Next step in Linear Algebra studies?

In summary, the conversation covers the topic of further study in linear algebra, specifically in multilinear algebra and tensor algebra. It is suggested that a background in abstract algebra is necessary for advanced linear algebra, with recommendations for books such as "Advanced linear algebra" by Steven Roman and "Linear algebra a beginning grad student ought to know". The direction of functional analysis and modules is also discussed, with suggestions for books such as "Basic algebra" by Knapp and "Functional analysis" by Kreyszig. The possibility of taking a representation theory course is also mentioned. Additionally, there are suggestions for introductory books in abstract algebra for those with a background in linear algebra, such as "Algebra from a geometric viewpoint" by Ted Shifrin and "
  • #1
Broccoli21
80
1
Hey folks,
I just finished reading Axler's Linear Algebra Done Right, and Halmos' Finite Dimensional Vector Spaces, as well as doing pretty much all the problems in both of them. I really like linear algebra, and would like to keep learning more. I am interested in multilinear algebra and tensor algebra. Is there a nice intro to this subject out there, especially one that doesn't rely on abstract algebra (as I don't know much of that)?
Thanks,
Broccoli 21
 
Physics news on Phys.org
  • #2
From what I have seen, advanced linear algebra either requires some abstract algebra (e.g. module theory) or topology (e.g metric spaces). Two books I think seem good are Steven Roman's "Advanced linear algebra" and "The linear algebra a beginning grad student ought to know".
 
  • #3
You'll experience some tensors in most analysis on manifolds courses. Check out Calculus on Manifolds by Spivak or Analysis on Manifolds by Munkres. Typical prerequisites are real analysis and rigorous linear algebra.
 
  • #4
Broccoli21 said:
Hey folks,
I just finished reading Axler's Linear Algebra Done Right, and Halmos' Finite Dimensional Vector Spaces, as well as doing pretty much all the problems in both of them. I really like linear algebra, and would like to keep learning more. I am interested in multilinear algebra and tensor algebra. Is there a nice intro to this subject out there, especially one that doesn't rely on abstract algebra (as I don't know much of that)?
Thanks,
Broccoli 21

from there I would say there are two directions you could go. there's functional analysis which is sort of like infinite-dimensional linear algebra or there's modules, which is like linear algebra where the field is replaced by a ring. if you do modules you'd definitely get to tensors at some point, & probably multilinear algebra. before doing any of that though I think it would be a good idea to brush up on abstract algebra, especially rings.
 
  • #5
Hmm. Both of these seem to require math that I haven't taken. Would Baby Rudin be sufficient background for functional analysis?
I have only basic abstract algebra under my belt. Maybe I'll wait till later to tackle modules and the like.
 
  • #6
To go any further, your will need to learn Abstract Algebra. As you go along, Linear Algebra and Abstract Algebra merge in the study of modules and algebras. There are a number of solid algebra textbooks, but different ones are better depending on your starting level. Have you done an abstract course, or just read on your own?
espen180 said:
Two books I think seem good are Steven Roman's "Advanced linear algebra" and "The linear algebra a beginning grad student ought to know".
Roman's book is excellent, and a good supplement to a straight algebra book.
 
  • #7
fourier jr said:
from there I would say there are two directions you could go. there's functional analysis which is sort of like infinite-dimensional linear algebra or there's modules, which is like linear algebra where the field is replaced by a ring. if you do modules you'd definitely get to tensors at some point, & probably multilinear algebra. before doing any of that though I think it would be a good idea to brush up on abstract algebra, especially rings.

Which direction would be most favourable for a physicist to take? I am in a similar position as the OP. Also, could someone recommend a beginners/an introductory book in abstract algebra, i.e. an abstract algebra book for people who have just finished linear algebra? Thanks in advance.
 
  • #8
Broccoli21 said:
Hmm. Both of these seem to require math that I haven't taken. Would Baby Rudin be sufficient background for functional analysis?
I have only basic abstract algebra under my belt. Maybe I'll wait till later to tackle modules and the like.

If you want to do functional analysis, then check out Kreyszig. This is a wonderful book that doesn't demand much prereqs. You don't need baby Rudin in order to read that book as it develops analysis from the beginning. (familiarity with a book like Spivaks calculus is necessary though)

If you want to go the algebra route, then you should read "basic algebra" by Knapp. It's an extremely beautiful and inspiring book.
 
  • #9
Pattern said:
Which direction would be most favourable for a physicist to take? I am in a similar position as the OP. Also, could someone recommend a beginners/an introductory book in abstract algebra, i.e. an abstract algebra book for people who have just finished linear algebra? Thanks in advance.

Algebra wouldn't hurt but functional analysis would probably be better. Kreyszig has a good book which keeps the topology & measure theory to a minimum. I haven't seen a lot of intro algebra books but iirc one of Herstein's is good & has lots of examples.
 
  • #10
If you want to do functional analysis, then check out Kreyszig. This is a wonderful book that doesn't demand much prereqs. You don't need baby Rudin in order to read that book as it develops analysis from the beginning. (familiarity with a book like Spivaks calculus is necessary though)
Thanks! I have taken intro to analysis (using baby Rudin), so I think I'll check out Kreyszig. Functional Analysis seems pretty cool, and like Pattern, I'm interested in Physics.

Also, after I take abstract algebra I (groups, rings and fields), I can take representation theory. Would that be sufficient background to study Roman's "Advanced linear algebra"? If not, then what other math should I take?

Note: on the representation theory class description it says:
"The topics covered will include group rings, characters, orthogonality relations, induced representations, applications of representation theory, and other select topics from module theory."
 
  • #11
I think you'll be fine for Roman if you know about groups, rings and fields. He does cover the preliminaries briefly, though.
 
  • #12
Ted Shifrin wrote a book on abstract algebra for people that have just had linear algebra, called algebra from a geometric viewpoint.werner greub has two books on linear algebra, one called linear algebra and one called multilinear algebra. I would suggest the second one for the OP's original request.
 
  • #13
mathwonk said:
Ted Shifrin wrote a book on abstract algebra for people that have just had linear algebra, called algebra from a geometric viewpoint.


werner greub has two books on linear algebra, one called linear algebra and one called multilinear algebra. I would suggest the second one for the OP's original request.

Thanks! In the description of "Multilinear Algebra", it says that his "Linear Algebra" book is a prerequisite. "Linear Algebra" seems to be a graduate-level book. Should I study that first?
You can see the table of contents here (look inside)
https://www.amazon.com/dp/0387901108/?tag=pfamazon01-20
 
  • #14
it's written in a less informal style & from a more abstract point of view than axler's. you could try working on a book on multilinear algebra (northcott does another one btw), say flip through a copy at your library & see if you can follow it. if not I still think more abstract algebra would probably make it easier.
 
  • #15
Since you've already studied Halmos' Finite Dimensional Vector Spaces, you might enjoy working the problems in his Linear Algebra Problem Book.
 

1. What is the next step after learning the basics of Linear Algebra?

After learning the basics of Linear Algebra, the next step is usually to delve deeper into the subject and explore more advanced topics such as vector spaces, linear transformations, and eigenvalues and eigenvectors. It is also important to gain a strong understanding of the fundamental theorems and concepts in Linear Algebra.

2. What are some good resources for studying the next level of Linear Algebra?

Some good resources for studying the next level of Linear Algebra include textbooks such as "Linear Algebra: A Modern Introduction" by David Poole and "Linear Algebra Done Right" by Sheldon Axler. Online resources such as MIT OpenCourseWare and Khan Academy also offer comprehensive lectures and practice problems for advanced Linear Algebra topics.

3. How can I apply Linear Algebra to real-world problems?

Linear Algebra has numerous applications in fields such as engineering, physics, computer science, and economics. Some examples of real-world problems that can be solved using Linear Algebra include optimization problems, image and signal processing, and data analysis. It is important to have a strong understanding of Linear Algebra concepts in order to effectively apply them to real-world problems.

4. What are some common challenges when studying advanced Linear Algebra?

Some common challenges when studying advanced Linear Algebra include understanding abstract concepts such as vector spaces and linear transformations, as well as mastering the various theorems and proofs. It is also important to have a solid foundation in basic algebra and calculus, as these concepts are often used in advanced Linear Algebra problems.

5. What are some tips for success in advanced Linear Algebra studies?

Some tips for success in advanced Linear Algebra studies include practicing regularly, seeking help from professors or tutors when needed, and actively engaging with the material by asking questions and working through challenging problems. It can also be helpful to make connections between different concepts and to apply what you have learned to real-world problems.

Similar threads

  • Science and Math Textbooks
Replies
13
Views
2K
  • Science and Math Textbooks
Replies
5
Views
1K
  • Science and Math Textbooks
Replies
6
Views
1K
  • Science and Math Textbooks
Replies
11
Views
2K
  • Science and Math Textbooks
Replies
3
Views
3K
  • Science and Math Textbooks
Replies
16
Views
2K
  • Science and Math Textbooks
Replies
4
Views
2K
  • Sticky
  • Science and Math Textbooks
Replies
9
Views
4K
  • STEM Academic Advising
Replies
11
Views
1K
  • Science and Math Textbooks
Replies
4
Views
1K
Back
Top