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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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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. –  Mark Meckes Mar 4 '10 at 14:47
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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. –  Dan Ramras Mar 4 '10 at 18:43
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I'm curious what book you ended up picking. –  Harry Gindi Mar 10 '10 at 19:07
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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 –  Michael Hoffman May 23 '10 at 0:23
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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. –  Victor Protsak May 23 '10 at 1:33

16 Answers 16

up vote 7 down vote accepted

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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In the end, this is the book I decided to go with. –  Dan Ramras May 24 '10 at 19:18
    
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. –  Victor Protsak May 24 '10 at 22:22
    
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. –  Dan Ramras Sep 19 '10 at 23:49
    
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. –  Stefan Nov 2 '13 at 0:54
    
Presumably, since the proof requires projections, which are, I think, only defined for closed subspaces, Lay proves the Cauchy-Schwarz inequality only for subspaces of $\mathbb{R}^n$, though it is easy to prove for any inner product space. This is my opinion, but the definition of "vector space" includes unnecessary "closure" rules, which makes the intimidating list of rules even longer. The exponential of a matrix does not appear in the book at all. Consider that the book contains several semesters' worth of material, from what I have heard, this is a serious omission. –  Stefan Nov 2 '13 at 1:00

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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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. –  Mikael Vejdemo-Johansson Mar 3 '10 at 20:08
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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. –  Harry Gindi Mar 3 '10 at 20:48
    
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. –  Carl Mummert Jun 19 '10 at 11:49

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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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. –  Rune Jun 20 '10 at 2:13

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.

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I also first learned linear algebra from H&K and share fpqc's enthusiasm for it, but I don't think it counts as "relatively gentle". I think RH is right that H&K is pretty advanced in spirit if your students don't already have experience with rigorous proof-based math. Having taught from Axler, which does have many nice points, I can also confirm the problems with it that fpqc describes. –  Mark Meckes Mar 4 '10 at 14:41
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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. –  darij grinberg Mar 10 '10 at 18:50
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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. –  darij grinberg Mar 10 '10 at 18:53
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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. –  Victor Protsak May 23 '10 at 1:31
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@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. –  Victor Protsak May 23 '10 at 6:49

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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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. –  Harry Gindi Mar 10 '10 at 19:11
    
@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. –  Andrew L Jun 19 '10 at 16:23

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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For a second there, I thought you said "Serge Lang's Algebra". I'm sure you can appreciate the humor in that. –  Harry Gindi Mar 3 '10 at 19:46
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Wait, so you mean there IS a readable book by Lang? –  darij grinberg Mar 3 '10 at 19:56
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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! –  Harry Gindi Mar 3 '10 at 20:55
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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). –  user2734 Mar 3 '10 at 22:30
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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. –  Harry Gindi Mar 10 '10 at 19:06

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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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. –  KConrad Mar 10 '10 at 19:33
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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. –  Brendan Cordy Apr 5 '10 at 3:02
    
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 –  Michael Hoffman May 23 '10 at 0:20
    
@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. –  Andrew L May 23 '10 at 3:45

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.

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"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: –  Harry Gindi Jun 19 '10 at 10:25
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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. –  Harry Gindi Jun 19 '10 at 10:31
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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. –  Harry Gindi Jun 19 '10 at 10:36
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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. –  Harry Gindi Jun 19 '10 at 10:43
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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. –  Victor Protsak Jun 20 '10 at 1:06

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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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. –  KConrad May 28 '13 at 12:36

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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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. –  Andrew L Jun 19 '10 at 22:36
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I agree with Andrew that Charles Curtis Linear Algebra book is superior to Hoffmann and Kunze. I took my first linear algebra course from Hoffmann and Kunze, and while Curtis's book has a better exposition of the theory, I do believe that the exercises in H&K are way better. –  Adrián Barquero Jun 20 '10 at 3:22

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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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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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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Haven't seen Nicholson's texts. I'll take a look. Thanks! –  Dan Ramras Mar 4 '10 at 3:35

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

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Here is a list of books that are good for linear algebra. Specifically the first link 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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