A First Course in Linear Algebra by Robert A Beezer Department of Mathematics and Computer Science University of Puget Sound Version 0.70 January 5, 2006 c 2004, 2005, 2006 Copyright c 2004, 2005, 2006 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 the Invariant Sections being “Preface”, no Front-Cover Texts, 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/ Preface This textbook is designed to teach the university mathematics student the basics of the 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 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 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 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 (the French would say gratis, at no cost) And it is However, it is more important that the book has freedom (the French would say libert´, liberty) It will never go “out of print” e 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 [219]) 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 and i ii column 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 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 usual definition using dot products of rows with columns And this delay emphasizes 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 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 canonical forms and matrix decompositions in the Core, with less central topics collected in a section of Topics 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, 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 advice mostly 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, Chapter A [685].) 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 Version 0.70 iii 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 • I had used TEX and the Internet for many years, so there was little to stand in the way of typesetting, distributing and “marketing” a free book With recreational and professional interests in software development, I had long been fascinated by the open-source software movement, as exemplified by the success of GNU and Linux, though public-domain TEX might also deserve mention Obviously, this book is an attempt to carry over 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 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 • I taught a single section of the course in the Spring 2005 semester, while my colleague, Professor Martin Jackson, graciously taught another section from the constantly shifting sands that was this project (version 0.30) His many suggestions have helped immeasurably For the Fall 2005 semester, I taught two sections of the course from version 0.50 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 of Mathematical 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 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 Version 0.70 iv 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 one Definition B, but there is also a Section B 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 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 discussed and illustrated in a fifty-minute session Consequently, 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 The Core is meant to carefully describe the basic ideas required of a first exposure to linear algebra In the final sections of the Core, one should ask the question: which previous Sections 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 most general 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 main idea will be to limit it to about forty sections Topics is meant to contain those subjects that are important in linear algebra, and which would make profitable detours from the Core for those interested in pursuing them Applications should illustrate the power and widespread 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 is encouraged by exercises that simply suggest the right properties to examine at the right time But what is more important, they are a repository that contains enough variety to provide abundant examples of key theorems, while also providing counterexamples to hypotheses 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 the habit 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 Version 0.70 v 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 before the lecture Given the flavor and purpose of these questions, including solutions seems foolish 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, for the 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 and applications of theorems So Solution MO.T34 is a solution to an exercise in Section MO that is theoretical in nature The number ‘34’ has no intrinsic meaning More on Freedom This book is freely-distributable under the terms of the GFDL, along with the underlying TEX code from which the book is built This arrangement provides many benefits unavailable with traditional texts • No cost, or low cost, to students With no physical vessel (i.e paper, binding), no transportation costs (Internet bandwidth being a negligible cost) and no marketing costs (evaluation and desk copies are free to all), anyone with an Internet connection can obtain it, and a teacher could make available paper copies in sufficient quantities for a class The cost to print a copy is not insignificant, but is just a fraction of the cost of a traditional textbook Students will not feel the need to sell back their book, 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 their own copy and use the book for as many years as they desire Further, the naming schemes for chapters, sections, theorems, etc is designed so that the addition of new material will not break any course syllabi or assignment list • With many eyes reading the book and with frequent postings of updates, the reliability should become very high Please report any errors you find that persist into the latest version • For those with a working installation of the popular typesetting program TEX, the book has been designed so that it can be customized Page layouts, presence of exercises, solutions, sections or chapters can all be easily controlled Furthermore, many variants of mathematical notation are achieved via TEX macros So by changing a single macro, one’s favorite notation can be reflected throughout the text For example, every transpose of a matrix is coded in the source as \transpose{A}, which when printed will yield At However by changing the definition of \transpose{ }, 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 Version 0.70 vi one and contributing it to one of the Topics chapters Does there need to be more exercises 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 no expectation that you pay for it You not even have a moral obligation to pay for the book Thomas Jefferson (1743 – 1826), the author of the United States Declaration of Independence, wrote, If nature has made any one thing less susceptible than all others of exclusive property, it is the action of the thinking power called an idea, which an individual may exclusively possess as long as he keeps it to himself; but the 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 the whole of it He who receives an idea from me, receives instruction himself without lessening mine; as he who lights his taper at mine, receives light without darkening me That ideas should freely spread from one to another over the globe, for the moral and mutual instruction of man, and improvement of his condition, seems to have been peculiarly and benevolently designed by nature, when she made them, like fire, expansible over all space, without lessening their density in any point, and like the air in which we breathe, move, and have our physical being, incapable of confinement or exclusive appropriation Letter to Isaac McPherson August 13, 1813 However, if you feel a royalty is due the author, or if you would like to encourage the author, or if you wish to show others that this approach to textbook publishing can also bring financial gains, then donations are gratefully received Moreover, non-financial forms of help can often be even more valuable A simple note of encouragement, submitting a report of an error, or contributing some exercises or perhaps an entire section for the Topics or Applications chapters are all important ways you can acknowledge the freedoms accorded to this work by the copyright holder and other contributors Conclusion Foremost, I hope that students find their time spent with this book profitable 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 the myriad ways they can help (as listed on the book’s website at linear.ups.edu) Robert A Beezer Tacoma, Washington January, 2006 Version 0.70 Contents Preface i Contents Definitions Theorems Notation Examples Proof Techniques Computation Notes Contributors GNU Free Documentation License APPLICABILITY AND DEFINITIONS VERBATIM COPYING COPYING IN QUANTITY MODIFICATIONS COMBINING DOCUMENTS COLLECTIONS OF DOCUMENTS AGGREGATION WITH INDEPENDENT WORKS TRANSLATION TERMINATION 10 FUTURE REVISIONS OF THIS LICENSE ADDENDUM: How to use this License for your documents Part C Core vii ix xi xiii xv xvii xix xxi xxiii xxiii xxv xxv xxv xxvii xxviii xxviii xxviii xxix xxix xxix Chapter SLE Systems of Linear Equations WILA What is Linear Algebra? LA “Linear” + “Algebra” A An application: packaging trail mix READ Reading Questions EXC Exercises SOL Solutions SSLE Solving Systems of Linear Equations PSS Possibilities for solution sets vii 3 11 13 15 17 viii Contents ESEO Equivalent systems and equation operations READ Reading Questions EXC Exercises SOL Solutions RREF Reduced Row-Echelon Form READ Reading Questions EXC Exercises SOL Solutions TSS Types of Solution Sets READ Reading Questions EXC Exercises SOL Solutions HSE Homogeneous Systems of Equations SHS Solutions of Homogeneous Systems MVNSE Matrix and Vector Notation for Systems of NSM Null Space of a Matrix READ Reading Questions EXC Exercises SOL Solutions NSM NonSingular Matrices NSM NonSingular Matrices READ Reading Questions EXC Exercises SOL Solutions Equations Chapter V Vectors VO Vector Operations VEASM Vector equality, addition, scalar multiplication VSP Vector Space Properties READ Reading Questions EXC Exercises SOL Solutions LC Linear Combinations LC Linear Combinations VFSS Vector Form of Solution Sets PSHS Particular Solutions, Homogeneous Solutions URREF Uniqueness of Reduced Row-Echelon Form READ Reading Questions EXC Exercises SOL Solutions SS Spanning Sets SSV Span of a Set of Vectors SSNS Spanning Sets of Null Spaces READ Reading Questions EXC Exercises Version 0.70 18 27 29 31 33 46 47 51 55 65 67 69 71 71 74 77 79 81 83 85 85 93 95 97 99 99 100 104 106 107 109 111 111 116 129 131 134 135 139 141 141 147 153 155 INDEX 793 MLT (subsection of LT), 512 MLTCV (theorem), 515 MLTLT (theorem), 524 MM (definition), 237 MM (section), 233 MM (subsection of MM), 237 MM.MMA (computation), 239 MMA (theorem), 243 MMCC (theorem), 244 MMDAA (theorem), 242 MMEE (subsection of MM), 239 MMIM (theorem), 241 MMIP (theorem), 244 MMNC (example), 238 MMSMM (theorem), 243 MMT (theorem), 245 MMZM (theorem), 241 MNEM (theorem), 479 MNSLE (example), 234 MO (section), 219 MOLT (example), 516 more variables than equations example OSGMD, 63 theorem CMVEI, 63 MPMR (example), 622 MR (definition), 613 MR (section), 613 MRBE (example), 663 MRCB (theorem), 659 MRCLT (theorem), 622 MRCM (example), 660 MRMLT (theorem), 621 MRS (subsection of CB), 659 MRSLT (theorem), 620 MSCN (example), 769 MSM (definition), 220 MSM (example), 220 MTV (example), 233 multiple equivalences technique ME, 92 MVNSE (subsection of HSE), 74 MVP (definition), 233 MVP (notation), 233 MVP (subsection of MM), 233 MVSLD (theorem), 171 MWIAA (example), 252 N (archetype), 748 N (subsection of O), 207 NDMS4 (example), 496 NEM (theorem), 476 NIAO (example), 543 NIAQ (example), 535 NIAQR (example), 543 NIDAU (example), 545 NKAO (example), 539 NLT (example), 510 NLTFO (subsection of LT), 523 NM (definition), 85 NOILT (theorem), 582 NOLT (definition), 582 NOLT (notation), 582 NOM (definition), 402 NOM (notation), 402 nonsingular columns as basis theorem CNSMB, 384 nonsingular matrices linearly independent columns theorem NSLIC, 172 nonsingular matrix Archetype B example NS, 86 column space, 287 equivalences theorem NSME1, 92 theorem NSME2, 172 theorem NSME3, 271 theorem NSME4, 288 theorem NSME5, 385 theorem NSME6, 405 theorem NSME7, 433 theorem NSME8, 470 theorem NSME9, 635 matrix inverse, 271 null space example NSNS, 88 nullity, 405 rank Version 0.70 794 INDEX theorem RNNSM, 405 row-reduced theorem NSRRI, 87 trivial null space theorem NSTNS, 88 unique solutions theorem NSMUS, 89 nonsingular matrix, row-reduced example NSRR, 87 norm example CNSV, 207 inner product, 207 notation, 207 notation AM, 76 CCCV, 203 CCN, 768 CSM, 281 CVA, 101 CVSM, 102 D, 395 DM, 428 IP, 204 KLT, 539 LNS, 305 LS, 76 LT, 507 M, 33 ME, 33 MI, 252 MVP, 233 NOLT, 582 NOM, 402 NSM, 77 NV, 207 RLT, 558 RO, 36 ROLT, 582 ROM, 402 RREFA, 55 RSM, 289 SSV, 141 TM, 222 V, 74 VE, 74 Version 0.70 VSCV, 99 ZM, 222 ZV, 75 notation for a linear system example NSE, 17 NRFO (subsection of MR), 620 NRREF (example), 39 NS (example), 86 NSAO (example), 562 NSAQ (example), 553 NSAQR (example), 561 NSC2A (example), 353 NSC2S (example), 353 NSC2Z (example), 353 NSDAT (example), 565 NSDS (example), 149 NSE (example), 17 NSEAI (example), 77 NSI (theorem), 271 NSLE (example), 76 NSLIC (theorem), 172 NSLIL (example), 174 NSM (definition), 77 NSM (notation), 77 NSM (section), 85 NSM (subsection of HSE), 77 NSM (subsection of NSM), 85 NSME1 (theorem), 92 NSME2 (theorem), 172 NSME3 (theorem), 271 NSME4 (theorem), 288 NSME5 (theorem), 385 NSME6 (theorem), 405 NSME7 (theorem), 433 NSME8 (theorem), 470 NSME9 (theorem), 635 NSMI (subsection of MINSM), 269 NSMS (theorem), 354 NSMUS (theorem), 89 NSNS (example), 88 NSRR (example), 87 NSRRI (theorem), 87 NSS (example), 88 NSSLI (subsection of LI), 173 NSTNS (theorem), 88 INDEX 795 Null space as a span example NSDS, 149 null space Archetype I example NSEAI, 77 basis theorem BNS, 173 computation example CNS1, 78 example CNS2, 79 isomorphic to kernel, 626 matrix definition NSM, 77 nonsingular matrix, 88 notation notation NSM, 77 singular matrix, 88 spanning set example SSNS, 148 theorem SSNS, 148 subspace theorem NSMS, 354 null space span, linearly independent Archetype L example NSLIL, 174 nullity computing, 403 injective linear transformation theorem NOILT, 582 linear transformation definition NOLT, 582 matrix, 403 definition NOM, 402 notation, 402, 582 square matrix, 404 NV (definition), 207 NV (notation), 207 O (archetype), 751 O (Property), 334 O (section), 203 OBC (subsection of PD), 418 OC (Property), 105 OD (subsection of SD), 500 OLTTR (example), 613 OM (definition), 272 OM (Property), 221 OM (subsection of MINSM), 272 OM3 (example), 272 OMI (theorem), 273 OMPIP (theorem), 275 one column vectors Property OC, 105 matrices Property OM, 221 vectors Property O, 334 ONFV (example), 214 ONS (definition), 214 ONTV (example), 214 OPM (example), 273 orthogonal linear independence theorem OSLI, 210 permutation matrix example OPM, 273 set example AOS, 209 set of vectors definition OSV, 209 size example OM3, 272 vector pairs definition OV, 209 orthogonal matrices columns theorem COMOS, 273 orthogonal matrix inner product theorem OMPIP, 275 orthogonal vectors example TOV, 209 orthonormal definition ONS, 214 matrix columns example OSMC, 274 orthonormal set Version 0.70 796 INDEX four vectors example ONFV, 214 three vectors example ONTV, 214 OSGMD (example), 63 OSIS (theorem), 270 OSLI (theorem), 210 OSMC (example), 274 OSV (definition), 209 OV (definition), 209 OV (subsection of O), 208 P (archetype), 754 P (chapter), 767 P (technique), 223 particular solutions example PSNS, 130 PC (definition), 39 PCVS (example), 343 PD (section), 413 PD (subsection of DM), 432 PEE (section), 469 PEEF (theorem), 311 PI (definition), 521 PI (subsection of LT), 520 PI (technique), 105 PIP (theorem), 208 pivot column definition PC, 39 PM (example), 441 PM (subsection of EE), 441 PMI (subsection of MISLE), 260 PMM (subsection of MM), 241 PMR (subsection of MR), 626 polynomial of a matrix example PM, 441 polynomial vector space dimension theorem DP, 400 practice technique P, 223 pre-image definition PI, 521 Version 0.70 kernel theorem KPI, 541 pre-images example SPIAS, 521 Property AA, 334 AAC, 104 AAM, 221 AC, 333 ACC, 104 ACM, 221 AI, 334 AIC, 104 AIM, 221 C, 334 CC, 104 CM, 221 DMAM, 221 DSA, 334 DSAC, 104 DSAM, 221 DVA, 334 DVAC, 104 O, 334 OC, 105 OM, 221 SC, 333 SCC, 104 SCM, 221 SMA, 334 SMAC, 104 SMAM, 221 Z, 334 ZC, 104 ZM, 221 PSHS (subsection of LC), 129 PSM (subsection of SD), 489 PSNS (example), 130 PSPHS (theorem), 129 PSS (subsection of SSLE), 17 PSSLS (theorem), 63 PTM (example), 238 PTMEE (example), 240 PWSMS (theorem), 269 INDEX 797 Q (archetype), 756 R (archetype), 760 R (chapter), 595 range full example FRAN, 560 isomorphic to column space theorem RCSI, 629 linear transformation example RAO, 558 notation, 558 of a linear transformation definition RLT, 558 pre-image theorem RPI, 564 subspace theorem RLTS, 559 surjective linear transformation theorem RSLT, 561 via matrix representation example RVMR, 630 rank computing theorem CRN, 403 linear transformation definition ROLT, 582 matrix definition ROM, 402 example RNM, 402 notation, 402, 582 of transpose example RRTI, 418 square matrix example RNSM, 404 surjective linear transformation theorem ROSLT, 582 transpose theorem RMRT, 417 rank+nullity theorem RPNC, 404 RAO (example), 558 RCLS (theorem), 60 RCSI (theorem), 629 RD (subsection of VS), 346 READ (subsection of B), 387 READ (subsection of CB), 677 READ (subsection of CRS), 295 READ (subsection of D), 406 READ (subsection of DM), 434 READ (subsection of EE), 459 READ (subsection of FS), 323 READ (subsection of HSE), 79 READ (subsection of ILT), 546 READ (subsection of IVLT), 587 READ (subsection of LC), 134 READ (subsection of LDS), 197 READ (subsection of LI), 174 READ (subsection of LT), 527 READ (subsection of MINSM), 276 READ (subsection of MISLE), 263 READ (subsection of MM), 246 READ (subsection of MO), 227 READ (subsection of MR), 636 READ (subsection of NSM), 93 READ (subsection of O), 215 READ (subsection of PD), 422 READ (subsection of PEE), 480 READ (subsection of RREF), 46 READ (subsection of S), 362 READ (subsection of SD), 500 READ (subsection of SLT), 566 READ (subsection of SS), 153 READ (subsection of SSLE), 27 READ (subsection of TSS), 65 READ (subsection of VO), 106 READ (subsection of VR), 606 READ (subsection of VS), 346 READ (subsection of WILA), reduced row-echelon form analysis notation notation RREFA, 55 definition RREF, 38 example NRREF, 39 example RREF, 39 extended definition EEF, 310 notation Version 0.70 798 INDEX example RREFN, 55 unique theorem RREFU, 132 reducing a span example RSC5, 188 relation of linear dependence definition RLD, 369 definition RLDCV, 165 REM (definition), 36 REMEF (theorem), 40 REMES (theorem), 37 REMRS (theorem), 290 RES (example), 196 RLD (definition), 369 RLDCV (definition), 165 RLT (definition), 558 RLT (notation), 558 RLT (subsection of SLT), 558 RLTS (theorem), 559 RMRT (theorem), 417 RNLT (subsection of IVLT), 582 RNM (example), 402 RNM (subsection of D), 402 RNNSM (subsection of D), 404 RNNSM (theorem), 405 RNSM (example), 404 RO (definition), 36 RO (notation), 36 ROLT (definition), 582 ROLT (notation), 582 ROM (definition), 402 ROM (notation), 402 ROSLT (theorem), 582 row operations definition RO, 36 notation, 36 row reduce mathematica, 45 ti83, 46 ti86, 45 row space Archetype I example RSAI, 289 as column space, 293 basis Version 0.70 example RSB, 382 theorem BRS, 292 matrix, 289 notation, 289 row-equivalent matrices theorem REMRS, 290 subspace theorem RSMS, 362 row-equivalent matrices definition REM, 36 example TREM, 36 row space, 290 row spaces example RSREM, 291 theorem REMES, 37 row-reduce the verb definition RR, 45 row-reduced matrices theorem REMEF, 40 RPI (theorem), 564 RPNC (theorem), 404 RPNDD (theorem), 583 RR (definition), 45 RR.MMA (computation), 45 RR.TI83 (computation), 46 RR.TI86 (computation), 45 RREF (definition), 38 RREF (example), 39 RREF (section), 33 RREFA (notation), 55 RREFN (example), 55 RREFU (theorem), 132 RRTI (example), 418 RS (example), 383 RSAI (example), 289 RSB (example), 382 RSC5 (example), 188 RSLT (theorem), 561 RSM (definition), 289 RSM (notation), 289 RSM (subsection of CRS), 288 RSMS (theorem), 362 RSNS (example), 355 RSREM (example), 291 INDEX 799 RSSC4 (example), 195 RT (subsection of PD), 417 RVMR (example), 630 S (archetype), 763 S (definition), 349 S (example), 86 S (section), 349 SAA (example), 42 SAB (example), 41 SABMI (example), 251 SAE (example), 44 SAN (example), 562 SAR (example), 555 SAV (example), 556 SC (Property), 333 SC (subsection of S), 361 SC3 (example), 349 SCAA (example), 144 SCAB (example), 146 SCAD (example), 151 scalar closure column vectors Property SCC, 104 matrices Property SCM, 221 vectors Property SC, 333 scalar multiple matrix inverse, 262 scalar multiplication canceling scalars theorem CSSM, 345 canceling vectors theorem CVSM, 345 zero scalar theorem ZSSM, 342 zero vector theorem ZVSM, 342 zero vector result theorem SMEZV, 344 scalar multiplication associativity column vectors Property SMAC, 104 matrices Property SMAM, 221 vectors Property SMA, 334 SCB (theorem), 662 SCC (Property), 104 SCM (Property), 221 SD (section), 487 SE (technique), 20 SEE (example), 439 SEEF (example), 310 SER (theorem), 489 set equality technique SE, 20 set notation technique SN, 58 shoes, 260 SHS (subsection of HSE), 71 SI (subsection of IVLT), 579 SIM (definition), 487 similar matrices equal eigenvalues example EENS, 490 eual eigenvalues theorem SMEE, 490 example SMS3, 488 example SMS5, 487 similarity definition SIM, 487 equivalence relation theorem SER, 489 singular matrix Archetype A example S, 86 null space example NSS, 88 product with theorem PWSMS, 269 singular matrix, row-reduced example SRR, 87 SLE (chapter), SLE (definition), 16 SLELT (subsection of IVLT), 585 SLEMM (theorem), 234 SLSLC (theorem), 115 Version 0.70 800 INDEX SLT (definition), 553 SLT (section), 553 SLTB (theorem), 564 SLTD (subsection of SLT), 565 SLTD (theorem), 565 SLTLT (theorem), 523 SM (definition), 427 SM (subsection of SD), 487 SM32 (example), 359 SMA (Property), 334 SMAC (Property), 104 SMAM (Property), 221 SMEE (theorem), 490 SMEZV (theorem), 344 SMLT (example), 525 SMS (theorem), 224 SMS3 (example), 488 SMS5 (example), 487 SMZD (theorem), 432 SMZE (theorem), 470 SN (technique), 58 SNSCM (theorem), 272 socks, 260 SOL (subsection of B), 391 SOL (subsection of CB), 681 SOL (subsection of CRS), 301 SOL (subsection of D), 409 SOL (subsection of DM), 437 SOL (subsection of EE), 463 SOL (subsection of FS), 329 SOL (subsection of HSE), 83 SOL (subsection of ILT), 549 SOL (subsection of IVLT), 591 SOL (subsection of LC), 139 SOL (subsection of LDS), 201 SOL (subsection of LI), 181 SOL (subsection of LT), 531 SOL (subsection of MINSM), 279 SOL (subsection of MISLE), 267 SOL (subsection of MM), 249 SOL (subsection of MO), 231 SOL (subsection of MR), 641 SOL (subsection of NSM), 97 SOL (subsection of PD), 425 SOL (subsection of PEE), 485 Version 0.70 SOL (subsection of RREF), 51 SOL (subsection of S), 365 SOL (subsection of SD), 503 SOL (subsection of SLT), 569 SOL (subsection of SS), 159 SOL (subsection of SSLE), 31 SOL (subsection of TSS), 69 SOL (subsection of VO), 109 SOL (subsection of VR), 611 SOL (subsection of WILA), 13 solution set theorem PSPHS, 129 solution sets possibilities theorem PSSLS, 63 solution vector definition SV, 75 solving homogeneous system Archetype A example HISAA, 72 Archetype B example HUSAB, 72 Archetype D example HISAD, 73 solving nonlinear equations example STNE, 15 SP4 (example), 352 span basic example ABS, 141 basis theorem BS, 193 definition SS, 356 improved example IAS, 292 notation, 141 reducing example RSSC4, 195 reduction example RS, 383 removing vectors example COV, 190 reworking elements example RES, 196 set of polynomials INDEX 801 example SSP, 358 subspace theorem SSS, 356 span of columns Archetype A example SCAA, 144 Archetype B example SCAB, 146 Archetype D example SCAD, 151 span:definition definition SSCV, 141 spanning set definition TSVS, 374 matrices example SSM22, 376 more vectors theorem SSLD, 395 polynomials example SSP4, 374 SPIAS (example), 521 SQM (definition), 85 SRR (example), 87 SS (definition), 356 SS (example), 427 SS (section), 141 SS (subsection of B), 373 SS (theorem), 260 SSCV (definition), 141 SSLD (theorem), 395 SSLE (section), 15 SSM22 (example), 376 SSNS (example), 148 SSNS (subsection of SS), 147 SSNS (theorem), 148 SSP (example), 358 SSP4 (example), 374 SSRLT (theorem), 563 SSS (theorem), 356 SSSLT (subsection of SLT), 563 SSV (notation), 141 SSV (subsection of SS), 141 STLT (example), 524 STNE (example), 15 subspace as null space example RSNS, 355 characterized example ASC, 602 definition S, 349 in P4 example SP4, 352 not, additive closure example NSC2A, 353 not, scalar closure example NSC2S, 353 not, zero vector example NSC2Z, 353 testing theorem TSS, 351 trivial definition TS, 354 verification example SC3, 349 example SM32, 359 subspaces equal dimension theorem EDYES, 417 surjective Archetype N example SAN, 562 example SAR, 555 not example NSAQ, 553 example NSAQR, 561 not, Archetype O example NSAO, 562 not, by dimension example NSDAT, 565 polynomials to matrices example SAV, 556 surjective linear transformation bases theorem SLTB, 564 surjective linear transformations dimension theorem SLTD, 565 SUV (definition), 254 SUVB (theorem), 379 SUVOS (example), 209 Version 0.70 802 INDEX SV (definition), 75 SVP4 (example), 416 SYM (definition), 223 SYM (example), 223 symmetric matrices theorem SMS, 224 symmetric matrix example SYM, 223 system of equations vector equality example VESE, 100 system of linear equations definition SLE, 16 T (archetype), 763 T (part), 767 T (technique), 19 TCSD (example), 430 technique C, 39 CP, 59 CV, 62 D, 15 DC, 113 E, 59 GS, 27 L, 25 ME, 92 P, 223 PI, 105 SE, 20 SN, 58 T, 19 U, 89 theorem AISM, 343 AIU, 341 BCS, 285 BIS, 399 BNS, 173 BRS, 292 BS, 193 CB, 653 CCRA, 768 Version 0.70 CCRM, 768 CCT, 768 CFDVS, 602 CILTI, 546 CINSM, 258 CIVLT, 578 CLI, 603 CLTLT, 526 CMVEI, 63 CNSMB, 384 COB, 419 COMOS, 273 CRMA, 226 CRMSM, 226 CRN, 403 CRSM, 204 CRVA, 203 CSCS, 282 CSLTS, 566 CSMS, 361 CSNSM, 287 CSRN, 61 CSRST, 293 CSS, 603 CSSM, 345 CVSM, 345 DC, 492 DCM, 400 DCP, 475 DED, 497 DERC, 430 DLDS, 187 DM, 400 DMLE, 495 DMST, 428 DP, 400 DRMM, 432 DT, 432 EDELI, 469 EDYES, 417 EER, 667 EIM, 473 ELIS, 413 EMMVP, 236 EMNS, 450 INDEX 803 EMP, 239 EMRCP, 448 EMS, 449 EOMP, 471 EOPSS, 20 EPM, 472 ERMCP, 475 ESMHE, 443 ESMM, 471 ETM, 474 FS, 313 FTMR, 616 FVCS, 61 G, 414 GSPCV, 211 HMOE, 480 HMRE, 479 HMVEI, 73 HSC, 72 ICBM, 653 ICLT, 579 ICRN, 61 IFDVS, 602 IILT, 577 ILTB, 544 ILTD, 545 ILTIS, 577 ILTLI, 544 ILTLT, 576 IMILT, 634 IMR, 632 IPAC, 206 IPN, 207 IPSM, 206 IPVA, 205 IVSED, 581 KILT, 542 KLTS, 540 KNSI, 626 KPI, 541 LIVHS, 168 LIVRN, 170 LNSMS, 362 LTDB, 518 LTLC, 517 LTTZZ, 511 MBLT, 513 MCT, 227 ME, 477 MIMI, 261 MISM, 262 MIT, 261 MIU, 260 MLTCV, 515 MLTLT, 524 MMA, 243 MMCC, 244 MMDAA, 242 MMIM, 241 MMIP, 244 MMSMM, 243 MMT, 245 MMZM, 241 MNEM, 479 MRCB, 659 MRCLT, 622 MRMLT, 621 MRSLT, 620 MVSLD, 171 NEM, 476 NOILT, 582 NSI, 271 NSLIC, 172 NSME1, 92 NSME2, 172 NSME3, 271 NSME4, 288 NSME5, 385 NSME6, 405 NSME7, 433 NSME8, 470 NSME9, 635 NSMS, 354 NSMUS, 89 NSRRI, 87 NSTNS, 88 OMI, 273 OMPIP, 275 OSIS, 270 OSLI, 210 Version 0.70 804 INDEX PEEF, 311 PIP, 208 PSPHS, 129 PSSLS, 63 PWSMS, 269 RCLS, 60 RCSI, 629 REMEF, 40 REMES, 37 REMRS, 290 RLTS, 559 RMRT, 417 RNNSM, 405 ROSLT, 582 RPI, 564 RPNC, 404 RPNDD, 583 RREFU, 132 RSLT, 561 RSMS, 362 SCB, 662 SER, 489 SLEMM, 234 SLSLC, 115 SLTB, 564 SLTD, 565 SLTLT, 523 SMEE, 490 SMEZV, 344 SMS, 224 SMZD, 432 SMZE, 470 SNSCM, 272 SS, 260 SSLD, 395 SSNS, 148 SSRLT, 563 SSS, 356 SUVB, 379 TMA, 224 TMSM, 224 TSS, 351 TT, 225 TTMI, 255 VAC, 344 Version 0.70 VFSLS, 122 VRI, 600 VRILT, 601 VRLT, 595 VRRB, 386 VRS, 601 VSLT, 525 VSPCV, 104 VSPM, 221 ZSSM, 342 ZVSM, 342 ZVU, 341 theorems technique T, 19 ti83 matrix entry computation ME.TI83, 34 row reduce computation RR.TI83, 46 vector linear combinations computation VLC.TI83, 103 ti86 matrix entry computation ME.TI86, 34 row reduce computation RR.TI86, 45 transpose of a matrix computation TM.TI86, 225 vector linear combinations computation VLC.TI86, 103 TIVS (example), 602 TKAP (example), 540 TLC (example), 111 TM (definition), 222 TM (example), 223 TM (notation), 222 TM.MMA (computation), 225 TM.TI86 (computation), 225 TMA (theorem), 224 TMP (example), TMSM (theorem), 224 TOV (example), 209 trail mix example TMP, transpose INDEX 805 matrix scalar multiplication theorem TMSM, 224 example TM, 223 matrix addition theorem TMA, 224 matrix inverse, 261 notation, 222 scalar multiplication, 225 transpose of a matrix mathematica, 225 ti86, 225 transpose of a transpose theorem TT, 225 TREM (example), 36 trivial solution system of equations definition TSHSE, 72 TS (definition), 354 TS (subsection of S), 351 TSHSE (definition), 72 TSM (subsection of MO), 222 TSS (section), 55 TSS (subsection of S), 355 TSS (theorem), 351 TSVS (definition), 374 TT (theorem), 225 TTMI (theorem), 255 TTS (example), 17 typical systems, × example TTS, 17 U (archetype), 763 U (technique), 89 unique solution, × example US, 23 example USR, 37 uniqueness technique U, 89 unit vectors basis theorem SUVB, 379 definition SUV, 254 orthogonal example SUVOS, 209 URREF (subsection of LC), 131 US (example), 23 USR (example), 37 V (archetype), 764 V (chapter), 99 V (notation), 74 VA (example), 101 VAC (theorem), 344 VE (notation), 74 VEASM (subsection of VO), 100 vector addition definition CVA, 101 column definition CV, 74 equality definition CVE, 100 inner product definition IP, 204 norm definition NV, 207 notation, 74 of constants definition VOC, 75 product with matrix, 233, 237 scalar multiplication definition CVSM, 102 vector addition example VA, 101 vector form of solutions Archetype D example VFSAD, 117 Archetype I example VFSAI, 125 Archetype L example VFSAL, 127 example VFS, 119 theorem VFSLS, 122 vector linear combinations mathematica, 103 ti83, 103 ti86, 103 vector representation Version 0.70 806 INDEX example AVR, 386 example VRC4, 597 injective theorem VRI, 600 invertible theorem VRILT, 601 linear transformation definition VR, 595 theorem VRLT, 595 surjective theorem VRS, 601 theorem VRRB, 386 vector representations polynomials example VRP2, 599 vector scalar multiplication example CVSM, 102 vector space characterization theorem CFDVS, 602 definition VS, 333 infinite dimension example VSPUD, 402 linear transformations theorem VSLT, 525 vector space of column vectors notation, 99 vector space of functions example VSF, 337 vector space of infinite sequences example VSIS, 337 vector space of matrices example VSM, 335 vector space of matrices:definition definition VSM, 219 vector space of polynomials example VSP, 336 vector space properties column vectors theorem VSPCV, 104 matrices theorem VSPM, 221 vector space, crazy example CVS, 338 vector space, singleton Version 0.70 example VSS, 338 vector space:definition definition VSCV, 99 vector spaces isomorphic definition IVS, 580 theorem IFDVS, 602 vectory entry notation, 74 VESE (example), 100 VFS (example), 119 VFSAD (example), 117 VFSAI (example), 125 VFSAL (example), 127 VFSLS (theorem), 122 VFSS (subsection of LC), 116 VLC.MMA (computation), 103 VLC.TI83 (computation), 103 VLC.TI86 (computation), 103 VO (section), 99 VOC (definition), 75 VR (definition), 595 VR (section), 595 VR (subsection of B), 385 VRC4 (example), 597 VRI (theorem), 600 VRILT (theorem), 601 VRLT (theorem), 595 VRP2 (example), 599 VRRB (theorem), 386 VRS (theorem), 601 VS (chapter), 333 VS (definition), 333 VS (section), 333 VS (subsection of VS), 333 VSCV (definition), 99 VSCV (example), 335 VSCV (notation), 99 VSF (example), 337 VSIS (example), 337 VSLT (theorem), 525 VSM (definition), 219 VSM (example), 335 VSP (example), 336 VSP (subsection of MO), 221 INDEX 807 VSP (subsection of VO), 104 VSP (subsection of VS), 341 VSPCV (theorem), 104 VSPM (theorem), 221 VSPUD (example), 402 VSS (example), 338 W (archetype), 764 WILA (section), Z (Property), 334 ZC (Property), 104 zero matrix notation, 222 zero vector column vectors Property ZC, 104 definition ZV, 74 matrices Property ZM, 221 notation, 75 unique theorem ZVU, 341 vectors Property Z, 334 ZM (definition), 222 ZM (notation), 222 ZM (Property), 221 ZNDAB (example), 433 ZRM (definition), 39 ZSSM (theorem), 342 ZV (definition), 74 ZV (notation), 75 ZVSM (theorem), 342 ZVU (theorem), 341 Version 0.70 ... Linear transformation, polynomials to polynomials Linear transformation from a matrix Matrix from a linear transformation Matrix of a linear transformation Linear transformation... Defined on a Basis Sum of Linear Transformations is a Linear Transformation Multiple of a Linear Transformation is a Linear Transformation Vector Space of Linear Transformations... Matrices Build Linear Transformations Matrix of a Linear Transformation, Column Vectors Linear Transformations and Linear Combinations Linear Transformation