

HMEM5 High multiplicity eigenvalues, matrix of size 5 . . . 428

CEMS6 Complex eigenvalues, matrix of size 6 . . . 429

DEMS5 Distinct eigenvalues, matrix of size 5 . . . 431 Section PEE

(39)

EXAMPLES xl

Section SD

SMS5 Similar matrices of size 5 . . . 453

SMS3 Similar matrices of size 3 . . . 454

EENS Equal eigenvalues, not similar . . . 456

DAB Diagonalization of Archetype B . . . 456

DMS3 Diagonalizing a matrix of size 3. . . 458

NDMS4 A non-diagonalizable matrix of size 4 . . . 461

DEHD Distinct eigenvalues, hence diagonalizable . . . 462

HPDM High power of a diagonalizable matrix . . . 463

FSCF Fibonacci sequence, closed form . . . 464 Section LT

ALT A linear transformation . . . 474

NLT Not a linear transformation . . . 475

LTPM Linear transformation, polynomials to matrices . . . 476

LTPP Linear transformation, polynomials to polynomials . . . 477

LTM Linear transformation from a matrix . . . 478

MFLT Matrix from a linear transformation . . . 480

MOLT Matrix of a linear transformation . . . 482

LTDB1 Linear transformation defined on a basis . . . 484

SPIAS Sample pre-images, Archetype S . . . 487

STLT Sum of two linear transformations . . . 489

SMLT Scalar multiple of a linear transformation . . . 490

CTLT Composition of two linear transformations . . . 492 Section ILT

NIAQ Not injective, Archetype Q . . . 499

IAR Injective, Archetype R . . . 500

IAV Injective, Archetype V . . . 502

NKAO Nontrivial kernel, Archetype O . . . 503

TKAP Trivial kernel, Archetype P . . . 504

NIAQR Not injective, Archetype Q, revisited . . . 506

NIAO Not injective, Archetype O . . . 507

IAP Injective, Archetype P . . . 507

NIDAU Not injective by dimension, Archetype U . . . 509 Section SLT

NSAQ Not surjective, Archetype Q . . . 516

SAR Surjective, Archetype R . . . 517

SAV Surjective, Archetype V . . . 518

RAO Range, Archetype O . . . 520

FRAN Full range, Archetype N . . . 522

NSAQR Not surjective, Archetype Q, revisited . . . 523

NSAO Not surjective, Archetype O . . . 524

SAN Surjective, Archetype N . . . 524

(40)

EXAMPLES xli

NSDAT Not surjective by dimension, Archetype T . . . 527 Section IVLT

AIVLT An invertible linear transformation . . . 534

ANILT A non-invertible linear transformation . . . 535

CIVLT Computing the Inverse of a Linear Transformations . . . 539

IVSAV Isomorphic vector spaces, Archetype V . . . 542 Section VR

VRC4 Vector representation in C4 . . . 558

VRP2 Vector representations in P2 . . . 560

TIVS Two isomorphic vector spaces . . . 563

CVSR Crazy vector space revealed . . . 563

ASC A subspace characterized . . . 563

MIVS Multiple isomorphic vector spaces . . . 563

CP2 Coordinatizing in P2 . . . 565

CM32 Coordinatization in M32 . . . 565 Section MR

OLTTR One linear transformation, three representations . . . 570

ALTMM A linear transformation as matrix multiplication . . . 574

MPMR Matrix product of matrix representations . . . 578

KVMR Kernel via matrix representation . . . 582

RVMR Range via matrix representation . . . 585

ILTVR Inverse of a linear transformation via a representation . . . 587 Section CB

ELTBM Eigenvectors of linear transformation between matrices . . . 603

ELTBP Eigenvectors of linear transformation between polynomials . . . 604

CBP Change of basis with polynomials . . . 605

CBCV Change of basis with column vectors . . . 609

MRCM Matrix representations and change-of-basis matrices . . . 611

MRBE Matrix representation with basis of eigenvectors. . . 613

ELTT Eigenvectors of a linear transformation, twice . . . 617

CELT Complex eigenvectors of a linear transformation. . . 622 Section OD

ANM A normal matrix . . . 637 Section NLT

NM64 Nilpotent matrix, size 6, index 4 . . . 642

JB4 Jordan block, size 4 . . . 644

NJB5 Nilpotent Jordan block, size 5. . . 644

KPNLT Kernels of powers of a nilpotent linear transformation . . . 649

(41)

EXAMPLES xlii

Section IS

TIS Two invariant subspaces . . . 659

EIS Eigenspaces as invariant subspaces . . . 661

ISJB Invariant subspaces and Jordan blocks . . . 662

GE4 Generalized eigenspaces, dimension 4 domain . . . 664

GE6 Generalized eigenspaces, dimension 6 domain . . . 665

LTRGE Linear transformation restriction on generalized eigenspace . . . 668

ISMR4 Invariant subspaces, matrix representation, dimension 4 domain . . . 670

ISMR6 Invariant subspaces, matrix representation, dimension 6 domain . . . 671

GENR6 Generalized eigenspaces and nilpotent restrictions, dimension 6 domain . . . 673 Section JCF

JCF10 Jordan canonical form, size 10 . . . 685 Section CNO

ACN Arithmetic of complex numbers . . . 713

CSCN Conjugate of some complex numbers . . . 714

MSCN Modulus of some complex numbers . . . 715 Section SET

SETM Set membership . . . 716

SSET Subset . . . 716

CS Cardinality and Size . . . 718

SU Set union . . . 718

SI Set intersection . . . 718

SC Set complement . . . 719 Section PT

Section F

IM11 Integers mod 11 . . . 830

VSIM5 Vector space over integers mod 5 . . . 830

SM2Z7 Symmetric matrices of size 2 over Z7 . . . 831

FF8 Finite field of size 8 . . . 831 Section T

Section HP

HP Hadamard Product . . . 843 Section VM

VM4 Vandermonde matrix of size 4. . . 849 Section PSM

Section ROD

ROD2 Rank one decomposition, size 2 . . . 860

ROD4 Rank one decomposition, size 4 . . . 860 Section TD

(42)

EXAMPLES xliii

TDSSE Triangular decomposition solves a system of equations . . . 866

TDEE6 Triangular decomposition, entry by entry, size 6. . . 869 Section SVD

Section SR Section POD Section CF

PTFP Polynomial through five points . . . 884 Section SAS

(43)

Preface

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 the fundamental concepts and techniques of matrix algebra and abstract vector spaces, and to teach the techniques associated 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 exclusive right to control the making of copies and derivative works for many years (too many years in some cases). It also gives others limited rights, generally referred to as “fair use,” such as the right to quote sections in a review without seeking permission. However, the author licenses this book to anyone under the terms of the GNU Free Documentation License (GFDL), which gives you more rights than most copyrights (seeAppendix GFDL[821]). Loosely speaking, you may make as many copies as you like at no cost, and you may distribute these unmodified copies if you please. You may modify the book for your own use. The catch is that if you make modifications and you distribute the modified version, or make use of portions in excess of fair use in another work, then you must 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 exact details 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 saylibert´e, liberty). It will never go “out of print” nor will there ever be trivial updates designed only to frustrate the used book market. Those considering teaching a course with this book can examine it thoroughly in advance. Adding new exercises or new sections has been purposely made very easy, and the hope is that others will contribute these modifications 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[191]) is basically a course in matrix algebra, though the foundation of some more advanced ideas is also being formed in these early sections. Vectors are presented exclusively 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, column spaces 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 frequent definition using dot products of rows with columns. And this delay emphasizes

(44)

PREFACE xlv

that linear algebra is built upon vector addition and scalar multiplication. Of course, matrix inverses must wait for matrix multiplication, but this does not prevent nonsingular matrices from occurring sooner. Vector space properties are hinted at when vector and matrix operations are first defined, but the notion of a vector space is saved for a more axiomatic treatment later (Chapter VS [295]). Once bases and dimension have been explored in the context of vector spaces, linear transformations and their matrix representations follow. The goal of the book is to go as far as Jordan canonical form in the Core (Part C[2]), with less central topics collected in the Topics (Part T[828]). A third part contains contributed applications (Part A [884]), 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 standard calculus 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 definition is stated carefully, set apart from the text. Likewise, every theorem is carefully stated, and almost every one 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[viii],Theorems

[ix]), and alphabetical order in the index at the back. Along the way, there are discussions of some more 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 students majoring in mathematics, computer science, physics, chemistry and economics. Between January 1986 and June 2002, I taught this course seventeen times. For the Spring 2003 semester, I elected to convert my course notes to an electronic form so that it would be easier to incorporate the inevitable and nearly-constant revisions. Central to my new notes was a collection of stock examples that would be used repeatedly to illustrate new concepts. (These would become the Archetypes, Appendix A [732].) 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 this course. 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, the textbook 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 my notes and list of suggested exercises to conform with the changed numbering of the chapters and exercises. I had an almost identical experience with the third course I was teaching that semester. I also learned that in the next academic year I would be teaching a course where my textbook of choice had gone out of print. I felt there had to be a better alternative to having the organization of my courses buffeted by the economics of traditional textbook publishing.

(45)

PREFACE xlvi

• As a sabbatical project during the Spring 2004 semester, I embarked on the current project of creating a freely-distributable linear algebra textbook. (Notice the implied financial support of the University of Puget Sound to this project.) Most of the material was written from scratch since changes in notation and approach made 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 of Math 232.

• While in early 2005 the book was complete enough to build a course around and Version 1.0 was released. Work has continued since, filling out the narrative, exercises and supplements.

However, much of my motivation for writing this book is captured by the sentiments expressed by H.M. Cundy and A.P. Rollet in their Preface to the First Edition ofMathematical Models(1952), especially the final sentence,

This book was born in the classroom, and arose from the spontaneous interest of a Mathemat-ical Sixth in the construction of simple models. A desire to show that even in mathematics one could have fun led to an exhibition of the results and attracted considerable attention throughout the school. Since then the Sherborne collection has grown, ideas have come from many sources, and widespread interest has been shown. It seems therefore desirable to give permanent form to the lessons of experience so that 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 that begin with the acronym of the section. So Subsection XYZ.AB is the subsection AB in Section XYZ. Acronyms are unique within their type, so for example there is just oneDefinition B[343], but there is also aSection B [343]. At first, all the letters flying around may be confusing, but with time, you will begin to recognize the more important ones on sight. Furthermore, there are lists of theorems, examples, etc. in the front of the book, and an index that contains every acronym. If you are reading this in an electronic version (PDF or XML), you will see that all of the cross-references are hyperlinks, allowing you to click to a definition or example, and then use the back button to return. In printed versions, you must rely on the page numbers. However, note that page numbers are not permanent! Different editions, different margins, or different sized paper will affect what content is on each page. And in time, the addition of new material will affect the page numbering. Chapter divisions are not critical to the organization of the book, as Sections are the main organi-zational unit. Sections are designed to be the subject of a single lecture or classroom session, though there is frequently more material than can be discussed and illustrated in a fifty-minute session. Con-sequently, the instructor will need to be selective about which topics to illustrate with other examples and which topics to leave to the student’s reading. Many of the examples are meant to be large, such as using five or six variables in a system of equations, so the instructor may just want to “walk” a class through these examples. The book has been written with the idea that some may work through it independently, so the hope is that students can learn some of the more mechanical ideas on their own. The highest level division of the book is the three Parts: Core, Topics, Applications (Part C [2],

A First Course in Linear Algebra - Area