Number Theory, Linear & Abstract Algebra

In summary: Do you think it would be a good introductory book for someone who is new to linear algebra?Thanks for the suggestion, I'll go and check it out.
  • #1
BloodyFrozen
353
1
Are there any basic prerequisites before learning about these branches of mathematics?
 
Physics news on Phys.org
  • #2
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.
 
  • #3
How difficult are we speaking with matrices?
Normal det, or stuff like eigenvalues?
 
  • #4
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.
 
  • #5
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:
  • #6
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:
  • #7
Ok, thanks for all your help:)
 
  • #8
What edition do you suggest for Pinter? and for friedberg, is 4th edition ok?
 
  • #9
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.
 

1. What is Number Theory?

Number Theory is the branch of mathematics that deals with the properties and relationships of numbers, particularly integers. It includes topics such as prime numbers, divisibility, and modular arithmetic.

2. What is Linear Algebra?

Linear Algebra is the branch of mathematics that studies linear equations, vectors, matrices, and their properties. It is used to solve systems of equations, analyze geometric transformations, and understand the behavior of systems with multiple variables.

3. What is Abstract Algebra?

Abstract Algebra is the study of algebraic structures such as groups, rings, and fields. It focuses on the properties and relationships between these structures and their operations, rather than specific numerical values.

4. What are some practical applications of Number Theory, Linear Algebra, and Abstract Algebra?

Number Theory is used in cryptography, coding theory, and computer science. Linear Algebra is used in physics, engineering, and computer graphics. Abstract Algebra has applications in cryptography, coding theory, and quantum mechanics.

5. What are some common techniques used in these branches of mathematics?

Some common techniques used in Number Theory include modular arithmetic, the Euclidean algorithm, and the Sieve of Eratosthenes. In Linear Algebra, common techniques include Gaussian elimination, vector spaces, and eigenvalues. In Abstract Algebra, techniques such as group presentations, Cayley graphs, and homomorphisms are often used.

Similar threads

B What do "linear" and "abstract" stand for?
    • Linear and Abstract Algebra
Replies
12
Views
1K
Insights Introduction to the World of Algebras
    • Linear and Abstract Algebra
Replies
0
Views
2K
Book recommendations (abstract algebra and number theory)
    • Science and Math Textbooks
Replies
5
Views
2K
I Bra-Ket Notation Manipulations: Quantum State Expansion
    • Linear and Abstract Algebra
Replies
8
Views
1K
I What is the proper matrix product?
    • Linear and Abstract Algebra
Replies
9
Views
1K
I Question Regarding Definition of Tensor Algebra
    • Linear and Abstract Algebra
Replies
10
Views
359
MHB Resource for learning linear algebra
    • Linear and Abstract Algebra
Replies
5
Views
1K
MHB Either or statement from Abstract Algebra Book
    • Linear and Abstract Algebra
Replies
16
Views
2K
Book for abstract algebra (group and galois theory)
    • Linear and Abstract Algebra
Replies
4
Views
2K
I Law of Cosines in Linear Algebra: Understanding the Dot Product of Unit Vectors
    • Linear and Abstract Algebra
Replies
10
Views
1K

Hot Threads

Recent Insights

Back
Top