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I am looking for a linear algebra book which is as abstract as possible. But not an abstract algebra book. Something that is what Rudin's is for (beginning) analysis, that is, terse, rigorous and with beautiful exercises.

For example: Intro to LA, by Curtis, and LA by Hoffman-Kunze. Opinions?

(I know there are already very similar questions around, but I felt they are not quite the same.)

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  • $\begingroup$ Linear algebra texts have been discussed over and over at m.se: see math.stackexchange.com/search?q=linear+algebra+book . I personally have never read a linear algebra book in full (or even more than a chapter's worth), instead gathering my knowledge from various online sources; thus I cannot contribute much to this discussion. Treil ("Linear algebra done wrong") and Hefferon ("Linear algebra") seem to be good introductory texts and have the undeniable advantage that you won't waste any money. $\endgroup$ Jul 21, 2015 at 23:37
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    $\begingroup$ Also Lang - Serge Lang. $\endgroup$
    – Alec Teal
    Jul 22, 2015 at 11:50
  • $\begingroup$ there are books of analysis far more abstract than Rudin, by example the books of Amann and Escher. The book of Axler, as said in an answer below, is enough abstract, but probably it will exists some books more abstract than this. $\endgroup$
    – Masacroso
    Aug 2, 2017 at 3:24

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I am not sure what you are really looking for ("as abstract as possible" would suggest a Bourbakist approach, but this is not likely to involve many "beautiful exercises", though in part this depends on what you consider beautiful and what you consider and exercise). But since nobody else did yet, I'd like to mention Roger Godement's Cours d'Algèbre (1963), which would meet some of your criteria (and has been translated in English, too). It is not restricted to linear algebra, but those willing to admit some basic stuff about logic, set theory, groups, rings, and complex numbers can start reading at section 10 (modules and vector spaces), after which it is pretty much all stuff relevant to Linear Algebra until the final section 36 (Hermitian forms).

In the Bourbakist tradition vector spaces are of course introduced as a special case of modules over a ring. However the pursuit of generality is done only where it is painless, in the sense that it assumes the level of generality natural for the theorems being considered. This means for instance that in section 19 (the notion of dimension) the base ring is assumed to be a skew field (a.k.a. division ring; just "corps" in French where this term does not imply commutativity) whereas in the previous section (finiteness theorems) it was only assumed to be Noetherian or principal, depending on the exact results stated. In the lead-up to determinants (sections 21-24) the ring or field will be assumed commutative, and sections 34-36 where eigenvalue problems are discussed assume a true (commutative) field. One of the really nice things that I found in this approach is that not needlessly assuming commutativity forces some notational habits that I have found are useful even when working over rings that are commutative (for instance writing scalars at the opposite side of vectors than linear maps).

For the record, the book contains about 165 pages of exercises at the end, and though I cannot vouch for the beauty of all of them, there must be some nice ones to found there.

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  • $\begingroup$ Agreed, I was being rather vague. What I am looking for is, as the title says, something similar to Rudin's PMA (not that this is not vague!). More precisely: dense and proof-based exposition but with non-mechanical and "exploratory" exercises. I'm not quite sure if this makes things clearer. But thanks for your answer, I will have a look at Godement's. $\endgroup$
    – ZenoCosini
    Jul 22, 2015 at 9:48
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Axler's "Linear algebra done right" fits your criteria. It's terse, it's rigorous, and it has beautiful exercises.

I would warn, however, that its focus is ultimately on the coordinate-free underpinnings and does little to prepare you for the computational aspects.

Katznelson and Katznelson's "A Terse Introduction to Linear Algebra" is a more "down to earth" alternative.

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  • $\begingroup$ Certainly not the algebraic underpinnings! The book really takes the "algebra" out of linear algebra... and that is not a gain. $\endgroup$ Jul 21, 2015 at 23:38
  • $\begingroup$ I think "coordinate-free" underpinnings is what I'm going for $\endgroup$ Jul 21, 2015 at 23:40
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    $\begingroup$ @coldnumber: It tries to base linear algebra on eigenvectors. Think of it as a "spectral analysis" approach to linear algebra. This is not a very natural approach from an algebraic perspective, to say the least (it does not generalize to fields other than $\mathbb{R}$ and $\mathbb{C}$ -- hell, even over $\mathbb{R}$ it leads to complications); it also forces you to deal with subtleties such as algebraic and geometric multiplicities in basic concepts like the definition of a determinant. And it makes exact computation (for, say, exact matrices) impossible, since everything ... $\endgroup$ Jul 21, 2015 at 23:45
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    $\begingroup$ ... relies on finding roots of polynomials. (Unless you know how to do exact computations with algebraic numbers; but this is really not elementary stuff.) I think of the book as an interesting experiment in putting carts before horses. $\endgroup$ Jul 21, 2015 at 23:46
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    $\begingroup$ It is one of the tragedies of modern mathematics that the split between "algebraic" and "analytic" mathematicians (to replace the usual, loaded, distinction with an equally arbitrary but more neutral one) has become so wide that it already begins in the first semester: For the former, Linear Algebra is a precursor to Algebra; for the latter, to Functional Analysis (and to a lesser extent, Numerical Linear Algebra). Neither is really concerned with the needs of the other. To get to the point: Axler's book is an excellent text if (and only if) you fall into the latter class. $\endgroup$ Jul 22, 2015 at 10:28
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Edit: The order I presented the books is not a ranking simply the chronological order I went through them and this happened to be the order I wrote them in the review. I would recommend H&K.

I'll try to answer two questions in my reviews below: first, the quality of exposition and shortcomings, second the exercises. (full disclosure: This is my first time reviewing a maths book so apologies for any mediocrity and I also did not purchase them or borrow them from the library - I'll leave you to make your deductions).

1 LADR by Axler (recently brought out a 3rd edition but I haven't read this).

i) The pedagogical approach and concept of this book sounds good but in practice it was handled terribly. First, he treats polynomials as functions, rather than formally, and though this can be altered if you know what you're doing, it is hardly appropriate for a beginner. Secondly, he delays matrices far too long - I think after chapter 3, when linear maps were done, the concept of a matrix and determinant could have been beautifully integrated and streamlined into the exposition - needless to say this opportunity was wasted. Everything falls apart, material wise, after chapter 6, though I still got some benefit out of it.

ii) The problems are exceptionally easy. I mean this in reference to knowing the material of the book - as long as you can remember the material in the chapters, you will have no trouble doing them - I didn't spend more than a few seconds on each exercise before seeing what to do (except when coming up with counterexamples).

2 Hoffman & Kunze:

i) Matrices are introduces straight away and it covers more material than Axler. I like the practicality of the book, though its terseness is not for everyone. Addresses many shortfalls of Axler.

ii) Good problem set, some challenging problems, many good 'routine' questions to drill concepts.

3 Halmos: Linear Algebra Problem Book (with reference to FDVS) i) You actually answer questions to make your own exposition, though Halmos is as perfect as usual. Needs a secondary text for a first time student (FDVS would be good) and the material follows his classic examples format.

ii) Excellent, perfect set of questions. He goes over many examples, you are asked to prove many theorems and search for the kind of things beginning students trip up over. For instance, in the first chapter - an introduction to groups and fields- you are 'hand held' through showing how things like associativity etc. are non-trivial, which group axioms are independent, why it is important for multiplication and addition to interact 'sensibly', and then you can move all of this theory into the second chapter on vector spaces, again beautifully motivated with examples and uses the preceding material. This format goes on, and the questions are both illuminating and yet also I would say completely approachable for a student. There are even hints with solutions the following section, so you can peak at a hint, then the solution if really stuck. However, the questions are mainly easy, though about ten in the whole book did stump me.

Companion to LAPB: FDVS by Halmos.

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    $\begingroup$ I guess I should comment that the idea of treating polynomials formally is important when you consider fields other than C. For instance, he makes a big deal at the beginning to say - F can stand for R or C - ... then relies on the fact we are dealing in R or C to treat polynomials as functions and provide different definitions for treating eignevectors, depending on when working in R or C - so the use of F is really a misnomer and not generalisable to a field. A bit of a pigs ear out of things by not being sufficiently rigorous with his use of F and careful with polynomials. $\endgroup$ Jul 22, 2015 at 0:20
  • $\begingroup$ I'll say that Hoffman is the best. $\endgroup$
    – Hasan Saad
    Jul 22, 2015 at 9:35
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I think Halmos's book "Finite Dimensional Vector Spaces" fits this description.

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The standard is Linear Algebra by Serge Lang.

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Another book that fits your requirements is Lax's "Linear Algebra and its Applications"...and it has answers in the back of the book (quite rare for graduate-level texts).

It focuses on Linear Algebra as the study of linear transformations as opposed to matrix operations.

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  • $\begingroup$ This is an awesome book, and the proof (using annihilators) that $ A $ and $ A^T $ have the same rank is a great example of how an abstract approach can be the most enlightening. $\endgroup$
    – littleO
    Jul 22, 2015 at 4:00
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For beginners, most colleges use: David Lay's Linear Algebra & Applications. This book really teaches students how to do proof on matrices.

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    $\begingroup$ I don't know such a book. I know about Lay's "Linear algebra and its applications", and it is weak on proofs (although probably better than many other of those black-and-blue texts tailored to students of the sciences). $\endgroup$ Jul 21, 2015 at 23:31
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    $\begingroup$ Yes, I just had a look. I was actually looking for something proof-based. $\endgroup$
    – ZenoCosini
    Jul 21, 2015 at 23:32
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    $\begingroup$ @ThomasAndrews: That is a secret info....but it uses extensively the row echelon forms of a matrix to prove facts on rank inequality, and equalities of matrices. $\endgroup$
    – DeepSea
    Jul 21, 2015 at 23:39
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    $\begingroup$ @ThomasAndrews I for one can say that both universities I've attended use the text, if that means anything. $\endgroup$ Jul 21, 2015 at 23:41
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    $\begingroup$ @Omnomnomnom you are enough of a mathematician to know that a sample set of $2$ is not much of a sample. $\endgroup$ Jul 21, 2015 at 23:44

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