What is Linear and Abstract Analysis and How Are They Applied in Science?

In summary, the conversation is about two potential courses for the upcoming year: Linear and Abstract Algebra. The speaker has taken both courses and finds them indispensable for their thesis on effective field theories for hadrons. The courses are described as "Abstract Analysis" and "Linear Analysis" and are recommended to be taken in that order. The courses are discussed in terms of their usefulness in applied mathematics and theoretical physics. The speaker recommends discussing with faculty for more specific recommendations.
  • #1
Lonewolf
336
1
Next year I have the option of studying these two courses. What are they about, and what are they used for? Has anybody taken courses in these, and if so, what did you think of them?
 
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  • #2
Do you mean "Linear and Abstract Algebra"? If so, then that would make more sense!

My answer would depend on your major. If you are majoring in math or physics, then you will need to take both of them. If you are in engineering, you will only find Linear Algebra useful.

I took both of these courses, and as a physics grad student I find them indispensable. My thesis is on effective field theories for hadrons, and one must have a solid foundation of vector spaces and group theory for that.
 
  • #3
Nope, it's definitely analysis. We've got separate options for abstract algebra and linear algebra.
 
  • #4
How weird... could you post the course description? Probably it is something like a real analysis course, which I would highly recommend. But from those names it's hard to tell -- are you in the States?
 
  • #5
  • #6
HallsofIvy

Thanks for the links.

"Abstract Analysis" looks to me like a fairly standard analysis course (perhaps a second year of analysis- I didn't look for prerequisites.)

"Linear Analysis" looks more like what I would call "Functional Analysis", generally speaking more "abstract" and more difficult than "Abstract Analysis"!
 
  • #7
Thanks for the input. I think I'm going to take them both. Is there any use for either of them in applied mathematics? I don't really want to take something that can't be used outside pure maths. The linear analysis course seems the more interesting of the two at the moment, although that could change once I take it...
 
  • #8
Certainly, I would recommend taking the "abstract analysis" course first. It will definitely help you in taking the "linear analysis" course later. (Remembering that the best thing you can do is talk to the faculty who actually teach the courses for their recomendations!)

As to how much use they are in "Applied Mathematics", that depends strongly on what you mean by "applied mathematics"- an unfortunately vague term! Some people use "applied mathematics" to mean the theory behind the kind of mathematics used in applications. Certainly, if you consider yourself a mathematician of any sort you should have as much anaysis as possible. Functional Analysis (what, I think, is your "linear analysis") is used in developing the theories behind solution methods for differential equations, among other things, but are not really necessary just for using those methods. Again, think about exactly what YOU want to do and talk to the faculty.
 
  • #9
Thanks a lot for your advice. I'll go and arrange an appointment to discuss it with a member of the faculty. I meant applied to some area of science, e.g. are they of any use in theoretical physics?
 
  • #10
Yes! If you get advanced enough, most areas of math are useful in theoretical physics. But analysis is especially useful.
 
  • #11
Wow, that's pretty cool. At least the stuff I'm learning is only seemingly useless :wink:
 

What is Linear Analysis?

Linear analysis is a branch of mathematics that deals with the study of linear functions and their properties. It involves the use of algebraic techniques to solve problems related to linear equations, matrices, and vector spaces.

What is Abstract Analysis?

Abstract analysis is a branch of mathematics that focuses on the study of abstract structures and their properties. It involves the use of logic and abstract reasoning to understand and solve problems related to sets, groups, and other abstract objects.

What are the main differences between Linear and Abstract Analysis?

The main difference between linear and abstract analysis is their focus. Linear analysis deals with concrete objects, such as numbers and equations, while abstract analysis deals with abstract objects, such as sets and groups. Additionally, linear analysis uses algebraic techniques, while abstract analysis uses logic and abstract reasoning.

What are some real-world applications of Linear and Abstract Analysis?

Linear analysis is used in a variety of fields, including engineering, physics, and economics, to solve problems related to linear systems and equations. Abstract analysis is used in fields such as computer science, cryptography, and theoretical physics, to study and solve problems related to abstract structures and their properties.

What skills are required to excel in Linear and Abstract Analysis?

To excel in linear and abstract analysis, one needs to have a strong background in mathematics, particularly in algebra and calculus. Additionally, critical thinking skills and the ability to think abstractly are crucial for understanding and applying concepts in both branches of analysis.

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