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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?

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    $\begingroup$ Dan, it might be helpful to know what the audience for your class is. Are the students math majors or not? Have they had proof-based math already or not? In particular, some textbooks are written with the assumption that students are working with proofs for the first time and try to ease the transition; some assume students are already completely comfortable with proofs; and some don't care about proofs at all and just aim to show how to do calculations, like a typical calculus book. $\endgroup$
    – Mark Meckes
    Commented Mar 4, 2010 at 14:47
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    $\begingroup$ Hi Mark, I think there will be a range of students, mostly non-math majors, and all of them writing proofs for the first time. I feel convinced by now that Axler would not be the right choice. $\endgroup$
    – Dan Ramras
    Commented Mar 4, 2010 at 18:43
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    $\begingroup$ 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 $\endgroup$
    – Michael Hoffman
    Commented May 23, 2010 at 0:23
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    $\begingroup$ While I've had precisely that experience on several occasions, do you $\textit{really}$ want to be hated for the whole semester, as opposed to only the second half? The reason why many (most?) recent books start with matrices and linear systems is that at least this way students will learn something in the first half, rather than giving up early and closing their minds under the onslaught of abstraction. $\endgroup$
    – Victor Protsak
    Commented May 23, 2010 at 1:33
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    $\begingroup$ Mainly because computational linear algebra by hand is frustrating and pointless. $\endgroup$
    – Harry Gindi
    Commented Jun 19, 2010 at 11:14

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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 $\mathbb{R}^n$.

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    $\begingroup$ In the end, this is the book I decided to go with. $\endgroup$
    – Dan Ramras
    Commented May 24, 2010 at 19:18
  • $\begingroup$ Mark it as the right answer, then:) This book isn't perfect, but I liked it a lot and I hope that so will you and your students. $\endgroup$
    – Victor Protsak
    Commented May 24, 2010 at 22:22
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    $\begingroup$ Now that I'm a month into the course, I think I can heartily say that I'm happy with the book. No, it isn't perfect, but quite often the complaints I have are addressed in the author's preface for instructors (in the instructor version) and several times I've become convinced that Lay has a good point, and there's a good reason for doing things the way he does. It's very tempting to lay on tons of concepts early on in a linear algebra course. Lay's book is good about introducing concepts slowly, and then reinforcing them later with new viewpoints. $\endgroup$
    – Dan Ramras
    Commented Sep 19, 2010 at 23:49
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    $\begingroup$ Lay has some serious flaws. He calls the dot product of two vectors in $\mathbb{R}^n$ "the inner product", as though this were the only inner product. I have gathered that this usage is common in the applied math world, but it is inappropriate in an introductory linear algebra book because you might want the students to learn the correct meaning of "inner product". The Cauchy-Schwarz Inequality is proven using projections, which is absurd, because all you need is some algebra and basic properties of inner product. The proof (using projections) is also more difficult than the usual proof. $\endgroup$
    – Stefan
    Commented Nov 2, 2013 at 0:54
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    $\begingroup$ I thought the presentation of abstract material such as subspaces and inner product spaces was weak and relied excessively on matrix algebra. I know most students who aren't math majors hate this stuff and I don't know if there is any book that will make them like it.I admit I can't recommend another book. I used Strang's book once and, to put it positively, I'll say I much preferred Lay's (the only relative advantage is that Strang does cover the matrix exponential). $\endgroup$
    – Stefan
    Commented Nov 2, 2013 at 1:04
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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.

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    $\begingroup$ I'm quite fond of Halmos's FDVS. It certainly takes the abstract vector spaces approach with seriousness and gusto. I was planning on writing an answer of my own pointing towards Halmos, but endorsing this might do the trick just as well. $\endgroup$
    – Mikael Vejdemo-Johansson
    Commented Mar 3, 2010 at 20:08
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    $\begingroup$ I forgot about Halmos. You should add it as a separate answer, since I'd vote it up, but as you can see, I really think Axler's book goes about things the wrong way. $\endgroup$
    – Harry Gindi
    Commented Mar 3, 2010 at 20:48
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    $\begingroup$ I have taught out of "Linear Algebra Done Right" and I like it. The main drawback I saw was that I had to introduce more computational problems (for example, so that the students could explicitly compute changes-of-basis and such). The book stays with real and complex spaces, so it's not an upper level text. But the proofs are very nice. I'm planning to use it again the next time I teach a linear algebra course at that level. $\endgroup$
    – Carl Mummert
    Commented Jun 19, 2010 at 11:49
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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.

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    $\begingroup$ The problem with Axler is that he tries to avoid algebra at all cost - when it makes things easier and when it makes thing harder. Sometimes much harder, actually. Even if one goes into physics and never has to work over a ring different from R and C, one will realize one day that in order to compute the characteristic polynomial one doesn't have to bring the matrix in upper triangular form (no joke, this is how Axler defines the characteristic polynomial), and that often, the characteristic polynomial matters and upper diagonalization doesn't. $\endgroup$
    – darij grinberg
    Commented Mar 10, 2010 at 18:50
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    $\begingroup$ For someone who plans to work in algebra or algebraic geometry, linear algebra learnt from Axler is mostly wasted time. I don't understand what he has against the notion of determinant; this notion (with the sum-over-permutations formula that he seems to hate) comes out straightforwardly if one tries to apply Gaussian elimination to a general systems where the coefficients of the system are variables. $\endgroup$
    – darij grinberg
    Commented Mar 10, 2010 at 18:53
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    $\begingroup$ Maybe not everyone is going into algebraic geometry? I think that a lot of the proofs in Axler suggest the right way to think about things in functional analysis. Maybe it was just that my first linear algebra course was totally computational and had very little motivation, and my second exposure to linear algebra out of Axler showed me that there were actually some beautiful ideas there. $\endgroup$
    – Steven Gubkin
    Commented Mar 10, 2010 at 19:21
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    $\begingroup$ From the description given (starts with a $\textit{review}$ of matrices and fields, uses modules and exterior algebra, etc), it is patently obvious that this book (HK) is unsuitable to people without abstract algebra under their belt. Quite a bit of mathematical maturity that you cannot reasonably expect from non-math majors with or without prior proof experience is required as well. $\endgroup$
    – Victor Protsak
    Commented May 23, 2010 at 1:31
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    $\begingroup$ @Harry: Bear in mind that what works for $\textit{you}$ (and other math majors at UM), doesn't necessarily work for others. And you have just confirmed that you were comfortable with abstract algebra, at the level higher than most non-majors ever see, before starting out. Learning composition of morphisms without being able to multiply matrices is $\textit{truly}$ pointless. One imperfect analogy: it's possible to learn AG from EGA or Harstshorne ("it has been done"), but as first books they are nowhere near Shafarevich, Cox-Little-O'Shea, Reid, Mumford, and any number of other texts. $\endgroup$
    – Victor Protsak
    Commented May 23, 2010 at 6:49
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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.

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    $\begingroup$ Definitely the book I would recommend for non-math majors. It has plenty of examples to motivate topics, which is what non-mathematicians need in order to be interested in linear algebra. Vector space axioms are the absolute worst way to teach linear algebra to any group of people that is not wholly composed of math majors. $\endgroup$
    – Rune
    Commented Jun 20, 2010 at 2:13
  • $\begingroup$ If anyone is still looking for such a book, this book was rewritten as Gilbert Strang, Introduction to Linear Algebra, now in its fourth edition (as of 2016). $\endgroup$
    – Ben McKay
    Commented May 31, 2016 at 19:37
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    $\begingroup$ How suitable is Strang's book for mathematicians? I am not convinced. I considered it for teaching but the entire lack of formal definitions of vector spaces is really at odds with everything I believe in. $\endgroup$
    – shuhalo
    Commented Jul 31, 2017 at 21:26
  • $\begingroup$ With a background in mathematical physics, signal processing for advanced imaging sensors, and combinatorics, I certainly would have benefitted greatly from a course combining a good book on linear algebra and Strang's Intoduction to Applied Mathematics. $\endgroup$
    – Tom Copeland
    Commented Apr 3, 2018 at 13:10
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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.

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    $\begingroup$ H&K composes in the standard way. $ST(\vec{v})=S(T(\vec{v}))$. They do not cover applications, but I think that "real-world" applications have no place in a math book. $\endgroup$
    – Harry Gindi
    Commented Mar 10, 2010 at 19:11
  • $\begingroup$ @Jim I think Friedberg-Insel-Spence is the best LA book out there for a general class of math majors right now-it's the most balanced between rigor and applications and it probably covers the widest range of topics at this level.Still,I agree that I don't think The Great American LA Text has been written yet. $\endgroup$
    – The Mathemagician
    Commented Jun 19, 2010 at 16:23
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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.

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    $\begingroup$ "Mathematical education is still suffering from the enthusiams which the discovery of this isomorphism has aroused. The result has been that geometry was eliminated and replaced by computations. Instead of the intuitive maps of a space preserving addition and multiplication by scalars (these maps have an immediate geometric meaning), matrices have been introduced. From the innumerable absurdities -from a pedagogical point of view - let me point out one example and contrast it with the direct description: $\endgroup$
    – Harry Gindi
    Commented Jun 19, 2010 at 10:25
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    $\begingroup$ Matrix method: A product of a matrix $A$ and a vector $X$ (which is then an n-tuple of numbers) is defined; it is also a vector. Now the poor student has to swallow the following definition: A vector X is called an eigenvector if a number $\lambda$ exists such that $$AX=\lambda X.$$ Going through the formalism, the characteristic equation, one then ends up with theorems like: If a matrix A has n distinct eigenvalues, then a matrix $D$ can be found such that $DAD^{-1}$ is a diagonal matrix. The student will of course learn all this since he will fail the course if he does not. $\endgroup$
    – Harry Gindi
    Commented Jun 19, 2010 at 10:31
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    $\begingroup$ Instead one should argue like this: Given a linear transformation f of the space $V$ into itself. Does there exist a line which is kept fixed by $f$? In order to include the eigenvalue $0$ one should then modify the question by asking whether a line is mapped into itself. This means of course for a vector spanning the line that $$f(X)=\lambda X.$$ Having thus motivated the problem, the matrix A describing f will enter only for a moment for the actual computation of X. It should disappear again. $\endgroup$
    – Harry Gindi
    Commented Jun 19, 2010 at 10:36
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    $\begingroup$ Artin's view here is very much the view of Dieudonné (as expressed in his book on linear algebra). I think that Arnold simplifies the world into black and white and attacks a straw boogeyman named Bourbaki. Hoffman and Kunze gives a very nice account of these geometric aspects as well as the algebraic ones. $\endgroup$
    – Harry Gindi
    Commented Jun 19, 2010 at 10:43
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    $\begingroup$ Harry, don't get all worked up. Let me repeat the key point: $\textit{It's a harmful fallacy that conceptual understanding and applications are mutually exclusive.}$ There isn't any application in sight in your quotes, just comparisons between different formalisms. And for what it's worth, the last one is perfectly in line with Arnold's philosophy, while it doesn't conform well to Bourbaki's way of thinking. $\endgroup$
    – Victor Protsak
    Commented Jun 20, 2010 at 1:06
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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

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    $\begingroup$ Nooooo! I used this book when I taught a 2nd linear algebra course and that book is dry as dust. It has all kinds of neat applications, but it is really boring to read. (I really mean boring, not "too elementary"). Also, despite the abstraction over general fields in the main text, there is little compelling rationale given for needing linear algebra over something besides R or C. In particular, F_2 is used in the book only for weird counterexamples, even though linear algebra over F_2 is really useful in computer science. $\endgroup$
    – KConrad
    Commented Mar 10, 2010 at 19:33
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    $\begingroup$ I think this book is perfect. Sure it's dry, but that's fine. It has worked out examples exactly where they should be, the presentation and proofs are crystal clear, and there are tons of good exercises. I'm stuck teaching calculus from a book, which I'll not name now, that tries to "sell it" by attempting to be more readable, using poorly construed applications for motivation, and filling up empty space with colour pictures. I don't find this helps to convert anyone who isn't already interested. "Selling it" is my job as a teacher. When it comes to texts, I look for simplicity and clarity. $\endgroup$
    – Brendan Cordy
    Commented Apr 5, 2010 at 3:02
  • $\begingroup$ It's dry to be sure, but it works. The only complaint I have is that there are a lot of silly computational problems. BUT there's nothing that says that you have to give those problems as an instructor, and there are some good problems in there $\endgroup$
    – Michael Hoffman
    Commented May 23, 2010 at 0:20
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    $\begingroup$ @KConrad WHat the heck do you mean,dry,KC?!? It's loaded with beautiful examples and applications,some of which are rarely presented in a first course,like stochastic matrices! It's RIGOROUS without being Bourbakian,that's what I love about it. The section on the Jordan canonical form is a mess,though.Use Curtis for that and it'll be fine. $\endgroup$
    – The Mathemagician
    Commented May 23, 2010 at 3:45
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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).

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    $\begingroup$ There is a note after the preface of the English translation saying that Gelfand asked for the appendices not to be translated. I looked at them online (they're on perturbation theory) and couldn't figure out why the appendices would not be something to translate. Do you know if there was some awkward homage to Stalin in the 1950 edition? The appendices I looked at are from a more recent edition. $\endgroup$
    – KConrad
    Commented May 28, 2013 at 12:36
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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.

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    $\begingroup$ Look at the exposition of Jordan canonical forms in both Charles Curtis' LINEAR ALGEBRA:AN INTRODUCTORY APPROACH and in Anthony Knapp's BASIC ALGEBRA,Dan. I think you'll find both superior to H&K. H & K is just too abstract to be helpful long-term for most math majors-although,to be honest,the possibility of a course based on the union of H&K and Gilbert Strang's book has always intrigued me. $\endgroup$
    – The Mathemagician
    Commented Jun 19, 2010 at 22:36
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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.

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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

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

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  • $\begingroup$ lol there is a page with Gauss' face, It looks good. $\endgroup$
    – Isana Yashiro
    Commented Sep 16, 2018 at 12:24
  • $\begingroup$ This book seems excellent. I will soon teach linear algebra for the first time, following Axler, and would adopt it if it had open access. But I will definitely recommend it to students and use some of its stuff in the lectures. $\endgroup$
    – Daniel Teixeira
    Commented Dec 9 at 20:46
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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.

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    $\begingroup$ For a second there, I thought you said "Serge Lang's Algebra". I'm sure you can appreciate the humor in that. $\endgroup$
    – Harry Gindi
    Commented Mar 3, 2010 at 19:46
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    $\begingroup$ Wait, so you mean there IS a readable book by Lang? $\endgroup$
    – darij grinberg
    Commented Mar 3, 2010 at 19:56
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    $\begingroup$ You're too hard on Lang, darij. Algebra is good, the differential geometry book is good, the book on cyclotomic fields is pretty good. Sure, there are unreadable sections and undefined notation, but the books are generally readable, and some of them are even pretty good! $\endgroup$
    – Harry Gindi
    Commented Mar 3, 2010 at 20:55
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    $\begingroup$ I second Lang's Linear Algebra. I also found it very accessible, and it also seems to be a good preparation for the corresponding chapters of his "Algebra" (Chs. XIII - XV of the Springer edition). $\endgroup$
    – user2734
    Commented Mar 3, 2010 at 22:30
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    $\begingroup$ Most of the trouble I've had with Lang is the sheer number of mistakes in the book. There's nothing worse than a line that says "of course..." followed by a typo. It leaves you feeling like an idiot when you've not done anything wrong. $\endgroup$
    – Harry Gindi
    Commented Mar 10, 2010 at 19:06
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It is a mistake to use matrices immediately in linear algebra for several reasons, even if they are (incorrectly!) regarded as central to linear algebra.

The general concepts are actually easier to understand without discussion of matrices, and most of the important results are simpler and easier to prove without reference to matrices.

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. Jänich'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.

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  • $\begingroup$ I think this depends greatly on the audience of the course. In the US, I've never taught a beginning linear algebra course that has a significant percentage of math majors. The courses are largely populated by engineering students who absolutely need to understand how to do linear algebraic computations, and therefore need to understand matrix algebra. I'll also say that I don't think it's ever a mistake to introduce computational methods alongside theory. $\endgroup$
    – Dan Ramras
    Commented Oct 2, 2016 at 14:13
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    $\begingroup$ Of course computational techniques are essential. But these need to be explained, especially to non-mathematicians. I know of no book on linear algebra focussing primarlly on matrix calculations that does that. The sorts of questions that arise naturally, such as why matrices must both be mxn in order to be able to add them, yet any mxn matrix can be multiplied by any nxp matrix (in the sutable order), remain unanswered, beyond, perhaps the usual cop-out "it proves to be useful". $\endgroup$
    – user99154
    Commented Oct 3, 2016 at 21:43
  • $\begingroup$ I agree that far too often the computations are not explained. Conceptual explanations based on linear transformations can make a huge difference. $\endgroup$
    – Dan Ramras
    Commented Oct 4, 2016 at 3:07
  • $\begingroup$ In particular, this makes it clear that calculus in R^{n} is really linear algebra applied to infinite dimensional vectors spaces of the form F(X,R), where F(X,R) is the vector space of certain real-valued functions on the subset X of R^{b}, where F stands for continuous, or integrable, or differentiable, or ... The difficulty for computation is that, being infinitely generated, matrices are not available. But exploring the general linear algebra of both simultaneously, shows the intimate link between smooth and discrete systems. $\endgroup$
    – user99154
    Commented Oct 5, 2016 at 11:25
  • $\begingroup$ Is this perspective on calculus discussed in Jänichs' book? I think I looked at his book at some point, but maybe not very carefully. $\endgroup$
    – Dan Ramras
    Commented Oct 6, 2016 at 14:58
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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.

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    $\begingroup$ Apostol is my favorite reader! He writes everything in the most precise style with proper motivations, historical remarks and examples. :) $\endgroup$
    – Hosein Rahnama
    Commented Sep 8, 2016 at 14:42
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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.

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  • $\begingroup$ Haven't seen Nicholson's texts. I'll take a look. Thanks! $\endgroup$
    – Dan Ramras
    Commented Mar 4, 2010 at 3:35
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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).

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For all who can read German, Egbert Brieskorn's two volumes on Linear Algebra and Analytical Geometry are just awesome - in my opinion. A text written with great care and deepest insight. An extraordinary teacher and gifted lecturer.

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Here is a list of books that are good for linear algebra. Specifically the first link (Hoffman and Kunze) is kind of the gold standard. For students to gain an understanding/appreciation of linear algebra I prefer working backwards, start with posing a real problem, like Google's pagerank problem described here. This really gets students excited about why they need to learn abstract vector spaces and other stuff before they can do some real world applications with it.

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Several answers have been posted here, but here, my main aim is not to post just answer, but to catch attention of learners of Linear algebra, of graduates, who want more to know on linear algebra.

A modern Linear Algebra which I like much is the book by Charles Curtis. To mention few features of this book, not with style of writing, but with content, are following:

(0) Many basic concepts of Linear algebra are motivated with simple examples in algebra as well as school geometry; for, one can have overlook in exercises of all chapters.

(1) In my undergraduate, I was searching for different books to understand Jordan Canonical Forms, but I found no books in my Library, except little exposition in H&K. But long time after completion of graduation, I came across this book, and found its beautiful exposition on Jordan theory.

(2) If you search for Adjoint transformation in google books, mostly you will see that it is introduced in chapter with title Inner product spaces. But, this concept of adjoint transformation do not requires space to be inner product space, and this is the only book I saw explained it in this general setting, so as soon as we have a linear transformation between "vector spaces", we can quickly go to "what is adjoint of it", without considering whether what inner product is there.

(3) When I came across looking for Jordan decomposition of linear operators (=semisimple+nilpotent), then, much of the tools to prove it are hidden in primary decomposition theorem or Jordan canonical forms; this is the only book I saw which beautifully explains this decomposition. I didn't get this theorem even in books of Algebra or Linear Algebra by famous algebraists.

(4) The book first geometrically explains concept of determinant, which I rarely find in other books.

(5) Finally, when reading this book of Curtis, I found his language much beautiful, elegant, and not creating fear of any simple or difficult concept, which shows that the subject could be easily learned by anyone just with this book.

One can even find a different elegant exposition to other important concept of linear algebra in this book (principal axis theorem, symmetry); but I couldn't not mention it fully, instead leave the reader to see at least once the book.

A point to mention here: I was searching reviews before writing these points of the book, but I didn't get its review in MAA; so I tried to write my experience with this book, which kept me enjoying the subject any time.

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I've looked through a few linear algebra books, I studied most of the Horner book and found it good but it got a little hairy - Hermitian forms, etc, although necessary for my subject (engineering) its not enjoyable. I like Gilbert Strang's course, he teaches the basics well. I've got his book and I find some of the problems very difficult. My plan is to study that book and do the easier problems to get a handle on things then get into pure math with an intro to proofs as I have never been good with proofs. I am good with difficult problems, but I never got the knack of proving things well in math. Fortunately you don't have to do much of this in engineering. I got my degree twenty years ago and now see that engineering is done with linear algebra rather than calculus. Fortunately I know my DSP quite well.I'm curious about a good pure math book myself, to start doing proofs - calc or algebra - just a good book for self study.

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I am an electrical major. Let me try to answer this question from the perspective of a non math major student without a prior course on analysis. My first linear algebra course followed Hoffman and Kunz but at the time, the abstraction was too overwhelming and I completely failed to appreciate the beauty of the subject and was even appalled by it. When I had planned to take up another course that required linear algebra(machine learning), I self studied it from gilbert strang's "linear algebra and its applications" , I found the intuition that I gained by going through it very useful and started liking the subject and now I am working out through Hoffman and Kunz again because I wish to learn more about it.I am currently able to get a handle over the abstraction and appreciate/understand the theorems and proofs.

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  • $\begingroup$ I cannot find the word "axiom" in Strang's book; does it actually define vector spaces? $\endgroup$
    – darij grinberg
    Commented Aug 18, 2017 at 14:35
  • $\begingroup$ Strang definitely has its place as text. I've taught from it, and like some (not all) aspects. Unfortunately, I lent my copy to a student and never got it back, so it's been some years since I last looked at the book. $\endgroup$
    – Dan Ramras
    Commented Aug 18, 2017 at 17:40
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There’s a recent book by Meckes and Meckes (published by Cambridge) that might work. Vector spaces appear early, as do linear transformations and eigenvalues/vectors. Determinants are delayed but can be moved earlier if desired. I am using it for the first time and students seem to find it difficult but not impossible.

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A more modern one is godement's algebra!

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Dieudonne and Shafarevich! Linear algebra and geometry

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  • $\begingroup$ What book, specifically? Have these two collaborated on writing one? Is it called "Linear Algebra and Geometry"? $\endgroup$
    – Alex M.
    Commented Apr 18, 2019 at 13:55
  • $\begingroup$ Maybe Peter is referring to a book by Shafarevich and Remizov? amazon.com/Linear-Algebra-Geometry-Igor-Shafarevich/dp/… $\endgroup$
    – Dan Ramras
    Commented Apr 18, 2019 at 18:59
  • $\begingroup$ two books,same title $\endgroup$
    – Peter Polo
    Commented May 6, 2019 at 3:46

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