# Linear algebra book for early phd student

I understand this is a frequently asked question, but I'm posing it again, since I am not sure about the extent to which other people who have asked this wanted the same thing as I do.

I'm an early phd student in an engineering discipline. I am looking for a book on LA aimed at advanced ugrads/early grads to help me cover my gaps. I've taken an introductory course in LA whose exposition focused more on linear systems etc, as well as an abstract algebra course focusing on group theory. Now I want an LA book focusing more on the vector space aspects of it instead of matrix theory. The particular topics of focus are linear spaces and transformations, eigenvalues and eigendecompositions, inner product spaces, matrix norms and quadratic forms, as well as the geometric interpretation of these notions.

Despite not being a math student, I'm not looking for an applied book (like Strang). My area of research requires an excellent command of the aforementioned topics, so I want a theoretically sound, proof-based exposition that doesn't delve too much into functional analysis/operator theory. Also, challenging problems are definitely a plus.

• Friedberg Insel Spence is a classic (the book I learnt linear algebra from), Hoffman and Kunze is another one. There's also linear algebra done right/wrong. These are all purely linear algebra books. Another option is Chapters 1 and 2 of Loomis and Sternberg's Advanced Calculus (freely available online), which cover much of what you're after (and if you read through this and solve as many exercises as possible, you'll be in great shape; but be warned that Loomis is not an easy first read). Aug 27 '21 at 12:43
• Checked Loomis and Sternberg now. I don't think it's a particularly good fit. I've checked a bit LADR/LADW, since most people suggest those in similar topics. I guess, between the 2, LADW is better suited for me, since it's more advanced and covers pretty much all the topics I want. As for Hoffman and Kunze and Friedberg Insel Spence, I'm not familiar with those books, but will check them out. Aug 27 '21 at 12:52
• The first half of the book Algebra by Artin is mostly about linear algebra. It's better than most books for giving you geometric intuition and applications. There will be some groups mixed in, but I think that actually helps with understanding. Other options are Algebra by Godement (which also has a bit on abstract algebra - you can skip the part on logic and sets), Lectures on Linear Algebra by Gelfand, Finite-Dimensional vector Spaces by Halmos, Linear Algebra by Lang. Aug 27 '21 at 16:35
• "Matrix norms" is a bit of a specialized topic, unless I'm misunderstanding what you mean. Maybe you can explain by telling me where it is in H & K. But on inner product spaces and quadratic forms, Artin has Chapter 8, Godement has Chapter 36, Gelfand has most of Chapter 1 (of four), Halmos has Chapter 3, and Lang has Chapter 5. Aug 27 '21 at 17:55
• I second @Anonymous 's recommendation of Artin. OP-- take a closer look at the book. Do the first 9 (10) chapters in 1st (2nd) edtion) and skip the ones on group theory (since you've already studied that), plain and simple. The chapter on bilinear forms covers inner products, quadratic forms and then some. Addiitonally SVD is a problem in Arint ex 7.9.7 in 1st edition. Your synopsis on Artin is wrong. Also if you want challenging problems, get the 1st edition... Artin dumbed the problems down somewhat in the 2nd edition. Aug 27 '21 at 19:21

I suggest you look at Linear Algebra by Kenneth Hoffman and Ray Kunze. Maybe it is helpful.

The Linear Algebra a Beginning Graduate Student Ought to Know by Jonathan S. Golan should be a good pick.

• Looks a little bit more advanced than what I was expecting, but I like it. Will definitely give it a shot. Aug 27 '21 at 12:45

The book here: https://mtaylor.web.unc.edu/notes/linear-algebra-notes/ covers those topics. The problems are not all super challenging: Often I found reading the text to be harder than solving the problems (this might be a quirk of the way I read though; I like to read slowly and thoroughly).

I quite liked Algebra Chapter 0 by Paolo Aluffi. In itself it is not just a linear algebra book but it builds the theory lets say "from scratch", it has many motviations and a good amount of problems.

• This is way too theoretical from what I gather. It goes through like 5 chapters of abstract algebra before going into linear algebra. Not really my cup of tea. Aug 27 '21 at 13:02
• @susami1996 It really helps to know something about groups in studying linear algebra. This comes up in various ways. Aug 27 '21 at 16:36
• I am familiar with group theory to some extent, as I mentioned in my original post. That is why I asked for a more advanced textbook. Still, Aluffi's book is a bit of an overkill I think. Aug 27 '21 at 17:17

You did comment that another book struck you as surprisingly advanced, so here's a suggestion that goes in the opposite direction: Elementary linear algebra by Anton and Rorres (and more recent editions seem to have added Kaul as an author). There also seem to be versions with and without applications. I am commenting on the 10th edition "with supplemental applications".

Does have, from your list: vector spaces and transformations, eigenvalues and eigendecompositions, inner product spaces, quadratic forms, as well as the geometric interpretation of these notions. (You might consider the material on inner product spaces a bit thin, perhaps, but it certainly isn't limited to matrices and numerical vectors only.) In the comments you also mention SVD which is covered, but the Courant–Fischer–Weyl min-max principle is only indirectly touched on (when considering the intuition for use of the Rayleigh quotient to find the dominant eigenvalue via the power method). Has lots and lots of questions, not limited to number-crunching — plenty of abstract proof questions too, though they may be a bit short.

Doesn't have: matrix norms, though you'll find them in more specialist texts and I think it would be rare to find a book that covers both introduction to eigenvalues and the norm of a matrix. You may also think that it's short of really challenging questions, but that depends where you're at so is hard for me to comment. (Given that you're preparing for PhD, I suspect you may find the bulk of questions too easy. But my suggestion for the book would only be as a "warm-up" before hitting the harder stuff.)

What you might find it useful for: the topics you're listing sound (to me) comparatively elementary bearing in mind the level of mathematics you intend to study. If it's too big a leap to make in one go, you might use a lower-to-mid undergraduate text to improve your intuition on basic abstract concepts, before transitioning to a heavy-duty text for the more advanced theory.

Why I think you might like it: I know you say you're not so interested in an applications or matrix oriented text, but is it possible that's because "applications" to you has meant "vectors are a tuple of numbers that represent coordinates, matrices represent transformations or changes of coordinate systems" and you want to generalise out from that to something more abstract? This book has loads of applications, many being quite different to that — for example, where the vector space is of polynomials, or continuous functions. If you're from an applied background, I would be very surprised if you didn't find some stuff in there to grab your interest, and I'm pretty sure you won't have seen all the applications before. There's traditional undergraduate matrix number-crunching in there too, but you lose nothing by just skipping over those sections! If you're someone who finds that applications motivate understanding of more abstract concepts, making them more concrete, this might be a really good starter text. Older editions are available cheaply, and despite your protestations to the contrary, I think the "with applications" version may still have some value to you. You probably don't want to go back further than the 10th edition, as this version seems to have beefed up on the theory front.

Why I think you might not like it: if you're at the point in life where you really hate applications, never want to see one again, and just want to be fed raw theory, then this is not your book. Similarly if you want an all-in-one book that gets you straight up to PhD level. You may also decide it's a bit thick for the material you want covered, though I must add that it's a very readable book (there are linear algebra books half the size but written very densely which would probably be a tougher slog to get through than this one).

• I have experience with this book, and it is pitched at quite a low level of mathematical sophistication. It deals with matrices at great length before ever connecting them with vector spaces and linear mappings. I think this is very far from what the OP wants. Aug 28 '21 at 18:19
• @Anonymous I hope my answer made it clear why I, too, would not recommend this as the book the OP is actually looking for. But I still think it may have value as a "warm-up" book for someone with an applied background who wants to get into linear algebra for the first time, before hitting something more heavy-duty. It is not mathematically sophisticated and the OP would likely skip or flick through a lot of matrix-heavy sections (which the structure of the book allows one to do fairly easily). Aug 31 '21 at 20:16
• Nevertheless, neither is it a purely applied book - it is proof-based, & covers (in an intuitive and practically motivated way) almost everything on the OP's list of desired topics. Even if not to the OP's taste, it could be useful for others who arrive here via search engine, and whose search terms may well include items from the OP's topic wish-list. On the flip-side, I hope the more critical parts of my review (and indeed your comment underneath... +1!) may help steer away people for whom this book would not be helpful. Aug 31 '21 at 20:22
• (For what it's worth, this is the core linear algebra undergraduate text used at several very prestigious universities - in at least one case with its teaching split across two years - so I don't think it's out of place to at least review its merits here, even if it's not an active recommendation.) Aug 31 '21 at 20:33