a first course in linear algebra - robert a. beezer

column spaces are also presented early, simply as sets, saving most of their vector spaceproperties for later, so they are familiar objects before being scrutinized carefully.You cannot

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A First Course in Linear Algebra

by Robert A Beezer Department of Mathematics and Computer Science

University of Puget Sound

Version 0.70January 5, 2006c

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Copyright c

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

“GNU Free Documentation License”

Most recent version can be found at http://linear.ups.edu/

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This textbook is designed to teach the university mathematics student the basics ofthe subject of linear algebra There are no prerequisites other than ordinary algebra,but it is probably best used by a student who has the “mathematical maturity” of asophomore or junior

The text has two goals: to teach the fundamental concepts and techniques of matrixalgebra and abstract vector spaces, and to teach the techniques associated with under-standing 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 amonopoly — the exclusive right to control the making of copies and derivative works formany years (too many years in some cases) It also gives others limited rights, generallyreferred to as “fair use,” such as the right to quote sections in a review without seekingpermission However, the author licenses this book to anyone under the terms of the GNUFree Documentation License (GFDL), which gives you more rights than most copyrights.Loosely speaking, you may make as many copies as you like at no cost, and you maydistribute these unmodified copies if you please You may modify the book for your ownuse 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 licensethe 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 licenseitself for all 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 (theFrench would say gratis, at no cost) And it is However, it is more important that thebook 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 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 latestversion (and other news) at http://linear.ups.edu/

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 alsohave 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 and

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column spaces are also presented early, simply as sets, saving most of their vector spaceproperties for later, so they are familiar objects before being scrutinized carefully.You cannot do everything early, so in particular matrix multiplication comes laterthan usual However, with a definition built on linear combinations of column vectors,

it should seem more natural than the usual definition using dot products of rows withcolumns And this delay emphasizes that linear algebra is built upon vector addition andscalar multiplication Of course, matrix inverses must wait for matrix multiplication, butthis does not prevent nonsingular matrices from occurring sooner Vector space propertiesare hinted at when vector and matrix operations are first defined, but the notion of avector space is saved for a more axiomatic treatment later Once bases and dimensionhave been explored in the context of vector spaces, linear transformations and theirmatrix representations follow The goal of the book is to go as far as canonical formsand matrix decompositions in the Core, with less central topics collected in a section ofTopics

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 manyuniversity students this is their first exposure to careful definitions and theorems, andthe expectation that they fully understand them, to say nothing of the expectation thatthey 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 apartfrom the text Likewise, every theorem is carefully stated, and almost every one has acomplete proof Theorems usually have just one conclusion, so they can be referencedprecisely later Definitions and theorems are cataloged in order of their appearance inthe front of the book, and alphabetical order in the index at the back Along the way,there are discussions of some more important ideas relating to formulating proofs (ProofTechniques), which is advice mostly

and trends

• At the University of Puget Sound we teach a one-semester, post-calculus linearalgebra course to students majoring in mathematics, computer science, physics,chemistry and economics Between January 1986 and June 2002, I taught thiscourse 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 theinevitable and nearly-constant revisions Central to my new notes was a collection

of stock examples that would be used repeatedly to illustrate new concepts (Thesewould become the Archetypes, Chapter A [685].) It was only a short leap to thendecide to distribute copies of these notes and examples to the students in the twosections of this course As the semester wore on, the notes began to look less likenotes 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

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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 andexercises I had an almost identical experience with the third course I was teachingthat 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 theeconomics of traditional textbook publishing

• I had used TEX and the Internet for many years, so there was little to stand in theway of typesetting, distributing and “marketing” a free book With recreationaland professional interests in software development, I had long been fascinated by theopen-source software movement, as exemplified by the success of GNU and Linux,though public-domain TEX might also deserve mention Obviously, this book is anattempt to carry over that model of creative endeavor to textbook publishing

• As a sabbatical project during the Spring 2004 semester, I embarked on the currentproject of creating a freely-distributable linear algebra textbook (Notice the im-plied financial support of the University of Puget Sound to this project.) Most ofthe material was written from scratch since changes in notation and approach mademuch of my notes of little use By August 2004 I had written half the materialnecessary for our Math 232 course The remaining half was written during the Fall

2004 semester as I taught another two sections of Math 232

• I taught a single section of the course in the Spring 2005 semester, while my league, Professor Martin Jackson, graciously taught another section from the con-stantly shifting sands that was this project (version 0.30) His many suggestionshave helped immeasurably For the Fall 2005 semester, I taught two sections of thecourse from version 0.50

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

This book was born in the classroom, and arose from the spontaneous interest

of a Mathematical Sixth in the construction of simple models A desire toshow that even in mathematics one could have fun led to an exhibition ofthe results and attracted considerable attention throughout the school Sincethen the Sherborne collection has grown, ideas have come from many sources,and widespread interest has been shown It seems therefore desirable to givepermanent form to the lessons of experience so that others can benefit bythem and be encouraged to undertake similar work

but are instead referenced by acronyms This means that Theorem XYZ will always beTheorem XYZ, no matter if new sections are added, or if an individual decides to remove

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certain other sections Within sections, the subsections are acronyms that begin with theacronym 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 one Definition B, butthere is also a Section B At first, all the letters flying around may be confusing, but withtime, you will begin to recognize the more important ones on sight Furthermore, thereare lists of theorems, examples, etc in the front of the book, and an index that containsevery acronym If you are reading this in an electronic version (PDF or XML), you willsee 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! Differenteditions, different margins, or different sized paper will affect what content is on eachpage 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 arethe main organizational unit Sections are designed to be the subject of a single lecture

or classroom session, though there is frequently more material than can be discussedand illustrated in a fifty-minute session Consequently, the instructor will need to beselective 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” aclass through these examples The book has been written with the idea that some maywork through it independently, so the hope is that students can learn some of the moremechanical ideas on their own

The highest level division of the book is the three Parts: Core, Topics, Applications.The Core is meant to carefully describe the basic ideas required of a first exposure to linearalgebra In the final sections of the Core, one should ask the question: which previousSections could be removed without destroying the logical development of the subject?Hopefully, the answer is “none.” The goal of the book is to finish the Core with the mostgeneral representations of linear transformations (Jordan and rational canonical forms)and perhaps matrix decompositions (LU , QR, singular value) Of course, there will not

be universal agreement on what should, or should not, constitute the Core, but the mainidea will be to limit it to about forty sections Topics is meant to contain those subjectsthat are important in linear algebra, and which would make profitable detours from theCore for those interested in pursuing them Applications should illustrate the power andwidespread applicability of linear algebra to as many fields as possible The Archetypes(Chapter A [685]) cover many of the computational aspects of systems of linear equations,matrices and linear transformations The student should consult them often, and this isencouraged by exercises that simply suggest the right properties to examine at the righttime But what is more important, they are a repository that contains enough variety

to provide abundant examples of key theorems, while also providing counterexamples tohypotheses or converses of theorems

I require my students to read each Section prior to the day’s discussion on that section.For some students this is a novel idea, but at the end of the semester a few always report

on the benefits, both for this course and other courses where they have adopted thehabit To make good on this requirement, each section contains three Reading Questions.These sometimes only require parroting back a key definition or theorem, or they require

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performing a small example of a key computation, or they ask for musings on key ideas

or new relationships between old ideas Answers are emailed to me the evening beforethe lecture Given the flavor and purpose of these questions, including solutions seemsfoolish

Formulating interesting and effective exercises is as difficult, or more so, than building

a narrative But it is the place where a student really learns the material As such, forthe student’s benefit, complete solutions should be given As the list of exercises expands,over time solutions will also be provided Exercises and their solutions are referenced with

a section name, followed by a dot, then a letter (C,M, or T) and a number The letter ‘C’indicates a problem that is mostly computational in nature, while the letter ‘T’ indicates

a problem that is more theoretical in nature A problem with a letter ‘M’ is somewhere

in between (middle, mid-level, median, middling), probably a mix of computation andapplications of theorems So Solution MO.T34 is a solution to an exercise in Section MOthat is theoretical in nature The number ‘34’ has no intrinsic meaning

along with the underlying TEX code from which the book is built This arrangementprovides many benefits unavailable with traditional texts

• No cost, or low cost, to students With no physical vessel (i.e paper, binding), notransportation costs (Internet bandwidth being a negligible cost) and no marketingcosts (evaluation and desk copies are free to all), anyone with an Internet connectioncan obtain it, and a teacher could make available paper copies in sufficient quantitiesfor a class The cost to print a copy is not insignificant, but is just a fraction ofthe cost of a traditional textbook Students will not feel the need to sell back theirbook, and in future years can even pick up a newer edition freely

• The book will not go out of print No matter what, a teacher can maintain theirown copy and use the book for as many years as they desire Further, the namingschemes for chapters, sections, theorems, etc is designed so that the addition ofnew material will not break any course syllabi or assignment list

• With many eyes reading the book and with frequent postings of updates, the bility should become very high Please report any errors you find that persist intothe latest version

relia-• For those with a working installation of the popular typesetting program TEX, thebook has been designed so that it can be customized Page layouts, presence of exer-cises, solutions, sections or chapters can all be easily controlled Furthermore, manyvariants of mathematical notation are achieved via TEX macros So by changing asingle macro, one’s favorite notation can be reflected throughout the text For ex-ample, every transpose of a matrix is coded in the source as \transpose{A}, which

any desired alternative notation will then appear throughout the text instead

• The book has also been designed to make it easy for others to contribute material.Would you like to see a section on symmetric bilinear forms? Consider writing

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one and contributing it to one of the Topics chapters Does there need to be moreexercises about the null space of a matrix? Send me some Historical Notes?Contact me, and we will see about adding those in also

• You have no legal obligation to pay for this book It has been licensed with noexpectation that you pay for it You do not even have a moral obligation to payfor the book Thomas Jefferson (1743 – 1826), the author of the United StatesDeclaration of Independence, wrote,

If nature has made any one thing less susceptible than all others of sive property, it is the action of the thinking power called an idea, which

exclu-an individual may exclusively possess as long as he keeps it to himself; butthe moment it is divulged, it forces itself into the possession of every one,and the receiver cannot dispossess himself of it Its peculiar character,too, is that no one possesses the less, because every other possesses thewhole of it He who receives an idea from me, receives instruction him-self without lessening mine; as he who lights his taper at mine, receiveslight without darkening me That ideas should freely spread from one toanother over the globe, for the moral and mutual instruction of man, andimprovement of his condition, seems to have been peculiarly and benev-olently designed by nature, when she made them, like fire, expansibleover all space, without lessening their density in any point, and like theair in which we breathe, move, and have our physical being, incapable ofconfinement or exclusive appropriation

Letter to Isaac McPherson

August 13, 1813However, if you feel a royalty is due the author, or if you would like to encouragethe author, or if you wish to show others that this approach to textbook publishingcan also bring financial gains, then donations are gratefully received Moreover,non-financial forms of help can often be even more valuable A simple note ofencouragement, submitting a report of an error, or contributing some exercises orperhaps an entire section for the Topics or Applications chapters are all importantways you can acknowledge the freedoms accorded to this work by the copyrightholder and other contributors

prof-itable I hope that instructors find it flexible enough to fit the needs of their course And

I hope that everyone will send me their comments and suggestions, and also consider themyriad ways they can help (as listed on the book’s website at linear.ups.edu)

Robert A BeezerTacoma, Washington

January, 2006

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Definitions ix

Theorems xi

Notation xiii

Examples xv

Proof Techniques xvii

Computation Notes xix

Contributors xxi

GNU Free Documentation License xxiii

1 APPLICABILITY AND DEFINITIONS xxiii

2 VERBATIM COPYING xxv

3 COPYING IN QUANTITY xxv

4 MODIFICATIONS xxv

5 COMBINING DOCUMENTS xxvii

6 COLLECTIONS OF DOCUMENTS xxviii

7 AGGREGATION WITH INDEPENDENT WORKS xxviii

8 TRANSLATION xxviii

9 TERMINATION xxix

10 FUTURE REVISIONS OF THIS LICENSE xxix

ADDENDUM: How to use this License for your documents xxix

Part C Core 3 Chapter SLE Systems of Linear Equations 3 WILA What is Linear Algebra? 3

LA “Linear” + “Algebra” 3

A An application: packaging trail mix 4

READ Reading Questions 8

EXC Exercises 11

SOL Solutions 13

SSLE Solving Systems of Linear Equations 15

PSS Possibilities for solution sets 17

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viii Contents

ESEO Equivalent systems and equation operations 18

EXC Exercises 29

SOL Solutions 31

RREF Reduced Row-Echelon Form 33

EXC Exercises 47

SOL Solutions 51

TSS Types of Solution Sets 55

EXC Exercises 67

SOL Solutions 69

HSE Homogeneous Systems of Equations 71

SHS Solutions of Homogeneous Systems 71

MVNSE Matrix and Vector Notation for Systems of Equations 74

NSM Null Space of a Matrix 77

EXC Exercises 81

SOL Solutions 83

NSM NonSingular Matrices 85

EXC Exercises 95

SOL Solutions 97

Chapter V Vectors 99 VO Vector Operations 99

VEASM Vector equality, addition, scalar multiplication 100

VSP Vector Space Properties 104

EXC Exercises 107

SOL Solutions 109

LC Linear Combinations 111

VFSS Vector Form of Solution Sets 116

PSHS Particular Solutions, Homogeneous Solutions 129

URREF Uniqueness of Reduced Row-Echelon Form 131

EXC Exercises 135

SOL Solutions 139

SS Spanning Sets 141

SSV Span of a Set of Vectors 141

SSNS Spanning Sets of Null Spaces 147

EXC Exercises 155

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Contents ix

SOL Solutions 159

LI Linear Independence 165

LISV Linearly Independent Sets of Vectors 165

LINSM Linear Independence and NonSingular Matrices 171

NSSLI Null Spaces, Spans, Linear Independence 173

EXC Exercises 177

SOL Solutions 181

LDS Linear Dependence and Spans 187

LDSS Linearly Dependent Sets and Spans 187

COV Casting Out Vectors 190

EXC Exercises 199

SOL Solutions 201

O Orthogonality 203

CAV Complex arithmetic and vectors 203

IP Inner products 204

N Norm 207

OV Orthogonal Vectors 208

GSP Gram-Schmidt Procedure 211

EXC Exercises 217

Chapter M Matrices 219 MO Matrix Operations 219

MEASM Matrix equality, addition, scalar multiplication 219

TSM Transposes and Symmetric Matrices 222

MCC Matrices and Complex Conjugation 225

EXC Exercises 229

SOL Solutions 231

MM Matrix Multiplication 233

MVP Matrix-Vector Product 233

MMEE Matrix Multiplication, Entry-by-Entry 239

PMM Properties of Matrix Multiplication 241

EXC Exercises 247

SOL Solutions 249

MISLE Matrix Inverses and Systems of Linear Equations 251

IM Inverse of a Matrix 252

CIM Computing the Inverse of a Matrix 254

PMI Properties of Matrix Inverses 260

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x Contents

EXC Exercises 265

SOL Solutions 267

MINSM Matrix Inverses and NonSingular Matrices 269

NSMI NonSingular Matrices are Invertible 269

OM Orthogonal Matrices 272

EXC Exercises 277

SOL Solutions 279

CRS Column and Row Spaces 281

CSSE Column spaces and systems of equations 281

CSSOC Column space spanned by original columns 284

CSNSM Column Space of a Nonsingular Matrix 286

RSM Row Space of a Matrix 288

EXC Exercises 297

SOL Solutions 301

FS Four Subsets 305

LNS Left Null Space 305

CRS Computing Column Spaces 306

EEF Extended echelon form 310

FS Four Subsets 313

EXC Exercises 325

SOL Solutions 329

Chapter VS Vector Spaces 333 VS Vector Spaces 333

VS Vector Spaces 333

EVS Examples of Vector Spaces 335

RD Recycling Definitions 346

EXC Exercises 347

S Subspaces 349

TS Testing Subspaces 351

TSS The Span of a Set 355

SC Subspace Constructions 361

EXC Exercises 363

SOL Solutions 365

B Bases 369

LI Linear independence 369

SS Spanning Sets 373

B Bases 378

BRS Bases from Row Spaces 382

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Contents xi

BNSM Bases and NonSingular Matrices 384

VR Vector Representation 385

EXC Exercises 389

SOL Solutions 391

D Dimension 395

DVS Dimension of Vector Spaces 400

RNM Rank and Nullity of a Matrix 402

RNNSM Rank and Nullity of a NonSingular Matrix 404

EXC Exercises 407

SOL Solutions 409

PD Properties of Dimension 413

GT Goldilocks’ Theorem 413

RT Ranks and Transposes 417

OBC Orthonormal Bases and Coordinates 418

EXC Exercises 423

SOL Solutions 425

Chapter D Determinants 427 DM Determinants of Matrices 427

CD Computing Determinants 429

PD Properties of Determinants 432

EXC Exercises 435

SOL Solutions 437

Chapter E Eigenvalues 439 EE Eigenvalues and Eigenvectors 439

EEM Eigenvalues and Eigenvectors of a Matrix 439

PM Polynomials and Matrices 441

EEE Existence of Eigenvalues and Eigenvectors 443

CEE Computing Eigenvalues and Eigenvectors 447

ECEE Examples of Computing Eigenvalues and Eigenvectors 451

EXC Exercises 461

SOL Solutions 463

PEE Properties of Eigenvalues and Eigenvectors 469

ME Multiplicities of Eigenvalues 475

EHM Eigenvalues of Hermitian Matrices 479

EXC Exercises 483

SOL Solutions 485

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xii Contents

SD Similarity and Diagonalization 487

SM Similar Matrices 487

PSM Properties of Similar Matrices 489

D Diagonalization 491

OD Orthonormal Diagonalization 500

EXC Exercises 501

SOL Solutions 503

Chapter LT Linear Transformations 507 LT Linear Transformations 507

LT Linear Transformations 507

MLT Matrices and Linear Transformations 512

LTLC Linear Transformations and Linear Combinations 517

PI Pre-Images 520

NLTFO New Linear Transformations From Old 523

EXC Exercises 529

SOL Solutions 531

ILT Injective Linear Transformations 535

EILT Examples of Injective Linear Transformations 535

KLT Kernel of a Linear Transformation 539

ILTLI Injective Linear Transformations and Linear Independence 544

ILTD Injective Linear Transformations and Dimension 545

CILT Composition of Injective Linear Transformations 546

EXC Exercises 547

SOL Solutions 549

SLT Surjective Linear Transformations 553

ESLT Examples of Surjective Linear Transformations 553

RLT Range of a Linear Transformation 558

SSSLT Spanning Sets and Surjective Linear Transformations 563

SLTD Surjective Linear Transformations and Dimension 565

CSLT Composition of Surjective Linear Transformations 566

EXC Exercises 567

SOL Solutions 569

IVLT Invertible Linear Transformations 573

IV Invertibility 577

SI Structure and Isomorphism 579

RNLT Rank and Nullity of a Linear Transformation 582

SLELT Systems of Linear Equations and Linear Transformations 585

EXC Exercises 589

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Contents xiii

SOL Solutions 591

Chapter R Representations 595 VR Vector Representations 595

CVS Characterization of Vector Spaces 602

CP Coordinatization Principle 603

EXC Exercises 609

SOL Solutions 611

MR Matrix Representations 613

NRFO New Representations from Old 620

PMR Properties of Matrix Representations 626

EXC Exercises 637

SOL Solutions 641

CB Change of Basis 651

EELT Eigenvalues and Eigenvectors of Linear Transformations 651

CBM Change-of-Basis Matrix 653

MRS Matrix Representations and Similarity 659

CELT Computing Eigenvectors of Linear Transformations 667

EXC Exercises 679

SOL Solutions 681

Chapter A Archetypes 685 A 689

B 694

C 699

D 703

E 707

F 711

G 717

H 721

I 726

J 731

K 736

L 741

M 745

N 748

O 751

P 754

Q 756

R 760

S 763

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xiv Contents

T 763

U 763

V 764

W 764

Part T Topics 767 Chapter P Preliminaries 767 CNO Complex Number Operations 767

CNA Arithmetic with complex numbers 767

CCN Conjugates of Complex Numbers 768

MCN Modulus of a Complex Number 769

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Section WILA

Section SSLE

SLE System of Linear Equations 16

ES Equivalent Systems 18

EO Equation Operations 19

Section RREF M Matrix 33

AM Augmented Matrix 35

RO Row Operations 36

REM Row-Equivalent Matrices 36

RREF Reduced Row-Echelon Form 38

ZRM Zero Row of a Matrix 39

LO Leading Ones 39

PC Pivot Columns 39

RR Row-Reducing 45

Section TSS CS Consistent System 55

IDV Independent and Dependent Variables 58

Section HSE HS Homogeneous System 71

TSHSE Trivial Solution to Homogeneous Systems of Equations 72

CV Column Vector 74

ZV Zero Vector 74

CM Coefficient Matrix 75

VOC Vector of Constants 75

SV Solution Vector 75

NSM Null Space of a Matrix 77

Section NSM SQM Square Matrix 85

NM Nonsingular Matrix 85

IM Identity Matrix 86

Section VO VSCV Vector Space of Column Vectors 99

CVE Column Vector Equality 100

CVA Column Vector Addition 101

CVSM Column Vector Scalar Multiplication 102

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xvi Definitions

Section LC

LCCV Linear Combination of Column Vectors 111

Section SS SSCV Span of a Set of Column Vectors 141

Section LI RLDCV Relation of Linear Dependence for Column Vectors 165

LICV Linear Independence of Column Vectors 165

Section LDS Section O CCCV Complex Conjugate of a Column Vector 203

IP Inner Product 204

NV Norm of a Vector 207

OV Orthogonal Vectors 209

OSV Orthogonal Set of Vectors 209

ONS OrthoNormal Set 214

Section MO VSM Vector Space of m × n Matrices 219

ME Matrix Equality 219

MA Matrix Addition 220

MSM Matrix Scalar Multiplication 220

ZM Zero Matrix 222

TM Transpose of a Matrix 222

SYM Symmetric Matrix 223

CCM Complex Conjugate of a Matrix 226

Section MM MVP Matrix-Vector Product 233

Section MISLE MI Matrix Inverse 252

SUV Standard Unit Vectors 254

Section MINSM OM Orthogonal Matrices 272

A Adjoint 275

HM Hermitian Matrix 276

Section CRS CSM Column Space of a Matrix 281

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Definitions xvii

RSM Row Space of a Matrix 289

Section FS LNS Left Null Space 305

EEF Extended Echelon Form 310

Section VS VS Vector Space 333

Section S S Subspace 349

TS Trivial Subspaces 354

LC Linear Combination 355

SS Span of a Set 356

Section B RLD Relation of Linear Dependence 369

LI Linear Independence 369

TSVS To Span a Vector Space 374

B Basis 378

Section D D Dimension 395

NOM Nullity Of a Matrix 402

ROM Rank Of a Matrix 402

Section PD Section DM SM SubMatrix 427

DM Determinant of a Matrix 427

MIM Minor In a Matrix 429

CIM Cofactor In a Matrix 429

Section EE EEM Eigenvalues and Eigenvectors of a Matrix 439

CP Characteristic Polynomial 448

EM Eigenspace of a Matrix 449

AME Algebraic Multiplicity of an Eigenvalue 451

GME Geometric Multiplicity of an Eigenvalue 452

Section PEE Section SD SIM Similar Matrices 487

DIM Diagonal Matrix 491

DZM Diagonalizable Matrix 491

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xviii Definitions

Section LT

LT Linear Transformation 507

PI Pre-Image 521

LTA Linear Transformation Addition 523

LTSM Linear Transformation Scalar Multiplication 524

LTC Linear Transformation Composition 526

Section ILT ILT Injective Linear Transformation 535

KLT Kernel of a Linear Transformation 539

Section SLT SLT Surjective Linear Transformation 553

RLT Range of a Linear Transformation 558

Section IVLT IDLT Identity Linear Transformation 573

IVS Isomorphic Vector Spaces 580

ROLT Rank Of a Linear Transformation 582

NOLT Nullity Of a Linear Transformation 582

Section VR VR Vector Representation 595

Section MR MR Matrix Representation 613

Section CB EELT Eigenvalue and Eigenvector of a Linear Transformation 651

CBM Change-of-Basis Matrix 653

Section CNO CCN Conjugate of a Complex Number 768

MCN Modulus of a Complex Number 769

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Section WILA

Section SSLE

EOPSS Equation Operations Preserve Solution Sets 20

Section RREF 2 37

REMES Row-Equivalent Matrices represent Equivalent Systems 37

REMEF Row-Equivalent Matrix in Echelon Form 40

Section TSS RCLS Recognizing Consistency of a Linear System 60

ICRN Inconsistent Systems, r and n 61

CSRN Consistent Systems, r and n 61

FVCS Free Variables for Consistent Systems 61

PSSLS Possible Solution Sets for Linear Systems 63

CMVEI Consistent, More Variables than Equations, Infinite solutions 63

Section HSE HSC Homogeneous Systems are Consistent 72

HMVEI Homogeneous, More Variables than Equations, Infinite solutions 73

Section NSM NSRRI NonSingular matrices Row Reduce to the Identity matrix 87

NSTNS NonSingular matrices have Trivial Null Spaces 88

NSMUS NonSingular Matrices and Unique Solutions 89

NSME1 NonSingular Matrix Equivalences, Round 1 92

Section VO VSPCV Vector Space Properties of Column Vectors 104

Section LC SLSLC Solutions to Linear Systems are Linear Combinations 115

VFSLS Vector Form of Solutions to Linear Systems 122

PSPHS Particular Solution Plus Homogeneous Solutions 129

RREFU Reduced Row-Echelon Form is Unique 132

Section SS SSNS Spanning Sets for Null Spaces 148

Section LI LIVHS Linearly Independent Vectors and Homogeneous Systems 168

LIVRN Linearly Independent Vectors, r and n 170

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xx Theorems

Section LDS

Section O

Section MO

Section MM

Section MISLE

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Theorems xxi

Section MINSM

Section CRS

Section FS

Section VS

Section S

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xxii Theorems

Section B

Section D

Section PD

Section DM

Section EE

Section PEE

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Theorems xxiii

Section SD

Section LT

Section ILT

Section SLT

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xxiv Theorems

Section IVLT

Section VR

Section MR

Section CB

Section CNO

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Theorems xxv

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xxvi Theorems

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Section WILA

Section SSLE

Section RREF

Section TSS

Section HSE

Section NSM

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xxx Examples

Section VO

Section LC

Section SS

Section LI

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Examples xxxi

Section MO

Section MM

Section MISLE

Section MINSM

Section CRS

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xxxii Examples

Section FS

Section VS

Section S

Section B

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Section PD

Section DM

Section EE

Section PEE

Section SD

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xxxiv Examples

Section LT

Section ILT

Section SLT

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Examples xxxv

Section IVLT

Section VR

Section MR

Section CB

Section CNO

Trang 38

xxxvi Examples

Trang 40

xxxviii Proof Techniques

Tiêu đề	A First Course in Linear Algebra
Tác giả	Robert A. Beezer
Trường học	University of Puget Sound
Chuyên ngành	Mathematics
Thể loại	Textbook
Năm xuất bản	2006
Thành phố	Tacoma

Định dạng
Số trang	859
Dung lượng	4,89 MB