I have a couple questions regarding Advanced Linear Algebra vs Functional Analysis.

1) Do these courses help in understanding or have applications in:

  1. Machine Learning
  2. Quantitative Finance, eg. Stochastic Calculus
  3. Numerical Analysis/Optimization

2) If I could only pick one course, which one would be more useful?

To reiterate my question, I'm asking whether linear algebra at a high level, or functional analysis has more applications to the three aforementioned subjects and what those applications are.

PS. I'm a first time poster; I read the criteria for asking questions and they say not to ask course selection questions. However, I tried to get around that by asking about the applications of the subjects. If these sorts of questions are not welcome, I will modify the question or take it down. Thanks for your help.

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    $\begingroup$ You can't do functional analysis without first knowing a fair bit of linear algebra, so if you can only take one course, go for advanced linear algebra. I don't actually think functional analysis will help with any of those classes, except for stochastic calculus, but there, the bits of functional analysis you would want would be covered in any course on measure theory, which would be a much more important prerequisite (unless they taught the material in the class). Linear algebra is useful for all 3. $\endgroup$ – Aaron Jul 10 '16 at 20:31
  • $\begingroup$ @Aaron, thanks for the comment. I have completed a course on Linear Algebra already. It covered this. The functional analysis course doesn't have the advanced linear algebra course as a pre-req. Its only pre-req is measure theory, so I think I should be able to do functional analysis without advanced linear algebra. Taking this into account, do you still think I should do the linear algebra course over functional analysis? Thanks. $\endgroup$ – orbit-stabilizer Jul 10 '16 at 20:37
  • $\begingroup$ Looking more closely, the linear algebra course is a lot of "you know how to compute this stuff, let's show you how to actually think about it/understand it." While I like what they cover and think it important, it is probably of limited utility (though not none) for applications. However, except for Hilbert spaces and $L^2$, I don't expect functional analysis to be that useful for applications either. Much of the linear algebra course will be covered in an abstract algebra course, is that a possibility? $\endgroup$ – Aaron Jul 10 '16 at 20:45
  • $\begingroup$ I think (advanced) linear algebra should definitely come first. This will open Optimization, which in turn is foundational for parts of machine learning. To make further progress in machine learning and also in stochastic calculus, you need a solid background in probability theory and statistics. Added after looking at the course descriptions None of these two courses are especially geared towards preparing you further for the courses that you appear to be interested in. $\endgroup$ – Hans Engler Jul 10 '16 at 20:48
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    $\begingroup$ @io_cantor - you could try to take a course in numerical linear algebra instead of the rather abstract advanced linear algebra course that you listed. You could then learn about things like matrix factorizations, singular value decomposition, iterative solvers. E.g. how would one solve a system of equations with $10^9$ equations and unknowns? That said, if the curricukum at your institution is not set up to train students in the fields you are interested in, you will have some difficulties putting one together all by yourself. $\endgroup$ – Hans Engler Jul 10 '16 at 22:49

I'm not an expert in the fields you list, but I can give you some general information. All of the fields you list require an understanding of both linear algebra and functional analysis. However, in any of the fields you are likely to encounter functions over spaces which are not topologically trivial. For instance, if you are looking for patterns in a machine learning problem or finance problem, you might look for periodic trends in data, and periodic functions are naturally seen as functions on the circle (in a sense this is the special property of functions on a circle). Or you might need to use advanced calculus techniques on non-contractible spaces, where vector fields may not be conservative. The point I'm trying to make: a basic understanding of topology will probably be more useful than, say, knowledge of the tensor product, both to the fields you list and to mathematics in general. Then the functional analysis class seems to be more relevant to the classes you list.

  • $\begingroup$ Thanks for the reply. I will definitely look into taking a topology course. Would an intro course such as this suffice? Also, I am unclear as to what your last sentence means; what do you mean by, "Then the functional analysis class seems to be more relevant to the classes you list." $\endgroup$ – orbit-stabilizer Jul 10 '16 at 20:58
  • $\begingroup$ Take it! I based much of my undergraduate thesis on Rolfsen's Knots and Links, a legendary text in differential topology. I would jump at the opprotunity to learn basic topology from him. As for the then, I only meant "In conclusion, the function analysis class..." $\endgroup$ – ate Jul 10 '16 at 21:31
  • $\begingroup$ Ah, thanks for clearing up what you meant. Wow! I didn't realize that the professor was well known. Thank you so much for opening me up to learning about a new area of math that I previously wasn't even considering. $\endgroup$ – orbit-stabilizer Jul 10 '16 at 21:37
  • $\begingroup$ Of course, good luck! $\endgroup$ – ate Jul 10 '16 at 21:47

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