Number Theory, Linear & Abstract Algebra

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Discussion Overview

The discussion revolves around the prerequisites for learning number theory, linear algebra, and abstract algebra, as well as recommendations for textbooks in these areas. Participants explore the interrelation between these mathematical branches and the necessary background knowledge for each.

Discussion Character

  • Exploratory
  • Technical explanation
  • Debate/contested
  • Homework-related

Main Points Raised

  • Some participants suggest that there are generally no prerequisites for linear and abstract algebra, though specific textbooks may assume prior knowledge.
  • It is noted that a familiarity with proofs is important for studying linear and abstract algebra.
  • For number theory, a knowledge of abstract algebra is considered beneficial, with some books requiring knowledge of calculus as well.
  • Participants discuss the complexity of matrices in linear algebra, mentioning topics such as determinants, eigenvalues, and vector spaces.
  • One participant emphasizes that understanding vector spaces over arbitrary fields enhances the study of linear algebra, although prior abstract algebra coursework is not deemed necessary.
  • There is a suggestion that linear algebra and abstract algebra reinforce each other and may be best studied simultaneously.
  • Recommendations for textbooks include Friedberg for linear algebra and Pinter for abstract algebra, with some participants discussing the differences between editions and prices.
  • Another participant mentions Halmos's "Finite Dimensional Vector Spaces" as a highly regarded linear algebra book.

Areas of Agreement / Disagreement

Participants generally agree on the lack of strict prerequisites for linear and abstract algebra, but there are varying opinions on the necessity of prior knowledge in abstract algebra for number theory. The discussion on textbook recommendations shows a mix of personal preferences without a clear consensus on the best options.

Contextual Notes

Some discussions highlight the dependence on specific textbooks and their varying prerequisites, as well as the subjective nature of textbook quality and suitability for learners.

Who May Find This Useful

Readers interested in the foundational aspects of linear algebra, abstract algebra, and number theory, as well as those seeking textbook recommendations for these subjects.

BloodyFrozen
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Are there any basic prerequisites before learning about these branches of mathematics?
 
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BloodyFrozen said:
Are there any basic prerequisites before learning about these branches of mathematics?

For linear and abstract algebra: no. Although it kind of depends on the book. For example, if you use Hoffman & Kunze for linear algebra, then this will assume a knowledge of abstract algebra already. Conversely, there are some abstract algebra books which use matrices to a certain extent, so linear algebra could come in handy.

But in general, there are no prereqs before learning linear and abstract algebra. However, be sure that you're acquainted with proofs! (or at least be prepared to learn proofs along the way).

For number theory, a knowledge of abstract algebra comes in extremely handy. Again, it depends on the book. Sometimes, a knowledge of abstract algebra or a knowledge of calculus is required.
 
How difficult are we speaking with matrices?
Normal det, or stuff like eigenvalues?
 
BloodyFrozen said:
How difficult are we speaking with matrices?
Normal det, or stuff like eigenvalues?

We're talking about almost anything in linear algebra. Things like, determinants, eigenvalues, vector spaces, bilinear forms, Jordan normal forms, etc. can be (and mostly is) done without abstract algebra.

However, (and I highly recommend this:) linear algebra is best done when working with vector spaces over arbitrary fields. So knowing what a field and a group is (and some of their very basic properties) will be important. Of course, this doesn't mean that you should have taken an abstract algebra course before.

The way I see it, the abstract algebra and the linear algebra reinforce each other. Things done in abstract algebra will increase understanding of linear algebra and vice versa. So it's probably best to study them simultaneously.
 
Great:)
Sorry for wasting your time, but are there any good linear and abstract algebra books that would go in harmony? (I new to the non-standard HS issued books)
 
Last edited:
BloodyFrozen said:
Great:)
Sorry for wasting time, but are there any good linear and abstract algebra that would go in harmony? (I new to the non-standard HS issued books)

Check out my blog for a comprehensive answer: https://www.physicsforums.com/blog.php?b=3206

Basically, Friedberg is (what I feel) the best book out there to learn linear algebra. However, Friedberg lacks a bit of concrete exercises. That's why a Schaum's outline (or exercises online) should be a good supplement.

For abstract algebra, I still feel that Pinter's "a book on abstract algebra" is an excellent introdcution. It is quite elementary, but it also goes quite far! Fraleigh's book is a nice substitute though...
 
Last edited by a moderator:
Ok, thanks for all your help:)
 
What edition do you suggest for Pinter? and for friedberg, is 4th edition ok?
 
BloodyFrozen said:
What edition do you suggest for Pinter? and for friedberg, is 4th edition ok?

I don't think there is a remarkable difference between editions, really. The 4th edition will probably be ok.
 
  • #10
I was just wondering about Pinter because one is $115 and the other us $11. It's a large difference between the two. You have the second edition, right?
 
  • #11
BloodyFrozen said:
I was just wondering about Pinter because one is $115 and the other us $11. It's a large difference between the two. You have the second edition, right?

Oh, by all means, get the cheaper one. There's no need to pay $100 extra for virtually the same book!

I think I also bough the $11-version...
 
  • #12
Ok, Second version it is!
 
  • #13
I haven't read nearly as many algebra books as say, micromass, has, but the best linear algebra book I have ever read is Finite Dimensional Vector Spaces by Paul Halmos. Yes, it looks like it was written on a typewriter (it was done WAY before LaTeX), but it is VERY good, IMHO.
 
  • #14
Robert1986 said:
I haven't read nearly as many algebra books as say, micromass, has, but the best linear algebra book I have ever read is Finite Dimensional Vector Spaces by Paul Halmos. Yes, it looks like it was written on a typewriter (it was done WAY before LaTeX), but it is VERY good, IMHO.


Thanks for the suggestion, I'll go and check it out.
 

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