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I'm taking a graduate Linear Algebra course and have limited experience writing proofs (mostly from a discrete math class). Can anyone recommend good books to teach you how to write proofs for linear algebra? My impression is that they're very different from other kinds of math proofs.

Thanks in advance.

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  • $\begingroup$ I've always been a fan of Linear Algebra by Hoffman and Kunze. It's an excellent book which has been used often as an introductory text, and it covers the core subjects of linear algebra quite well. $\endgroup$ – EuYu Dec 18 '13 at 23:32
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    $\begingroup$ @Ike : you have to be careful because many, maybe even most, linear algebra books are aimed at engineering students, defer proofs until fairly late in the book, and I have seem some popular books by highly regarded authors that contain imprecise and/or poorly worded definitions and terminology, or use difficult proofs to prove things that should be easy to prove. My point is that you cannot take a random linear algebra book and take it for granted that it will help you write good proofs. I've heard good things about Linear Algebra Done Right... $\endgroup$ – Stefan Smith Dec 19 '13 at 7:11
  • $\begingroup$ ...I think it introduces vector spaces early, unlike most texts which defer them because they are an abstract topic that most students have a lot of trouble with. This approach makes sense because a primary reason for using matrices at all (perhaps the most important reason) is to manipulate linear transformations between vector spaces. $\endgroup$ – Stefan Smith Dec 19 '13 at 7:13
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To begin, you do have to understand the definitions. It's impossible to prove anything if you don't really understand the question.

Linear algebra has its own terminology, and many a student has gotten so tangled in it they could not proceed well. Besides the statement of the definitions, you have to understand what they mean and how they relate to other definitions. The best way to accomplish that is to work a lot of basic problems, dealing with vectors, matrices, systems of equations.

Next, there are a few fundamental theorems which come up over and over. There is a laundry list of about 12 statements which are all equivalent to saying that a set of vectors is independent. You should get your hands on such a list, and then make sure you know why each one is equivalent. If you can't find such a list (which I provided my own students), why not ask here. I'll bet you wind up with 30 equivalent statements. (Not that you necessarily want all 30, I know).

The next step would be to work many, many problems about vector independence and basis. If you get stuck, get help, and make sure you understand the answers.

The next big linear algebra topic is "spectral theory" which is about eigenvalues and eigenvectors. Study that separately, and again work a lot of problems. Stay away from generalized eigenvectors until you have mastered the basic material.

This short list is only a start on the subject, but it can help you understand what is most fundamental. I think it is always best to solidify foundations, after which moving forward is much easier.

Regarding books, you now have several recommendations. I find that different approaches work better for different people, so that someone may love a book that makes no sense to you at all. Try out various chapters that have been published online and see whose style appeals to you. A book that provides many problems of varying levels of difficulty would be the best.

The best way to learn the subject is to work many problems. Along the way, you'll get a sense of how to approach linear algebra proofs.

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  • $\begingroup$ Although I realize there is a great deal to learn with math proofs before I'll be able to tackle graduate linear algebra, it's great to get a summary of the steps needed to do proofs. Thanks for your very constructive post! $\endgroup$ – JIL Feb 20 '14 at 22:14
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If you have limited experience writing proofs, you should not be taking graduate-level linear algebra.

If you don't know linear algebra, check out Sheldon Axler's Linear Algebra Done Right. If you need a text that goes into more advanced material more rapidly, see Artin's Algebra.

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    $\begingroup$ To amplify: if you don't have much experience writing proofs, you shouldn't be taking any graduate-level math class. You will be lost in a heartbeat. $\endgroup$ – dfeuer Dec 18 '13 at 22:14
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Another book you might consider is Curtis' Abstract Linear Algebra. It claims that it could be used for a "first course," but is quite sophisticated (but I have to agree with the claim). It takes some care to introduce the more abstract topics not usually covered in undergrad classes. For instance, exact sequences of maps, defining the determinant via the exterior product, quotient spaces, etc. The majority of the proofs are very easy to follow and the book is a good reference (despite how slim it is).

There's also a book called "The Linear Algebra a Graduate Student Ought to Know." It looks pretty good.

To learn to write proofs, you might look through some of these short "articles" and then do a lot of problems. (I don't recommend a whole book on learning to write proofs). Be sure to have your solutions/proofs critiqued (here for instance!).

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Not sure how advanced your class is, but Sheldon Axler's book "Linear Algebra Done Right", covers the basics very well for a lower level grad course (like upper undergrad/Masters level).

A more advanced, and very thorough text for an upper level graduate linear algebra would be Evar Nehring's "Linear Algebra and Matrix Theory". Even though it is not an easy text, it is pretty much self contained and the proofs are well written

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