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I have a background of a one year course in linear algebra (covering most of K&H) and two years of calculus (the first year was a one real variable course at the level of Spivak's Calculus book and then one year of multivariable calculus starting from topology of $\mathbb{R}^n$ and up to Stokes' theorem).

I am looking for some interesting and not-so popular topics which I can understand and help me develop my understanding of the topics I already know.

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Real Analysis will help you understand calculus better. Abstract Algebra will help you understand Linear Algebra better. –  Arturo Magidin Nov 21 '11 at 4:48
    
That's why I said not-so popular. –  Vicfred Nov 21 '11 at 4:51
    
They aren't terribly popular among the general public... But there is a reason why they are the natural "next steps": it's because they are the natural way to try to get a better understanding of the two topics you do know. Other subjects may be accessible (e.g., Number Theory, Set Theory), but they will not connect back to Linear Algebra or to Calculus. In short, your two goals are mutually incompatible. –  Arturo Magidin Nov 21 '11 at 4:53
    
I feel other subjects can help understanding algebra without being its next steps, ex. some subject which gave birth to an idea which later become part of algebra. Also one can get intuition of algebra by studying different topics, so these goals are not mutually incompatible at all. Saying something like that is like saying algebra or calculus are independent from other subjects. –  Vicfred Nov 21 '11 at 5:01
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Well, I'm saying that your goals are mutually incompatible form the perspective of someone who is familiar with a lot of topics and whether or not they are accessible to you after learning one year of linear algebra and two of calculus. You are making your conclusion that they are not incompatible from the perspective of someone who does not know where to go next and who does not know any subjects other than one year of linear algebra and two years of calculus. Perhaps you should give me the benefit of the doubt? And, no, saying this is not saying they are "independent from other subjects." –  Arturo Magidin Nov 21 '11 at 5:04
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2 Answers

up vote 5 down vote accepted

The subjects that are reasonably accessible after Calculus and Linear Algebra are:

  1. Real Analysis.
  2. Abstract Algebra.
  3. Elementary Number Theory.
  4. Elementary Set Theory.
  5. Elementary Logic.
  6. Combinatorics and Discrete Mathematics.
  7. Geometry.
  8. Differential Equations.

Of these, 1-3 and 6 are "popular", so you want to discard them. 8 is popular among "users" of mathematics (engineers, for example).

While Elementary Set Theory and Elementary Logic will probably help with mathematics in general, they are unlikely to give you any particular insight into either calculus or linear algebra. The best possible insight into calculus will be given by Real Analysis; the best possible insight into Linear Algebra will be given by Abstract Algebra. Elementary Number Theory is the source from which many other areas of mathematics sprung, but it will give you no insight into either linear algebra or into calculus (from Number Theory and Real Analysis you can move on to Complex Analysis and Analytic Number Theory, which will likely be useful; from Number Theory you can also move to abstract algebra and from there to Algebraic Number Theory, which would also shed light on the development of a lot of Algebra).

Combinatorics and discrete math are quite fun and interesting, but again there is very little connection with calculus or with linear algebra. Likewise with Geometry.

Topology is a reach, without some real analysis "in the bag" to fall back on, or a lot more experience with abstraction than provided by one year of Linear Algebra.

It's not that real analysis and linear algebra are "independent of other subjects", it's that the gateways from linear algebra and real analysis to those "other subjects" that have intimate connections with them are precisely the subject that you want to avoid and discard because they are "popular." While "the road less traveled" may sound romantic, you may want to avail yourself of 100+ years of experience of mathematicians who have come up with, if not "royal roads", then at least well-traveled roads that help get you to your destination.

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I'm not avoiding real analysis or abstract algebra, I'm definitively taking them next semester (along with differential geometry, and I'm going to eventually take topology, complex analysis...) that's why I didn't want someone to recommend those topics. I asked this question because I've read that affine geometry and geometric algebra gives insight of ideas in linear algebra and I was wondering if there are topic I can aboard (and understand) and help me develop my understanding. –  Vicfred Nov 21 '11 at 5:30
    
Then (i) why not ask that? and (ii) Why complain about my suggestion that you take real analysis and abstract algebra, dismissing them as "popular" (and by implication, not what you want)? –  Arturo Magidin Nov 21 '11 at 5:32
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(I) I thought I was clear in my question, I see I was not. (II) I did not complain about that, if I complained about something it was about the idea that studying other subjects can't help understanding the ideas/motivation of linear algebra or calculus. –  Vicfred Nov 21 '11 at 5:37
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Some subjects have no connection to linear algebra or to calculus. Would you expect a course in medieval history to give you insight into the definition of vector spaces? Discrete mathematics, for example, is the very antithesis of calculus (with its focus on the continuum), and most of the connections with linear algebra flow in the other direction (linear algebra to discrete mathematics); it's a much more recent topic than linear algebra, so expecting insights into the "whys" of linear algebra from it is like expecting studying World War II to give you insights into the causes of WWI. –  Arturo Magidin Nov 21 '11 at 5:43
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I suggest at least looking over the following book. There's a 2nd edition available, but the 2nd edition is less theoretical and omits some of the linear algebra in the 1st edition. This book was very popular for those with approximately your background when I was an undergraduate (late 1970s), but I don't seem to hear about it much anymore (hence, it might fit your desire for something "not-so popular"). This book ties together and uses much of the calculus and linear algebra background that you have, and it also introduces/previews a few topics that come up in later math courses (e.g. some baby operator theory via exponentiation of matrices). Finally, at least in the U.S., you can find this book in most any college or university library (usually cataloged at QA3.P8).

Morris W. Hirsch and Stephen Smale, Differential Equations, Dynamical Systems, and Linear Algebra, Pure and Applied Mathematics #60, Academic Press, 1974, 358 pages.

http://www.amazon.com/dp/0123495504

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