Taking Real Analysis, Abstract Algebra, and Linear Algebra

[bacte2013]

Dear Physics Forum advisers,

I am a college sophomore in US with a major in mathematics, and an aspiring algebraic number theorist and cryptographer. I wrote this email to seek your advice about taking the Analysis I (Real Analysis I), Abstract Algebra I, and Linear Algebra with Proofs. At my university, it is recommended to take Linear Algebra with Proofs (basically LA course + Intro. Proof course) before taking both Analysis I and Abstract Algebra. However, I have been self-studying the introductory number theory (Niven), mathematical analysis (Apostol), and abstract algebra (Pinter, Herstein), and the number theory + abstract algebra textbooks gretly helped me to strengthening my proof methodology. Should I take all three courses together for Fall 2015 or should I take both abstract algebra and linear algebra first and then take analysis on Spring 2016? I am planning to read Friedberg and one more book for LA during this Summer along with my research and other books since I am very interested in learning about the LA and also to study ahead of the formal course.

My Analysis I course uses Strichartz & Rudin, abstract algebra course uses Dummit & Foote, and LA course uses Friedberg.




PK

 

[micromass]

I took those courses together in my freshman year, and I turned out ok. So I'd say to go for it.

 

[one]

There isn't really problem contentwise with taking all of those classes together. Your only concern might be time, depending on how many other classes you are taking. If you wanted to put off a class, analysis would probably be the best.

 

[bacte2013]

Thank you for the advice. I will be taking those courses plus an undergraduate research course.

 

[mathwonk]

i would find it hard to devote enough time to that many substantial math courses all at the same time to do well in all of them.

 

[bacte2013]

Thence should I take Abstract Algebra I, Theoretical Linear Algebra, and Probability or Geometry course first? Will abstract algebra a good preparation for the Analysis I (Real Analysis)?

 

[micromass]

Abstract algebra is useless for real analysis. Not even the proofs look very much alike. Sure, you still have the usual suspects: contradiction, induction, etc. But aside from that, the two courses are quite different.

 

[mathwonk]

it is true that abstract algebra has a very different feel from real analysis. the algebra arguments concern symmetry, and are finite in nature. the analysis arguments concern approximations, and the subtlety of using an infinite sequence of better and better approximations to infer something about the elusive "limiting" object. It may still be true however that more experience doing proofs of any kind will strengthen your preparation, since as micromass observes, the usual suspects: the basic logical concepts, are present in all subjects.

It may be that linear algebra will be more relevant to analysis eventually than abstract algebra, since differential analysis at least consists in using linear functions to approximate non linear functions. some geometry courses may also be useful. but there are basically three separate areas of mathematics, algebra, analysis, and geometry, each involving a different style of thinking, and you have to know something of all of them. they do interact. but each has its own special features one must learn. I am just suggesting you limit the number of different hard math courses you take at once to at most two, and many people do find analysis the most difficult of those, so some people would put it last or take it alone, not everyone of course.

Analysis is also useful in studying some aspects of abstract algebra, such as linear groups, where the three subjects, algebra, analysis, and geometry mix beautifully together. A good introduction to this topic is in Mike Artin's book Algebra. Algebra is also relevant to analysis, e.g. in the study of rings of power series, which has its formal algebra side, as explained say in Lang's book on complex analysis. It also comes up later in several complex variables and even real Banach algebras, where local rings and algebras are used, see e.g. thje abstract Stone Weierstras theorem and the related correspondence between subnalgebras of continuous functions and compactifications of nice spaces. Boolean algebra is also used in the formal theory of integratiion in real analysis, but this is not usually part of an abstract algebra course. But these interactions are more commonly foud in slightly more advanced treatmemnt of analysis as suggested here. There are fairly elementary examples of using algebra in analysis however, such as the lovely construction of the real numbers from the rationals, by taking the algebra of Cauchy sequences and modding out by the maximal ideal of null sequences (those converging to zero). A natural generalization allows one to complete any metric space, nd this is a standard real analysis topic.

why don't you take a look at the relevant books and see how the subjects strike you. Friedberg, Ansel, and Spence is a rather good book on linear algebra I think if that is what you are using. Dummitt and Foote I thikn is much more advanced, and I am not familiar with your analysis book. If it is not too advanced you might take linear algebra and analysis, and then Dummitt and Foote. I also have several free sets of linear algebra notes on m,y website. Not as good say as Sergei Treil's free book, but maybe having an advantage of being shorter than some other books. Actually I have at least three free linear algebra books, one 15 pages long, one about 100 pages, and one a full treatmemnt of abstract algebra a la Dummitt and Foote.

http://alpha.math.uga.edu/~roy/

 

[cpsinkule]

It's very wise to take a proof based course before analysis or abstract algebra. Abstract proofs are much easier than analysis proofs, you should be able to swing that one. I would definitely wait to take analysis after you take a proof course, you will get more out of it that way.

 

[mathwonk]

I have looked at the analysis books you will use, or rather i know rudin pretty well and i read some reviews of strichartz. They seem quite different, i.e. rudin is very hard to learn from and apparently strichartz is much better for that purpose, while some people feel rudin's organization makes it more useful as a reference.

I am also quite familiar with both dummitt and foote and friedberg insel and spence, indeed i have taught from all these books except strichartz. So in my opinion the sophistication level of these books is: friedberg: least, next rudin and strichartz, and most sophisticated and abstract: dummitt and foote.

however, in terms of writing style, friedberg, dummitt and foote, and strichartz are apparently carefully written in a user friendly style, the opposite of rudin. so in order to scale your classes from easier to harder, i would think a reasonable ordering would be to take the friedberg and strichartz courses first, and the dummitt and foote course second. just my opinion, i.e. even though well written, dummitt and foote is a more advanced course in terms of abstraction and sophistication. the difficulty in analysis is inherent in the infinite nature of limits, but the material is rather concrete.but i again recommend you sit down with these books and peruse them somewhat at length to see which ones you are happy spending time with and can learn from now. you might try some problems too. dummitt and foiote is also a rather long book, with at least enough material for two years of courses, one advanced undergrad and one graduate level. moreover with all its virtues dummitt and foote is not my favorite algebra book. it has excellent problem sets, and givesv ery clear explanations of the material it chooses to present. however there are places where the choices of proofs presented there are not the most enlightening ones in my opinion.

oh yes, consider which professors will teach the courses, because a good prof can make a course worth taking and a bad one can ruin the subject for you. e.g. since analysis can be quite hard to learn, if there is an especially good prof teaching thst course one might want to take it just to get him or her.

 

[Loststudent22]

I took those courses together in my freshman year, and I turned out ok. So I'd say to go for it.

Is this typical outside the U.S.? I'm going to be starting at my local state school as a junior and will just be taking abstract algebra and analysis that year.

Dummitt and Foote is also used as the masters textbook so I'd imagine that would be for an advanced class and not your first introduction.

 

[micromass]

Is this typical outside the U.S.? I'm going to be starting at my local state school as a junior and will just be taking abstract algebra and analysis that year.

It is certainly typical in my country (=belgium). I don't know about other countries.

 

[mathwonk]

I'm not familiar with education in Belgium, but it seems quite possible to me that pre college education there is much more advanced than in the US. Here in the US we have notoriously poor average high school preparation and pretty good college education, so our students often struggle even with what they encounter here in making that transition. The three courses desccribed here are more often met in junior or senior year in college in the US. The Rudin course, e.g. was offered at my university as a hybrid senior/graduate level course when I taught it, as I recall, and still considered quite difficult. One of my friends from Hungary once gave a research level talk here in which he remarked that the tool he wanted to use should be familiar to any freshman, and then paused and added: "or here in the US, perhaps to any graduate student."

Here is the webpage from UGA with a modest beginning abstract algebra course and an introductory analysis course, both described as "4000 level" i.e. senior undergrad, courses. the linear algebra is a 3000 level course, or junior level. The Dummitt and Foote course is usually for graduate students, Of course there is a huge variety of levels available to those who can take them. At Harvard in the 1960's at least the majority of students enrolled in the Loomis and Sternberg advanced honors calculus course were sometimes freshmen, and nowadays it may be entirely freshmen, since the prior Spivak style course no longer exists there. (This course covers abstract linear algebra including tensor products, metric space theory - e.g. the basic contraction lemma, some general topology e.g. Tychonoff, elementary Banach space theory, and calculus on finite dimensional manifolds. The general Stokes theorem is proved, but DeRham's theorem and Poincare duality may only be stated.) And even at UGA there has for years been a Spivak style calculus class, which covers intrioductory analysis to some extent, offered to selected freshmen.

 

[Infrared]

@mathwonk It's not so unusual in the U.S. either. I just finished my freshman year of undergrad and did well in these classes (LA, algebra, analysis). While taking them early isn't necessarily the standard track, there were plenty of us who did so.