I have a couple questions regarding Advanced Linear Algebra vs Functional Analysis.
1) Do these courses help in understanding or have applications in:
- Machine Learning
- Quantitative Finance, eg. Stochastic Calculus
- 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.
Best Answer
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.