Recommended Linear Algebra books

[srfriggen]

I've taken a course in linear algebra a couple years ago, but I didn't take away too much from it, and now I'm looking to take some higher level math courses and have been brushing up on the subject, spending about an hour to two hours a day going through my old notes and textbook. However, the class I took wasn't very proof driven, and was more computational. We didn't get down to much of the theory behind the subject.

I don't feel like taking another linear algebra course and would rather just learn the subject the best I can on my own.

Any recommendations of books or other methods that would really give me a good insight into the subject?


Also, as far as pure math is concerned, in which other areas of study does the knowledge of linear algebra become most useful?, i.e. analysis, topology, etc.

Not sure if this is relevant, but this fall I'll be taking a course in abstract algebra and an introduction to analysis.

 

[micromass]

Books which I highly recommend are (ranked from easy to hard):

"Linear algebra" by Friedberg, Insel, Spence: https://www.amazon.com/dp/0130084514/?tag=pfamazon01-20
"Linear algebra" by Serge Lang: https://www.amazon.com/dp/1441930817/?tag=pfamazon01-20
"Linear algebra" by Hoffman, Kunze: https://www.amazon.com/dp/0135367972/?tag=pfamazon01-20

Each of this books will give you a a solid and rigorous introduction to linear algebra.

Linear algebra is important in most pure math contexts:

- In abstract algebra, we study algebraic structures like modules over a ring. These are direct generalization of the vector spaces of linear algebra. In some sense, the vector spaces are the ideal situation on which everything is modeled. Modules are essential in abstract algebra and are used in, for example, representation theory where one tries to model groups as matrices.

- In analysis, linear algebra is very important as the basic idea of analysis is to represent nonlinear rough maps as linear nice maps. As soon as you leave the one-dimensional picture, linear algebra sets in everywhere.

- Even more fun is the infinite-dimensional picture, where one tries to generalize linear algebra to infinite dimensions, this is the study of Banach spaces and Hilbert spaces. Again, linear algebra is the ideal situation on which everything is modeled.

- Differential equations is studied easiest with linear algebra. For example, if you are given a system of differential equations, then one can solve it easiest by linear algebra.

- In applied mathematics, matrices are used truly everywhere. So it's no surprise that linear algebra pops up everywhere here as well.

In short, name whatever field of mathematics and you will meet linear algebra in it at some point. So knowing it well is important.

 

[srfriggen]

Books which I highly recommend are (ranked from easy to hard):

"Linear algebra" by Friedberg, Insel, Spence: https://www.amazon.com/dp/0130084514/?tag=pfamazon01-20
"Linear algebra" by Serge Lang: https://www.amazon.com/dp/1441930817/?tag=pfamazon01-20
"Linear algebra" by Hoffman, Kunze: https://www.amazon.com/dp/0135367972/?tag=pfamazon01-20

Each of this books will give you a a solid and rigorous introduction to linear algebra.

Linear algebra is important in most pure math contexts:

- In abstract algebra, we study algebraic structures like modules over a ring. These are direct generalization of the vector spaces of linear algebra. In some sense, the vector spaces are the ideal situation on which everything is modeled. Modules are essential in abstract algebra and are used in, for example, representation theory where one tries to model groups as matrices.

- In analysis, linear algebra is very important as the basic idea of analysis is to represent nonlinear rough maps as linear nice maps. As soon as you leave the one-dimensional picture, linear algebra sets in everywhere.

- Even more fun is the infinite-dimensional picture, where one tries to generalize linear algebra to infinite dimensions, this is the study of Banach spaces and Hilbert spaces. Again, linear algebra is the ideal situation on which everything is modeled.

- Differential equations is studied easiest with linear algebra. For example, if you are given a system of differential equations, then one can solve it easiest by linear algebra.

- In applied mathematics, matrices are used truly everywhere. So it's no surprise that linear algebra pops up everywhere here as well.

In short, name whatever field of mathematics and you will meet linear algebra in it at some point. So knowing it well is important.

Thanks so much micromass. much appreciated!

 

[robertjford80]

Whatever you do don't read Gilbert Strang's book. That book is terrible. I'm using Richard Hill's book and I find it rather good but there are still a few points that he does not explain well. Gilbert Strang's lectures on youtube on the other hand are ok. Just put in Linear Algebra MIT lecture and you will find them.

 

[a7d07c8114]

I haven't read it myself, but I've seen Axler's "Linear Algebra Done Right" recommended by quite a few posters when I searched for this same question.

 

[mathwonk]

some earlier answers to this question:

https://www.physicsforums.com/showthread.php?t=450903&highlight=primer+linear+algebra