--- product_id: 5392334 title: "Linear Algebra Done Right (Undergraduate Texts in Mathematics)" price: "2013 kr" currency: NOK in_stock: true reviews_count: 13 url: https://www.desertcart.no/products/5392334-linear-algebra-done-right-undergraduate-texts-in-mathematics store_origin: NO region: Norway --- # Linear Algebra Done Right (Undergraduate Texts in Mathematics) **Price:** 2013 kr **Availability:** ✅ In Stock ## Quick Answers - **What is this?** Linear Algebra Done Right (Undergraduate Texts in Mathematics) - **How much does it cost?** 2013 kr with free shipping - **Is it available?** Yes, in stock and ready to ship - **Where can I buy it?** [www.desertcart.no](https://www.desertcart.no/products/5392334-linear-algebra-done-right-undergraduate-texts-in-mathematics) ## Best For - Customers looking for quality international products ## Why This Product - Free international shipping included - Worldwide delivery with tracking - 15-day hassle-free returns ## Description This text for a second course in linear algebra, aimed at math majors and graduates, adopts a novel approach by banishing determinants to the end of the book and focusing on understanding the structure of linear operators on vector spaces. The author has taken unusual care to motivate concepts and to simplify proofs. For example, the book presents - without having defined determinants - a clean proof that every linear operator on a finite-dimensional complex vector space has an eigenvalue. The book starts by discussing vector spaces, linear independence, span, basics, and dimension. Students are introduced to inner-product spaces in the first half of the book and shortly thereafter to the finite- dimensional spectral theorem. A variety of interesting exercises in each chapter helps students understand and manipulate the objects of linear algebra. This second edition features new chapters on diagonal matrices, on linear functionals and adjoints, and on the spectral theorem; some sections, such as those on self-adjoint and normal operators, have been entirely rewritten; and hundreds of minor improvements have been made throughout the text. Review: Forgive him for the title - What an awful title. I guess Sheldon Axler decided on the ungrammatical "Linear Algebra Done Right" to avoid "Linear Algebra Done Properly" or something similar, which would have sounded intolerably arrogant. He justifies this title repeatedly by rather obnoxiously flaunting his determinant-free proof that operators on complex (or real, odd-dimensional) vector spaces have an eigenvalue. (It's pretty cool, but I've seen cooler constructions, and I'm not even a mathematician.) He also often makes other snarky jabs at the unnamed body of traditional linear algebra texts. Read the book, and you will forgive him on all counts. Other reviewers have already been thorough in their praise/criticism of Axler's elegant exposition that deprecates matrices and determinants. The highlight in my view is how Axler cleans up proofs by simplifying notation and carefully abstracting common algorithms into lemmas (like 2.4, his Linear Dependence Lemma) that are used over and over. This greatly improves readability and promotes the development of intuition. Some of his nonstandard choices of notation are used to such great pedagogical effect that they seem to threaten to redefine what is standard. The prose is correspondingly clear, concise, and full of useful motivation for difficult points. The formatting is impeccable - definitions, equations, inequalities, and theorems/lemmas are all given a uniform numbering system, making them easy and unambiguous to cite. Subsidiary comments are relegated to the margins of the book, keeping the main line of exposition free of digressions. The text is quite shockingly free of errors. Finally, the layout has a clean but cheerful flower-power look that reminds the reader that math is about beauty and fun - not just intimidating formalism. Axler's refusal to refer directly to others for inspiration (he seemingly proudly omits a bibliography) does cause some warts. For instance, when looking at orthogonal projections for optimization, he asks the reader to do inner-product gymnastics in polynomial space on [0,1] instead of on [-1,1]. The latter choice gives rise to the all-important Legendre polynomials, whose symmetry properties are much clearer. Also, while the pristine algebraic presentation was remarkable, I'd have liked to see more geometric insight in places. I got into this book because my undergraduate linear algebra experience, with Apostol Vol. 2, was so frustrating - all of the sweeping and magical structure theorems of self-adjoint operators and so forth seemed to reduce to incomprehensible index-pushing. For me, what finally cleared up these notions to me was drawing, on graph paper, the fate of vectors in R^2 under various linear operators. This was not in the book, but Axler's inclusion of the theory of polar and singular-value decompositions did give some important tools to help unravel these beautiful but elusive issues. Finally, the crystal clarity of the exposition rolls off in Chapters 8 and 9 when getting into the structure theory of general operators on real and complex vector spaces. The symbols get more abstruse, and the arguments get more murky. But I've never seen another author make anything but a mess of, say, the proof of Jordan form. It is hard stuff, and it is not fair to be too hard on authors for failing to make it look easy. The end-of-chapter problems are abundant enough to give a good feel for the material, with an appropriate range of difficulties for an advanced undergraduate book. There are enough of the routine computations and simple proofs that familiarize readers with the new machinery they are learning, but at least a proof or two in each chapter require creative constructions to complete. I just finished the last of the 224 problems, a task that took me five years' worth of sporadic effort in my free time and vacations as a high school math teacher and then as a graduate student in chemistry. A few problems took me the better part of a year to figure out, though this was without the benefit of collaboration. I found the equivalent of at one sequence of problems (problems 6-8 in Chapter 6) as a starred problem in a graduate functional analysis text. I consider myself a good but not award-winning math student, so this indicates that the problems are consistently tractable but can get pretty tough in places. Axler does not mark his most difficult problems as such; for the teacher assigning Axler's problems for a course, then, it is imperative to work through the problems beforehand. All told, this is quite a remarkable book. I now feel like I understand linear algebra, something I couldn't say when I first studied the subject eight years ago. The title does not do it justice. Review: Efficient and elegant - First things first: this is intended for a second course in the subject. This could not be made more clear, being written on the back cover and the foreword. Still, about half of the one and two star reviewers somehow missed this and (surprise) didn't enjoy the book. Anyway, this is one of several sources from which I learned Linear Algebra, the others being Lay and Leon for my first exposure, and Shilov as a companion to this book for my second exposure. Top that off with Professor Strang's recorded lectures and I really should know much more than I actually do! Although the word is used a lot, this book really is elegant. The first three or four chapters cover the basics and are done well, but I think the book really takes off at the start of chapter 5 (I know a lot about determinants from Shilov, so I haven't looked much at chapter 10). Sure, there isn't an abundance of worked out examples, but there are plenty of sources for that. If you want a clean, efficient, and helpful presentation of the theory of Linear Algebra, this is your book. I also appreciate the fact that Axler gives the reader a taste of the main idea and the motivation for a new topic in a few simple sentences. This, along with his ability to make the subject seem deceptively simple, with straightforward proofs that don't rely on any "tricks", led to a number of aha! moments while reading. I don't have any complaints, but there is a lot of material in Shilov that isn't in this book (Shilov's is about twice as long), so it may be possible for a good student to go right to that after a first course. It is harder, provides less motivation, and relies heavily on determinants throughout, especially early on, so beware of that. Skipping between Shilov and Axler simultaneously worked well for me. ## Technical Specifications | Specification | Value | |---------------|-------| | Best Sellers Rank | #1,456,122 in Books ( See Top 100 in Books ) #278 in Linear Algebra (Books) #1,286 in Algebra & Trigonometry | | Customer Reviews | 4.2 out of 5 stars 117 Reviews | ## Images ![Linear Algebra Done Right (Undergraduate Texts in Mathematics) - Image 1](https://m.media-amazon.com/images/I/612qA6JPN5L.jpg) ## Customer Reviews ### ⭐⭐⭐⭐⭐ Forgive him for the title *by R***N on February 28, 2011* What an awful title. I guess Sheldon Axler decided on the ungrammatical "Linear Algebra Done Right" to avoid "Linear Algebra Done Properly" or something similar, which would have sounded intolerably arrogant. He justifies this title repeatedly by rather obnoxiously flaunting his determinant-free proof that operators on complex (or real, odd-dimensional) vector spaces have an eigenvalue. (It's pretty cool, but I've seen cooler constructions, and I'm not even a mathematician.) He also often makes other snarky jabs at the unnamed body of traditional linear algebra texts. Read the book, and you will forgive him on all counts. Other reviewers have already been thorough in their praise/criticism of Axler's elegant exposition that deprecates matrices and determinants. The highlight in my view is how Axler cleans up proofs by simplifying notation and carefully abstracting common algorithms into lemmas (like 2.4, his Linear Dependence Lemma) that are used over and over. This greatly improves readability and promotes the development of intuition. Some of his nonstandard choices of notation are used to such great pedagogical effect that they seem to threaten to redefine what is standard. The prose is correspondingly clear, concise, and full of useful motivation for difficult points. The formatting is impeccable - definitions, equations, inequalities, and theorems/lemmas are all given a uniform numbering system, making them easy and unambiguous to cite. Subsidiary comments are relegated to the margins of the book, keeping the main line of exposition free of digressions. The text is quite shockingly free of errors. Finally, the layout has a clean but cheerful flower-power look that reminds the reader that math is about beauty and fun - not just intimidating formalism. Axler's refusal to refer directly to others for inspiration (he seemingly proudly omits a bibliography) does cause some warts. For instance, when looking at orthogonal projections for optimization, he asks the reader to do inner-product gymnastics in polynomial space on [0,1] instead of on [-1,1]. The latter choice gives rise to the all-important Legendre polynomials, whose symmetry properties are much clearer. Also, while the pristine algebraic presentation was remarkable, I'd have liked to see more geometric insight in places. I got into this book because my undergraduate linear algebra experience, with Apostol Vol. 2, was so frustrating - all of the sweeping and magical structure theorems of self-adjoint operators and so forth seemed to reduce to incomprehensible index-pushing. For me, what finally cleared up these notions to me was drawing, on graph paper, the fate of vectors in R^2 under various linear operators. This was not in the book, but Axler's inclusion of the theory of polar and singular-value decompositions did give some important tools to help unravel these beautiful but elusive issues. Finally, the crystal clarity of the exposition rolls off in Chapters 8 and 9 when getting into the structure theory of general operators on real and complex vector spaces. The symbols get more abstruse, and the arguments get more murky. But I've never seen another author make anything but a mess of, say, the proof of Jordan form. It is hard stuff, and it is not fair to be too hard on authors for failing to make it look easy. The end-of-chapter problems are abundant enough to give a good feel for the material, with an appropriate range of difficulties for an advanced undergraduate book. There are enough of the routine computations and simple proofs that familiarize readers with the new machinery they are learning, but at least a proof or two in each chapter require creative constructions to complete. I just finished the last of the 224 problems, a task that took me five years' worth of sporadic effort in my free time and vacations as a high school math teacher and then as a graduate student in chemistry. A few problems took me the better part of a year to figure out, though this was without the benefit of collaboration. I found the equivalent of at one sequence of problems (problems 6-8 in Chapter 6) as a starred problem in a graduate functional analysis text. I consider myself a good but not award-winning math student, so this indicates that the problems are consistently tractable but can get pretty tough in places. Axler does not mark his most difficult problems as such; for the teacher assigning Axler's problems for a course, then, it is imperative to work through the problems beforehand. All told, this is quite a remarkable book. I now feel like I understand linear algebra, something I couldn't say when I first studied the subject eight years ago. The title does not do it justice. ### ⭐⭐⭐⭐⭐ Efficient and elegant *by H***L on November 28, 2010* First things first: this is intended for a second course in the subject. This could not be made more clear, being written on the back cover and the foreword. Still, about half of the one and two star reviewers somehow missed this and (surprise) didn't enjoy the book. Anyway, this is one of several sources from which I learned Linear Algebra, the others being Lay and Leon for my first exposure, and Shilov as a companion to this book for my second exposure. Top that off with Professor Strang's recorded lectures and I really should know much more than I actually do! Although the word is used a lot, this book really is elegant. The first three or four chapters cover the basics and are done well, but I think the book really takes off at the start of chapter 5 (I know a lot about determinants from Shilov, so I haven't looked much at chapter 10). Sure, there isn't an abundance of worked out examples, but there are plenty of sources for that. If you want a clean, efficient, and helpful presentation of the theory of Linear Algebra, this is your book. I also appreciate the fact that Axler gives the reader a taste of the main idea and the motivation for a new topic in a few simple sentences. This, along with his ability to make the subject seem deceptively simple, with straightforward proofs that don't rely on any "tricks", led to a number of aha! moments while reading. I don't have any complaints, but there is a lot of material in Shilov that isn't in this book (Shilov's is about twice as long), so it may be possible for a good student to go right to that after a first course. It is harder, provides less motivation, and relies heavily on determinants throughout, especially early on, so beware of that. Skipping between Shilov and Axler simultaneously worked well for me. ### ⭐⭐⭐⭐ Thought Provoking *by S***I on September 26, 2007* I have no doubt that this is one of the most thought provoking math books that I have come across. I used this book for a linear algebra course last fall '07 and I learned a ton. Specifically about the structure of vector spaces and linear operators. However, the most important function that this book serves is to move students towards the methodology of mathematics, which means proof construction and counter examples. It also trains students to let go of their intuitions. But you can not self-study this book, there are no answers and more importantly the structure of the course begs for instruction. I would recommend before taking this course doing what i didn't do and have had to do since, make sure you have your first course of linear algebra solidly under your belt, and that doesn't mean having gotten an A in the prior class is sufficient. Go through the most difficult proof driven exercises in your first text, that should serve as practice for easiest homework problems in this book. All that said, there are serious limitations to this book. It would be nice if the author worked out 1 comprehensive semi-difficult exercise in each chapter of the text. While struggling to solve the problems can be enlightening, there is only so many times I can read the same sections over and over again, looking for some insight from the kiddie exercises provided by the author. It would also help if some of the kiddie exercises were accompanied with graphs, especially when describing the sums of vector spaces. 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