Lectures70EvansHall,TuTh 12:40-2PM.SyllabusGroup theory, including the Jordan-Holder theorem and the Sylowtheorems. Basic theory of rings and their ideals. Uniquefactorization domains and principal ideal domains. Modules. Chainconditions. Fields, including fundamental theorem of Galoistheory, theory of finite fields, and transcendence degree.TextbookAlgebra, 3rd rev. ed.bySerge Lang;this is volume 211 in theSpringerGraduateTexts in Mathematicsseries.Lang's book is the classic algebra textbookfor graduate courses. I used anearlier editionwhen I wasan undergraduate at Brown Universityand a graduate student atHarvard.You can look at some unofficialcompanionmaterialfor Lang'sbook that was written byoneofmycolleagues.See, for instance, theerrata to printings past and present.Graduate Student InstructorThe GSI for this course will be Chu-Wee Lim.Chu-Wee will hold office hours, grade homework and offer review sessionsbefore exams. He may possibly alsogive two or three guest lectures during the semester.See hisMath 250A pagefor more information.GradingLetter grades were a function of each student's composite numericalgrade. This grade was calculated asa linear combination of the four course components:the two midterms (15% each), the final (40%), and thehomework (30%).The table below shows the distribution of scores for the 25 registeredUC students in the course. It includes a line for the fictional student"Gauss," who had perfect scores. There were also three students inthe course who do not have student IDs.SID mod 100MT1MT2HW totalFinal ExamComposite ScoreLetter grade0303042040100.00Gauss31923383.673280.41A111472942253.50B1430253533688.71A+171218297.672258.26B+2617102772053.29B295172591342.50C4024173122466.79A-4314213662467.64A-45115195728.93D4822264113487.36A+5812113531349.71B-6330233423080.93A6318214082371.64A6519273882474.71A6814194001459.07B+6826173412873.86A711271332039.00C7721154043985.86A+7917213523579.14A822225349.333684.45A+8519263623684.36A+9019133501758.00B+9218112381950.50B-9596263.331036.31C-98128284.671848.33B-ExaminationsWe will have three examinations in this course: First midterm exam, September 30[questions and possible answers;scores] Last midterm exam, November 4[questions and possible answers;scores] Final examination, December 20, 2004, 12:30-3:30PMin C125 Cheit[questions and possible answers;scores]I taught this course three years ago.You can look atthe web page for my old coursefor more information (including exam questions and solutions).HomeworkHomework will be assigned weekly.Assignment due September 7:Chapter I, 4 (ignore the hypothesis that K normalizes H), 5, 6, 7, 8, 9.In Problem 8, there is a pair of misprints: as the problem iswritten, there are three union signs, where the indices are respectively c,x_c and x_c. The first union should be over elements x_c; these form aset of representatives for the coset space H/H", where H" is the intersectionof H and the conjugate of H' by c. The second union is over elements c.The third union is as written; namely, it's a union over the sameset of elements x_c that appeared in the first union. In short,one needs to exchangethe indices in the first and second union signs.(Possible solutions, written by Ribet.)Assignment due September 14:Chapter I, problems 13, 14, 16, 15, 17, 19, 20, 22, 23bc.I wanted to assign Problem 12, but it's sort of a mess.I wrote up some comments on that problemand took photos of the two pages (p. 76 andp. 77 where the problems appear.(Possible solutions, written by Ribet.)Assignment due September 21; this waslast updated on September 19, 2004.(Possible solutions, written by Ribetand updated on September 22, 2004.)Assignment due October 5:Chapter I, problems 24, 25, 26a, 28, 29, 3039, 40, 41, 46, 47, 48, 50, 52(Possible solutions, written by Ribet).Assignment due October 12:Chapter II, problems 1-7(possible solutions, written by Chu-Wee Lim).Assignment due October 19:Chapter II, problems on Dedekind rings (13-19).There are some relevantcomments on thecomments pageand nowpossible solutions, written by Ribet.Assignment due October 26.For the last problem, you should probably read Chapter VII throughto the statement of Proposition 1.1, which occurs at the top ofthe third page of the chapter.Also, note that an "integral ideal" of a ring is the same thing asan "ideal" of the ring; people sometimes use the adjective "integral"to stress that they're not talking about fractional ideals.(Possible solutions, written by Ribet.) The assignment due November 9 is likeHW #4 in that it represents three lectures, rather than two.Note that there arepossible solutions, written by Chu-Wee.For a slightly different discussion of problem 13c in Chapter III, you couldconsult page 11 (and page 12) of"Introductionto Algebraic K-Theory" by John Milnor.In the book,the main theorem that emerges in problem 13cis attributed to Steinitz.Assignment due November 16:Chapter IV, problems 3, 5, 6, 7, 18.In problem 7, p is the characteristic of k, so that q is a power of p.(Possible solutions, written by Ribet.)Assignment due November 30:Chapter V, problems 3, 5, 7, 9, 11 (all parts).(Possible solutions, written by Ribet.)Last assignment, due December 9:Prove Corollary 1.4 on page 263. (It's not "obvious," contrary towhat our author says. You can appeal to results that come after thecorollary if you have checked that the statement of the corollary isnot used in the proofs of the subsequent results.)Also, do the following problems from Chapter VI:1 (a through e), 5, 6, 7, 9, 11, 15.(Possible solutions, written by Ribet.)Anonymous FeedbackPlease let me know what I'm doing right and what I'm doing wrong.Constructive feedback is always welcome;don't hesitate to propose changes.You might be inspired by some previous comment pages:Math 250A (Fall, 2001),Math H113 (Spring, 2003),Math H110 (Fall, 2003),Math 114 (Spring, 2004).You can readthe comments that have beensubmitted for this course so far.Added Christmas Day, 2004: comments are all finished now. Thanks foryour helpful feedback and questions.Last Updated: document.write(document.lastModified);
From a pedagogical as well as strictly mathematical perspective, which one of Lang's Linear algebra and Introduction to linear algebra would you recommend to an undergraduate with not much experience with linear algebra, but a fairly good grip of some abstract algebra and real analysis, who wants to gain a rigorous and precise knowledge of the topic? Are these two books really different? How do they compare with each other?
Caveat: this answer is predicated on the student having "a fairly good grip of some abstract algebra and real analysis [and] who wants to gain a rigorous and precise knowledge of the topic [of linear algebra]." Therefore, I would tend to recommend a book more in the vein of Lang's Linear Algebra than his Introduction to Linear Algebra, unless one has not seen any linear algebra at all. Usually, the difference between such books is that the introductory type focuses on vectors, matrices, computations and concrete examples, whereas the more advanced books take a more abstract approach, emphasizing structures, mappings and proofs. 2b1af7f3a8