Linear Algebra Texts? 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? 
 A: 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the part "Introduction to tensors" is far from perfect; (2) the proof of the Jordan normal form theorem is dramatically outdated (keep in mind that the only English translation of the book is that of the 1950s edition - the latest editions contain a proof that totally makes sense).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).
A: 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. 
A: 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. 
A: 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
A: Although I have not lectured from it, I like very much Klaus Jänich's Linear Algebra book.
A: 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.
A: 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.
A: 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.
A: 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.
A: 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).
A: 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. 
A: 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.
A: 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.
A: 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 $\mathbb{R}^n$.
A: 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. 
A: 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.
A: 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.
A: 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. 
A: 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.
A: 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
A: 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.
A: 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.
A: 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.
A: A more modern one is godement's algebra!
A: Dieudonne and Shafarevich! Linear algebra and geometry
