Linear Algebra - Joshua

1 Linear ALGEBRAJim HefferonThird ,R+,Rnreal numbers, positive reals,n-tuples of realsN,Cnatural numbers{0,1,2,..}, complex numbers( ),[ ]open interval, closed interval .. sequence (a list in which order matters)hi,jrowiand columnjentry of matrixHV,W,Uvector spaces~v,~0,~0 Vvector, zero vector , zero vector of a spaceVPn,Mn mspace of degreenpolynomials,n mmatrices[S]span of a set B,D ,~ ,~ basis, basis vectorsEn= ~e1, ..,~en standard basis forRnV =Wisomorphic spacesM Ndirect sum of subspacesh,ghomomorphisms ( Linear maps)t,stransformations ( Linear maps from a space to itself)RepB(~v), RepB,D(h)representation of a vector , a mapZn morZ,In norIzero matrix, identity matrix|T|determinant of the matrixR(h),N(h)range space, null space of the mapR (h),N (h)generalized range space and null spaceGreek letters with pronounciationcharacternamecharactername alphaAL-fuh nuNEW betaBAY-tuh , xiKSIGH , gammaGAM-muhoomicronOM-uh-CRON , deltaDEL-tuh , piPIE epsilonEP-suh-lon rhoROW zetaZAY-tuh , sigmaSIG-muh etaAY-tuh tauTOW (as in cow)

2 , thetaTHAY-tuh , upsilonOOP-suh-LON iotaeye-OH-tuh , phiFEE, or FI (as in hi) kappaKAP-uh chiKI (as in hi) , lambdaLAM-duh , psiSIGH, or PSIGH muMEW , omegaoh-MAY-guhCapitals shown are the ones that differ from Roman book helps students to master the material of a standard US undergraduatefirst course in Linear material is standard in that the subjects covered are Gaussian reduction, vector spaces, Linear maps, determinants, and eigenvalues and standard is book s audience: sophomores or juniors, usually witha background of at least one semester of calculus . The help that it gives tostudents comes from taking a developmental approach this book s presentationemphasizes motivation and naturalness, using many developmental approach is what most recommends this book so I willelaborate.

3 Courses at the beginning of a mathematics program focus less ontheory and more on calculating. Later courses ask for mathematical maturity: theability to follow different types of arguments, a familiarity with the themes thatunderlie many mathematical investigations such as elementary set and functionfacts, and a capacity for some independent reading and thinking. Some programshave a separate course devoted to developing maturity but in any case a LinearAlgebra course is an ideal spot to work on this transition. It comes early in aprogram so that progress made here pays off later but it also comes late enoughso that the classroom contains only students who are serious about material is accessible, coherent, and elegant.

4 And, examples are readers with their transition requires taking the mathematics seri-ously. All of the results here are proved. On the other hand, we cannot assumethat students have already arrived and so in contrast with more advancedtexts this book is filled with illustrations of the theory, often quite texts that assume a not-yet sophisticated reader begin with matrixmultiplication and determinants. Then, when vector spaces and Linear mapsfinally appear and definitions and proofs start, the abrupt change brings thestudents to an abrupt stop. While this book begins with Linear reduction, fromthe start we do more than compute. The first chapter includes proofs, such asthe proof that Linear reduction gives a correct and complete solution set.

5 Withthat as motivation the second chapter does vector spaces over the reals. In theschedule below this happens at the start of the third student progresses most in mathematics by doing exercises. The problemsets start with routine checks and range up to reasonably involved proofs. Ihave aimed to typically put two dozen in each set, thereby giving a selection. Inparticular there is a good number of the medium-difficult problems that stretcha learner, but not too far. At the high end, there are a few that are puzzles takenfrom various journals, competitions, or problems collections, which are markedwith a ? (as part of the fun I have worked to keep the original wording).That is, as with the rest of the book, the exercises are aimed to both buildan ability at, and help students experience the pleasure of, should see how the ideas arise and should be able to picture themselvesdoing the same type of and computing are interesting and vital aspects of thesubject.

6 Consequently, each chapter closes with a selection of topics in thoseareas. These give a reader a taste of the subject, discuss how Linear Algebracomes in, point to some further reading, and give a few exercises. They arebrief enough that an instructor can do one in a day s class or can assign themas projects for individuals or small groups. Whether they figure formally in acourse or not, they help readers see for themselves that Linear Algebra is a toolthat a professional must book is Free. See this book s web the license details. That page also has the latest version,exercise answers, beamer slides, lab manual, additional material, and LATEX source. This book is also available in a professionally printed and bound edition,from standard publishing sources, for very little cost.

7 See the web lesson of software development is that complex projectshave bugs, and need a process to fix them. I am grateful for reports from bothinstructors and students. I periodically issue revisions and acknowledge in thebook s source all of the reports that I use. My current contact information is onthe web page am grateful to Saint Michael s College for supporting this project over manyyears, even before the idea of open educational resources became familiar. I alsothank Adobe Color CC userclaflin61for the cover , I cannot thank my wife Lynne enough for her unflagging book s emphasis on motivation and development, and its availability,make it widely used for self-study.

8 If you are an independent student then goodfor you, I admire your industry. However, you may find some advice an experienced instructor knows what subjects and pace suit theirclass, this semester s timetable (graciously shared by George Ashline) may helpyou plan a sensible rate. It presumes that you have already studied the materialof Section , the elements of , , , , , , Thanksgiving break , enrichment, you could pick one or two extra things that appeal to you, fromthe lab manual or from the Topics from the end of each chapter. I like the Topicson Voting Paradoxes, Geometry of Linear Maps, and Coupled Oscillators. You llget more from these if you have access to software for calculations.

9 I recommendSage, freely available the table of contents I have marked a few subsections as optional if someinstructors will pass over them in favor of spending more time that in addition to the in-class exams, students in the above course dotake-home problem sets that include proofs, such as a verification that a set is avector space. Computations are important but so are the main advice is: do many exercises. I have marked a good sample withX s in the margin. Do not simply read the answers you must try the problemsand possibly struggle with them. For all of the exercises, you must justify youranswer either with a computation or with a proof. Be aware that few peoplecan write correct proofs without training; try to find a knowledgeable person towork with , a caution for all students, independent or not: I cannot overemphasizethat the statement, I understand the material but it is only that I have troublewith the problems shows a misconception.

10 Being able to do things with theideas is their entire point. The quotes below express this sentiment admirably (Ihave taken the liberty of formatting them as poetry). They capture the essenceof both the beauty and the power of mathematics and science in general, and ofLinear Algebra in know of no better tacticthan the illustration of exciting principlesby well-chosen particulars. Stephen Jay GouldIf you really wish to learnyou must mount a machineand become acquainted with its tricksby actual trial. Wilbur WrightJim HefferonMathematics, Saint Michael s CollegeColchester, Vermont USA 05439 s a good exercise, one that enlightens as well as tests,is a creative act, and hard work.