Linear Algebra Texts? - MathOverflow

Linear Algebra Texts?

2010-03-03T19:14:23Z

Can anyone suggest a relatively gentle linear algebra text that integrates vector spaces and matrix algebra right from the start? I've found in the past that students react in very negative ways to the introduction of abstract vector spaces mid-way through a course. Sometimes it feels as though I've walked into class and said "Forget math. Let's learn ancient Greek instead." Sometimes the students realize that Greek is interesting too, but it can take a lot of convincing! Hence I would really like to let students know, right from the start, what they're getting themselves into.

Does anyone know of a text that might help me do this in a not-too-advanced manner? One possibility, I guess, is Linear Algebra Done Right by Axler, but are there others? Axler's book might be too advanced.

Or would anyone caution me against trying this, based on past experience?

Answer by Harry Gindi for Linear Algebra Texts?

2010-03-03T19:21:00Z

Hands-down, my favorite text is Hoffman and Kunze's Linear Algebra. Chapter 1 is a review of matrices. From then on, everything is integrated. The abstract definition of a vector space is introduced in chapter 2 with a review of field theory. Chapter 3 is all about abstract linear transformations as well as the representation of such transformations as matrices. I'm not going to recount all of the chapters for you, but it seems to be exactly what you want. It's also very flexible for teaching a course. It includes sections on modules and derives the determinant both classically and using the exterior algebra. Normed spaces and inner product spaces are introduced in the second half of the book, and do not depend on some of the more "algebraic" sections (like those mentioned above on modules, tensors, and the exterior algebra).

From what I've been told, H&K has been the standard linear algebra text for the past 30 or so years, although universities have been phasing it out in recent years in favor of more "colorful" books with more emphasis on applications.

Edit: One last thing. I have not heard great things about Axler. While the book achieves its goals of avoiding bases and matrices for almost the entire book, I have heard that students who have taken a course modeled on Axler have a very hard time computing determinants and don't gain a sufficient level of competence with explicit computations using bases, which are also important. Based on your question, it seems like Axler's approach would have exactly the same problems you currently have, but going in the "opposite direciton", as it were.

Answer by Andrea Ferretti for Linear Algebra Texts?

2010-03-03T19:42:20Z

Serge Lang's Linear Algebra does not cover much material, but is very nice for a first introduction. It does not emphasize particularly matrices and computations, so one understands immediately that matrices only come as representations of linear maps, but it's also not too abstract.

Answer by RH for Linear Algebra Texts?

2010-03-03T19:58:58Z

I rather like Linear Algebra Done Right, and depending on the type of students you are aiming the course for, I would recommend it over Hoffman and Kunze. Since you seemed worried that Axler might be too advanced, my feeling is that Hoffman and Kunze will definitely be (especially if these are students who have never been taught proof-based mathematics).

Of course, the big caveat here being that Axler avoids determinants at all costs, and this will put more on you to introduce them comprehensively.

I've never looked at it, but another one worth considering might be Halmos's Finite Dimensional Vector Spaces.

Answer by Jason Polak for Linear Algebra Texts?

2010-03-03T20:00:06Z

There's also Nicholson's Elementary Linear Algebra or the slightly more advanced Linear Algebra: With Applications. If your students react negatively to the intro of abstract vector spaces, I don't think Hoffman and Kunze's book would be good for them. While I love that book myself it might be a little too daunting for your class. Also I think that if you want to introduce abstract vector spaces from the start there's no reason you can't cover the chapter on abstract vector spaces first.

Answer by Franz Lemmermeyer for Linear Algebra Texts?

2010-03-03T20:16:41Z

There were times when I was rather fond of Strang's Linear Algebra and Its Applications. I haven't looked at it for a long time, but back then I found it very clear and appealing. Even if you don't follow the book chapter by chapter, it might still give you ideas.

Answer by Álvaro Lozano-Robledo for Linear Algebra Texts?

2010-03-04T20:31:34Z

If you are looking for a gentle introduction, that uses matrices from the beginning, I would suggest you consider "Linear Algebra" by Friedberg, Insel and Spence. I haven't used this book myself, but somebody (I trust) recommended this book to me. I now own it, and it looks very nice and gentle (but covering all the topics I would like to include), and matrices are introduced in page 8.

Alvaro

Answer by Andrew L for Linear Algebra Texts?

2010-03-10T18:14:51Z

My old mentor Nick Metas was part of the teams of graduate students who worked over the drafts of H&K when they were writing it for the linear algebra course at MIT in the 1960's.that being said,despite its' rigor and beauty,I think a "pure" linear algebra course is just as big a mistake as a pure theoretical calculus course no matter how good the students are. It's like teaching music students all about pentamer,note grammer and acuostics and never teaching them how to play a single note.I don't go for this whole pure/applied distinction,it's an idiotic consequence of this age of specialization.I love rigor,but applications should never be denied or ignored. That's why my overall favorite LA text is Friedberg,Insel and Spence-it's the only one I've seen that aims for and hits a terrific balance between algebraic theory and applications. I also love Curtis for similar reasons,but it's coverage isn't as broad. I love books that aim for that Grand Mean Balance-sadly,in America,there aren't anywhere near enough such texts.

Answer by Jim Humphreys for Linear Algebra Texts?

2010-03-10T18:59:20Z

There is no ideal text for a beginning one semester course as taught in the US to first or second year college students. Older books like H&K treat only the abstract theory, in a fairly conceptual way and (if I recall correctly) with maps written on the right contrary to what students do in calculus. A later generation of books like the original Anton are also pure math books but start by overemphasizing unrealistic manipulations with small matrices and vectors; then there is an abrupt shift to abstraction. Determinants are presented in a purely computational mode, as though they were really used for this purpose; then eigenvalues occur very late and again in oversimplified small examples. Fortunately the newer texts tend to mix pure and applied throughout, but as a result they contain far too much material for a first course. And eigenvalue theory still gets introduced very late. Strang is attractive in many ways, but too loosely written down and not suitable for an inexperienced reader without a reliable guide at hand. Aside from Strang, the emphasis in most US textbooks remains placed on unrealistic integer calculations with very small matrices rather than on the geometry of subspaces, etc. The pervasive role of geometric thinking in the subject is mostly downplayed in texts, as is the role of analysis. For self-study, something like Friedberg-Insel-Spence may be the best compromise choice.

Answer by Vladimir Dotsenko for Linear Algebra Texts?

2010-04-05T02:12:48Z

My personal pick is I.M.Gelfand's "Lectures on linear algebra" (link to a copy on Google Books), accompanied by two warnings: (1) the part "Introduction to tensors" is far from perfect; (2) the proof of the Jordan normal form theorem is dramatically outdated (keep in mind that the only English translation of the book is that of the 1950s edition - the latest editions contain a proof that totally makes sense). Then again, many linear algebra textbooks simply avoid Jordan normal forms completely (which I think is a mild disaster).

Answer by hypercube for Linear Algebra Texts?

2010-05-22T19:01:35Z

Newer Books

Matrix Analysis and Applied Linear Algebra by Meyer is very well written with clear cut examples and exercises. I think this would make an excellent first course.

I agree also that Axler's books is a great text for the more mature.

Classics

Finite-Dimensional Vector Spaces by P. R. Halmos is an absolute essential for the budding mathematician in my opinion. This is because of the exercises (My recommendation: solve all of them).

As mentioned above Linear Algebra (2nd Edition) by Kenneth M Hoffman and Ray Kunze. This may be my favorite text because of its volume of content.

More Advanced

Advanced Linear Algebra by Steven Roman

Matrix Analysis

Matrix Analysis and Topics in Matrix Analysis by Roger A. Horn and Charles R. Johnson

Matrix Analysis by Rajendra Bhatia

Answer by Michael Hoffman for Linear Algebra Texts?

2010-05-23T00:23:52Z

Why does no one go over applied linear algebra, or more, why is there no book that actually talks seriously about the computational end and about the theory. By computational end I mean the REAL computational end, that which is actually done on a computer or at least is the background to understand those algorithms. If there were a nice undergraduate version of Demmel then I'd defer to that book, but so far as I know such a book doesn't exist. If you're going to split linear algebra at all it would seem to be

Theoretical Linear Algebra

and

Computational Linear Algebra

Answer by Victor Protsak for Linear Algebra Texts?

2010-05-23T01:46:21Z

For teaching the type of course that Dan described, I'd like to recommend David Lay's "Linear algebra". It is very thoroughly thought out and well written, with uniform difficulty level, some applications, and several possible routes/courses that he explains in the instructor's edition. Vector spaces are introduced in Chapter 4, following the chapters on linear systems, matrices, and determinants. Due to built-in redundancy, you can get there earlier, but I don't see any advantage to that. The chapter on matrices has a couple of sections that "preview" abstract linear algebra by studying the subspaces of $\mathbb{R}^n$.

Answer by Anirbit for Linear Algebra Texts?

2010-06-19T10:28:58Z

The best thing about Hoffman and Kunze's book is its beautiful exposition of Jordan Forms. If a course is planning to get to Jordan Forms as a target then I can't think of any better approach than that in Hoffman and Kunze.

Sections on linear algebra in Artin and Herstein's book's are also very good but then Hoffman and Kunze win hands down if the objective is Jordan Form.

Explanation of concepts like conductors and annihilators, invariant polynomials and variations/equivalence between notions of semi-simplicity and myriad of different ways to test diagonalizability of a linear transformation are I would say the claim to fame for Hoffman and Kunze's book. And all this merges beautifully in their writing of Jordan forms, as if everything else was written just to make this concept clear.

Very importantly this books gives instructive numerical examples after every bunch of concepts.

Answer by José Figueroa-O'Farrill for Linear Algebra Texts?

2010-06-20T00:44:38Z

Although I have not lectured from it, I like very much Klaus Jänich's Linear Algebra book.

Answer by Andrei Halanay for Linear Algebra Texts?

2010-07-08T13:43:16Z

A very good textbook is Shilov's. It is actually the first (or perhaps Volume 0) of his textbook in Mathematical Analysis. It covers more than the standard material, but is very clear written with many examples and exercises (many solved).