Advanced Linear Algebra Content

[YAHA]

I have seen a few recommendations on this forum that said a decent math preparation for a physics grad school should include advanced LA. What exactly would that include? What books would you suggest?

Is this book one of them? https://www.amazon.com/dp/048663518X/?tag=pfamazon01-20

 

[micromass]

I suggest Friedberg's book. If you know that, then you'll know more linear algebra then you'll ever need!

See my blog as well: https://www.physicsforums.com/blog.php?b=3206

Shilov is a decent book, but I'm not a fan because he introduces determinants too early...

 

[YAHA]

I see :) Interestingly, I found the Shilov's exposition of determinants absolutely lucid. Although I lack the experience to determine the proper order of exposition, so I will take your word on it. I am going through my first LA class right now and found the Chapters 1-3 in Shilov to be a great supplement to the class textbook taking me to a deeper level.
The book that is used for my current class is Anton and Rorres. Is that any decent? I don't remember seeing it mentioned in your blog post.

 

[micromass]

I see :) Interestingly, I found the Shilov's exposition of determinants absolutely lucid. Although I lack the experience to determine the proper order of exposition, so I will take your word on it. I am going through my first LA class right now and found the Chapters 1-3 in Shilov to be a great supplement to the class textbook taking me to a deeper level.
The book that is used for my current class is Anton and Rorres. Is that any decent? I don't remember seeing it mentioned in your blog post.

Well, if you find Shilov lucid enough, then by all means follow the book. It's an extremely good book and it'll teach you everything you need to know about linear algebra. I just didn't like the book very much (mainly because it might be too difficult for first time students), but if you like it and don't have any problems with it, then study from the book!

Anton and Rorres strikes me as a book that is too easy and too application oriented. This is good if you never seen matrices before, but (as you understood Shilov's determinants), then you might want a more challenging book.
If I were to study from Anton and Rorres, I would be very bored...

In any case, choosing a math book that suits you is quite difficult. In my blog I just recommend books which are good and which I like very much. But you need to choose a book that you like

 

[YAHA]

Awesome :) yeah, i think i started looking for a supplement book in the first place because I was bored with Anton and Rorres.

 

[ahsanxr]

On the topic of Advanced LA, for the course in our school, the book used by the professor last year was "Advanced Linear Algebra" by Steve Roman and I've heard its very rigorous and difficult. However the professor this year is using Friedberg's book. I want to know which would be more suitable for a second course in Linear Algebra and am afraid whether the course has lost some of its rigor by the professor's choice to use that book. I don't want to be going over the same stuff we went over in the Elementary LA course.

 

[micromass]

On the topic of Advanced LA, for the course in our school, the book used by the professor last year was "Advanced Linear Algebra" by Steve Roman and I've heard its very rigorous and difficult. However the professor this year is using Friedberg's book. I want to know which would be more suitable for a second course in Linear Algebra and am afraid whether the course has lost some of its rigor by the professor's choice to use that book. I don't want to be going over the same stuff we went over in the Elementary LA course.

Advanced linear algebra by Roman is a graduate text. It's a very good text, but quite difficult. Knowledge of abstract algebra should be expected before embarking on that book.

What kind of stuff did you see in your first course of linear algebra? Maybe I can suggest a book that better suits your purposes??

 

[ahsanxr]

Well it was very computation based. It used Linear Algebra with Applications by Leon and our course covered

1. Matrices
2. Determinants
3. Vector Spaces
4. Linear Transformations
5. Inner Product Spaces
6. Eigenvalues and Eigenvectors.

Most of it just had to do with calculations with a proof in every other homework.

That's the text which people who have already taken the course said they used, although this year its Friedberg. Its a 500-level class.

 

[micromass]

Well it was very computation based. It used Linear Algebra with Applications by Leon and our course covered

1. Matrices
2. Determinants
3. Vector Spaces
4. Linear Transformations
5. Inner Product Spaces
6. Eigenvalues and Eigenvectors.

Most of it just had to do with calculations with a proof in every other homework.

That's the text which people who have already taken the course said they used, although this year its Friedberg. Its a 500-level class.

Hmmm, with your current knowledge I can almost certainly say that Roman will be too difficult for you. On the other hand, Friedberg will be a bit too easy for a 500-level class. But you won't find Friedberg boring however, it contains lots of stuff that you didn't cover yet.

Maybe try reading Hoffman & Kunze. It's more difficult than Friedberg, and it might suit your purposes...

 

[ahsanxr]

Hmmm, with your current knowledge I can almost certainly say that Roman will be too difficult for you. On the other hand, Friedberg will be a bit too easy for a 500-level class. But you won't find Friedberg boring however, it contains lots of stuff that you didn't cover yet.

Maybe try reading Hoffman & Kunze. It's more difficult than Friedberg, and it might suit your purposes...

But how would I benefit from reading that text if the class isn't using it? Would it prepare me more for say, an Abstract Algebra course the next semester? On the other hand how well should one be acquainted with proofs to do well with Friedberg? I don't have much experience with proofs.

 

[micromass]

But how would I benefit from reading that text if the class isn't using it?

Oh, I thought you were asking for advice on texts you could read outside of the assigned text. A benefit could be that it gives you a different perspective on the matter. But it's also more work/

Would it prepare me more for say, an Abstract Algebra course the next semester?

Not necessarily, but it will increase your "mathematical maturity". You don't really need much linear algebra to do good in abstract algebra.

On the other hand how well should one be acquainted with proofs to do well with Friedberg? I don't have much experience with proofs.

I think it's fair to say that you can learn proofs along the way. But why are you doing a 500-level class if you don't have experience with proofs?? It will be highly unlikely that the lecturer will teach you proofs. You will have to learn on your own here...

 

[ahsanxr]

Oh, I thought you were asking for advice on texts you could read outside of the assigned text. A benefit could be that it gives you a different perspective on the matter. But it's also more work/

Well no, my concern was that whether the course had necessarily been made easier as I don't want to be going over the algorithmic sort of math that we have to do in a first course in linear algebra. Do you think I'll still be challenged by Friedberg?

Not necessarily, but it will increase your "mathematical maturity". You don't really need much linear algebra to do good in abstract algebra.

Good to know.

I think it's fair to say that you can learn proofs along the way. But why are you doing a 500-level class if you don't have experience with proofs?? It will be highly unlikely that the lecturer will teach you proofs. You will have to learn on your own here...

Well the course numberings at my school are kind of weird. There are hardly any 400-level courses. It just goes from 300-level ones to 500-level ones. I have a copy of "How to Prove It" by Velleman, so will I have much problems if I just learn from that along the way?

 

[micromass]

Well no, my concern was that whether the course had necessarily been made easier as I don't want to be going over the algorithmic sort of math that we have to do in a first course in linear algebra. Do you think I'll still be challenged by Friedberg?

Certainly yes! You won't be doing much algorithmic stuff in Friedberg. It's certainly quite a good book for a second course!
It's good for you that the course has been made easier, as you would probably have had a lot of trouble with Roman (judging from the fact that you don't have experience with proofs).

Well the course numberings at my school are kind of weird. There are hardly any 400-level courses. It just goes from 300-level ones to 500-level ones. I have a copy of "How to Prove It" by Velleman, so will I have much problems if I just learn from that along the way?

No, you won't have much problems if you learn side-by-side with Velleman. Maybe reading a bit before the course starts could be benificial. But apart from that, you should be ready to start the course!

 

[ahsanxr]

Certainly yes! You won't be doing much algorithmic stuff in Friedberg. It's certainly quite a good book for a second course!
It's good for you that the course has been made easier, as you would probably have had a lot of trouble with Roman (judging from the fact that you don't have experience with proofs).

I'm glad to hear that.

No, you won't have much problems if you learn side-by-side with Velleman. Maybe reading a bit before the course starts could be benificial. But apart from that, you should be ready to start the course!

Do you think the same would hold true for a course using baby Rudin, or an introductory Differential Geometry course?

 

[micromass]

Do you think the same would hold true for a course using baby Rudin, or an introductory Differential Geometry course?

Baby Rudin is quite difficult for a first-time proof course I'm afraid You can try it, but expect to work VERY hard!
If you do want to use baby Rudin for a course right now, be sure to supplement it with a book like Abbot's "Understanding Analysis".

You shouldn't have much difficulty with an introductory differential geometry class. But this depends on the lecturer. He can make it very difficult or he can make it moderately easy. Better ask other people what they thought of it...

 

[ahsanxr]

Baby Rudin is quite difficult for a first-time proof course I'm afraid You can try it, but expect to work VERY hard!
If you do want to use baby Rudin for a course right now, be sure to supplement it with a book like Abbot's "Understanding Analysis".

You shouldn't have much difficulty with an introductory differential geometry class. But this depends on the lecturer. He can make it very difficult or he can make it moderately easy. Better ask other people what they thought of it...

Since we've gotten a bit off-topic I've moved our discussion to messages. Thanks a lot for answering my questions so far.