Strang Gilbert. Introduction To Linear Algebra

[Curtis Cassel]

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.

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 $\mathbbR^n$.

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).

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 direction", as it were.

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.

There is no ideal text for a beginning one semester course as taughtin 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 recallcorrectly) with maps written on the right contrary to what students do incalculus. A later generation of books like the original Anton are alsopure math books but start by overemphasizing unrealistic manipulations withsmall matrices and vectors; then there is an abrupt shift to abstraction.Determinants are presented in a purely computational mode, as though theywere really used for this purpose; then eigenvalues occur very late and againin oversimplified small examples. Fortunately the newer texts tend to mixpure and applied throughout, but as a result they contain far too much materialfor 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. Asidefrom Strang, the emphasis in most US textbooks remains placed on unrealisticinteger calculations with very small matrices rather than on the geometry ofsubspaces, etc. The pervasive role of geometric thinking in the subject ismostly downplayed in texts, as is the role of analysis. For self-study,something like Friedberg-Insel-Spence may be the best compromise choice.

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 grammar and acoustics 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.

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.

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).

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.

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.

I apologize for plugging my own text, but I think that "Introduction to Linear, Ideas and Applications" by Richard Penney might be exactly what the questioner is looking for. It is relatively gentle and it does integrate vector spaces and matrix algebra from the get go. When I have taught from it the question of "what is a vector space" has never been an issue.

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.

Matrices are a useful computational tool in the case of finitely generated vector spaces, but they are not natural. That is why there are different rules for transforming matrices, depending on whether they represent linear transformations or bilinear forms. Matrices also hide the fact that calculus is nothing more or less than linear algebra, but with infinitely generated vector spaces.

If I were to recommend a single book for a first course, it would be K. Jnich's Linear Algebra, which is a first semester first year text for German students of mathematics and/or physics. It is thoroughly modern and readable.

I have taught Linear Algebra a few times, at both basic and advanced levels, and the introductory text which served me best for precisely the goal the OP is aiming at is, surprisingly, the first five chapters of the second volume of the late Tom M. Apostol's classic Calculus. It is a concise, no-nonsense and down-to-earth first course in linear algebra which starts from abstract vector spaces right at the first chapter (with lots of examples, of course) and moves to matrices in the second chapter right after introducing linear transformations. I find Apostol's approach quite refreshing because it greatly illuminates the matrix operations involved in solving linear systems (Gauss-Jordan elimination, etc.).

Notice, though, that this is really a first encounter with linear algebra, so only real vector spaces are discussed and a tad more advanced topics like the Jordan canonical form are not treated. For the latter, I agree with The Mathemagician's answer that a purely algebraic approach might not be advisable to a broader audience. For instance, I particularly enjoy Filippov's proof of the Jordan canonical form using matrix exponentials as fundamental solutions to linear autonomous ODE systems, which is the one used in G. Strang's Linear Algebra and its Applications. Such an argument would fit perfectly in Chapter 7 (on linear ODE systems) of Apostol's volume, where matrix exponentials are discussed in depth (including Putzer's algorithm, presented as an application of the Cayley-Hamilton theorem), so in retrospect I feel somehow it was a missed opportunity.

c80f0f1006