Abstract Linear Algebra vs Real Analysis

[jzavala]

Hey guys, I wanted some advice on which theory class I should take. I am an undergraduate Economics major with a minor in mathematics. My math coursework has been calc 1,2,3, statistics, linear algebra, ordinary differential equations, applications of linear algebra, and probability theory. I have also taken econometrics 1,2, and advanced econometrics.

For next semester I am currently registered for partial differential equations and a theoretical linear algebra course. I am debating whether I should substitute the theoretical linear algebra course with real analysis. I did not want to take real analysis alongside PDE because it seemed like a big load along with my undergraduate thesis and other econ courses. However, it seems this may help me in the long run. My goal is to get into a good financial engineering or math finance program maybe after 2-3 years of work. The thought of a masters in economics or PhD in economics crossed my mind as well. For this I figured my work experience should be research related and the real analysis would help. Any thoughts/ ideas? Does a course that proves theorems from linear algebra help in applying to math finance programs, or will real analysis be more impressive. Thanks in advance for the responses.

-Juan

 

[Daniel Duffy]

"Does a course that proves theorems from linear algebra help in applying to math finance programs, or will real analysis be more impressive."

my 2 cents
Numerical linear algebra is more relevant in many parts of QF.

And hard analysis is always useful.

 

[Yike Lu]

You should be taking both. Application wise, theoretical lin al looks redundant with the lin al course you already have, so I'm leaning towards real analysis.

 

[dstefan]

Real Analysis.

Also, numerical linear algebra is MUCH more useful for financial applications than the commonly taught linear algebra.

 

[jzavala]

thanks guys, yea i figured the linear algebra course would look redundant. It just seemed that maybe takin PDE and real analysis along with an undergraduate thesis, math econ course, and risk & uncertainty course would be quite a workload for my last semester in college, but hey thats what hard work is right.

 

[Daniel Duffy]

Here is a set of lectures by Prof. Gil Strang. Brilliant!

http://ocw.mit.edu/courses/mathematics/18-06-linear-algebra-spring-2010/video-lectures/

 

[bigbadwolf]

The last thing I remember learning was a field extension where essentially you were adding a "pseudo-square root of 2" to the integers which meant you could really force square root of 2 to be an integer?

Maybe it wasn't taught properly. The easiest way is to start with the rationals, which form a field. Then add the square root of -1 to this field. You now have the field of Gaussian numbers, which contains Gaussian integers. Now you can prove Fermat's last theorem that p congruent to 1(mod 4) iff p = x^2 + y^2. For example 13 = 4+9; 29 = 25+ 4, while 19, 23, etc., cannot be expressed as the sum of two squares. Note also in passing that the field of Gaussian numbers considered as a vector space has dimension 2 over the usual rationals.

If you want to take these classes, it is really essential you have toned your logic skills before hand by taking at least one course on logic because both courses are 100% theorem-proof based and the proofs are more than just "prove 4 is an even number" so just make sure you have some understanding of mathematical theory.

Logic goes into other things and is unnecessary. It is important one understands the reason for proofs -- and this is often left out. For example many number fields do not have unique factorisation: it was the implicit assumption that they do which led to faulty proofs of Fermat's Last Theorem in the 19th century. So we need to prove which of our number fields do have unique factorisation -- e.g., Gaussian numbers have unique factorisation.

Proofs that the square root of 2 is irrational or that there exist an infinity of primes are the best ways of seeing why they are necessary. We can keep calculating further digits in the decimal expansion of the square root of 2 and we can keep finding another prime number -- but these calculations do not prove that the square root of 2 is irrational nor that the primes continue forever. Something more abstract is needed.