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Linear Algebra Done Right
Intended for a second course in pursuing mathematics, this volume discusses topics such as the existence of eigenvalues on complex vector spaces, upper-triangular matrices and orthogonal projections.
357 pages, Hardcover
First published November 29, 1995
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Displaying 1 - 30 of 94 reviews
July 9, 2016
Excellent textbook for a second look at linear algebra from a strictly theoretical standpoint. It size is small enough so that one may comfortably carry it around and promptly, effortlessly smack around fools that utter:
Their education is the responsibility of us all, and how often we forget the old ways...
The proofs are clear but do require the reader to fill in some gaps. This is intended. Open to any page and witness the clarity that so often escape the best efforts of a certain class of instructor; Axler will not suffer any unmotivated concepts. Everything builds from previous definitions until there's just enough structure to flesh out the chapter objectives, thus there's little fat distract the reader.
Moreover, Axler is so badass that he does away with determinants until the last chapter of the book, he's so pimp he just didn't need any stinking determinants in his proofs. That's right, the last chapter introduces trace and determinants and proceeds to bring everything together into a magnificent mic drop:
I now finally understand why determinants are inextricably tied to notions of volume, and why we must multiply by the Jacobian when performing change of variables in multi-variable integrals, and so on.
A newcomer to linear algebra will get very little of use here, save for the clearest definitions I've ever seen regarding the structure of vector spaces, subspaces and linear operators. For a more applied/introductory approach to linear algebra, one can do much worse than Strang.
I now feel much more comfortable moving onto a graduate-level Linear algebra course after visiting Axler's book, as such, it will be an invaluable reference moving forward.
"linear algebra! that's just y = mx + b!!! LOLZ".
Their education is the responsibility of us all, and how often we forget the old ways...
The proofs are clear but do require the reader to fill in some gaps. This is intended. Open to any page and witness the clarity that so often escape the best efforts of a certain class of instructor; Axler will not suffer any unmotivated concepts. Everything builds from previous definitions until there's just enough structure to flesh out the chapter objectives, thus there's little fat distract the reader.
Moreover, Axler is so badass that he does away with determinants until the last chapter of the book, he's so pimp he just didn't need any stinking determinants in his proofs. That's right, the last chapter introduces trace and determinants and proceeds to bring everything together into a magnificent mic drop:
I now finally understand why determinants are inextricably tied to notions of volume, and why we must multiply by the Jacobian when performing change of variables in multi-variable integrals, and so on.
A newcomer to linear algebra will get very little of use here, save for the clearest definitions I've ever seen regarding the structure of vector spaces, subspaces and linear operators. For a more applied/introductory approach to linear algebra, one can do much worse than Strang.
I now feel much more comfortable moving onto a graduate-level Linear algebra course after visiting Axler's book, as such, it will be an invaluable reference moving forward.
October 14, 2019
[Second reading]
I read this the second time at the beginning of my physics PhD. It was much more pleasant to read. I am a slow learner, so even if the subject is for beginning undergraduates, a few concepts only make sense to me now. I can also appreciate a little better the whole thing about generalized eigenvectors, nilpotency, Jordan form, and various decomposition (in particular singular value and polar decomposition). I am sure I will forget them soon, considering that I don't use them often in my research, but this probably means that I am not as blind as I was in the past, and I probably don't need to re-read the book for the third time from cover to back.
Highly recommended linear algebra textbook, more so than the standard "Friedberg" one actually.
---
[First reading] Very good book for linear algebra, one which attempts to avoid the rather unmotivated concepts of determinants to prove many important results. The book focuses more on operators which are more abstract and more general than matrices themselves, providing easy proofs of theorems such as Cayley-Hamilton or that operators in complex vector space always has an eigenvalue.
The first two chapters are easy to grasp and it forces us to understand linear independence and basis without explicitly using matrix concepts of basis vectors.
Chapter 3 gives good motivation of matrix product, that matrix is a book-keeping device for linear mapping process. Full understanding of range-null formula and composition of maps enables us to easily get back the usual Rank-Nullity formula and matrix products. Chapter 4's short explanations on polynomials are rather enlightening for those who did not delve deeply into it - the factorization theorem is particularly illuminating.
Chapter 5 is where the difficulty starts - the concept of invariant subspaces, projections of subspaces. However, the statements on how we can always find upper triangular matrices and diagonal matrices for some operators are useful. The proof that every operator in complex vector space has eigenvalue is extremely simple, compared to the use of determinant version.
Chapter 6 is the usual standard teaching of inner product space. A useful concept taught would be how the concept can be used to compute better approximations of some functions within an interval than its own Taylor expansion. The Cauchy-Schwarz inequality proof is very intuitive and simple instead of using rather random construction. I found it difficult on digesting linear functionals, however.
Chapter 7's usefulness should be well known to those who have done linear algebra - the spectral theorems. It gives clear decomposition of vector spaces into eigenspaces, allowing us to always find diagonal matrices for normal/hermitian operators depending on whether the vector space is complex or real. It also covers block diagonal matrices and isometry, and also polar and singular value decomposition quite clearly.
Chapter 8 and 9 I find it difficult to follow, but the rough idea is indeed simple. Cayley-Hamilton theorem is easy to prove using tools developed in this book. Many results are generalized using generalized eigenvectors and block diagonal matrices. It covers Jordan form in the final section of chapter 8. Characteristic polynomials are well developed here, as well as why characteristic polynomials are basis-independent.
Last chapter on trace and determinant is in my opinion one of the best chapters. The determinant is no longer felt as if it came out of nowhere, since even the origin of signed elementary products are shown clearly. One can follow why determinant can be defined that way, though quite a bit of preliminary work to be done (about 2 pages of A4 paper) before things flow smoothly. One can also see clearly the relation between traces and determinants with characteristic polynomials. Many basic properties of determinants are beautifully proven here without much explicit use of matrices.
Overall, the book presents much cleaner proofs and explanations as compared to the usual introductory linear algebra which uses matrices mostly. While clean proofs do not imply easier proofs, but it clarifies and generalizes many concepts. This is especially useful for mathematics majors who are more into abstract thinking rather than computational procedures, though if well understood, the book can tell many results which provide computational advantages, such as under what conditions the matrices have many zeros. Many of these concepts give much clearer motivation than the usual matrix explanations. However, for those who only need linear algebra insofar as computation and Gaussian elimination is concerned, then perhaps this book is sort of overkill.
I would recommend the book for those more into abstract thinking, or more theoretical students such as theoretical physics students or mathematics students.
I read this the second time at the beginning of my physics PhD. It was much more pleasant to read. I am a slow learner, so even if the subject is for beginning undergraduates, a few concepts only make sense to me now. I can also appreciate a little better the whole thing about generalized eigenvectors, nilpotency, Jordan form, and various decomposition (in particular singular value and polar decomposition). I am sure I will forget them soon, considering that I don't use them often in my research, but this probably means that I am not as blind as I was in the past, and I probably don't need to re-read the book for the third time from cover to back.
Highly recommended linear algebra textbook, more so than the standard "Friedberg" one actually.
---
[First reading] Very good book for linear algebra, one which attempts to avoid the rather unmotivated concepts of determinants to prove many important results. The book focuses more on operators which are more abstract and more general than matrices themselves, providing easy proofs of theorems such as Cayley-Hamilton or that operators in complex vector space always has an eigenvalue.
The first two chapters are easy to grasp and it forces us to understand linear independence and basis without explicitly using matrix concepts of basis vectors.
Chapter 3 gives good motivation of matrix product, that matrix is a book-keeping device for linear mapping process. Full understanding of range-null formula and composition of maps enables us to easily get back the usual Rank-Nullity formula and matrix products. Chapter 4's short explanations on polynomials are rather enlightening for those who did not delve deeply into it - the factorization theorem is particularly illuminating.
Chapter 5 is where the difficulty starts - the concept of invariant subspaces, projections of subspaces. However, the statements on how we can always find upper triangular matrices and diagonal matrices for some operators are useful. The proof that every operator in complex vector space has eigenvalue is extremely simple, compared to the use of determinant version.
Chapter 6 is the usual standard teaching of inner product space. A useful concept taught would be how the concept can be used to compute better approximations of some functions within an interval than its own Taylor expansion. The Cauchy-Schwarz inequality proof is very intuitive and simple instead of using rather random construction. I found it difficult on digesting linear functionals, however.
Chapter 7's usefulness should be well known to those who have done linear algebra - the spectral theorems. It gives clear decomposition of vector spaces into eigenspaces, allowing us to always find diagonal matrices for normal/hermitian operators depending on whether the vector space is complex or real. It also covers block diagonal matrices and isometry, and also polar and singular value decomposition quite clearly.
Chapter 8 and 9 I find it difficult to follow, but the rough idea is indeed simple. Cayley-Hamilton theorem is easy to prove using tools developed in this book. Many results are generalized using generalized eigenvectors and block diagonal matrices. It covers Jordan form in the final section of chapter 8. Characteristic polynomials are well developed here, as well as why characteristic polynomials are basis-independent.
Last chapter on trace and determinant is in my opinion one of the best chapters. The determinant is no longer felt as if it came out of nowhere, since even the origin of signed elementary products are shown clearly. One can follow why determinant can be defined that way, though quite a bit of preliminary work to be done (about 2 pages of A4 paper) before things flow smoothly. One can also see clearly the relation between traces and determinants with characteristic polynomials. Many basic properties of determinants are beautifully proven here without much explicit use of matrices.
Overall, the book presents much cleaner proofs and explanations as compared to the usual introductory linear algebra which uses matrices mostly. While clean proofs do not imply easier proofs, but it clarifies and generalizes many concepts. This is especially useful for mathematics majors who are more into abstract thinking rather than computational procedures, though if well understood, the book can tell many results which provide computational advantages, such as under what conditions the matrices have many zeros. Many of these concepts give much clearer motivation than the usual matrix explanations. However, for those who only need linear algebra insofar as computation and Gaussian elimination is concerned, then perhaps this book is sort of overkill.
I would recommend the book for those more into abstract thinking, or more theoretical students such as theoretical physics students or mathematics students.
December 26, 2013
This book was a real page-turner! Great approach towards linear algebra. Sheldon Axler doesn't introduce determinants until the end, which was a true delight in my opinion because it surely kept me reading until the very last chapter in order to learn about those nasty determinants. The section about Jordan Canonical forms had me hooked for days. The way the Jordan bases were formed was absolutely mind blowing! I did not see it coming at all! Axler truly outdid himself with this one. The book definitely lives up to its title.
September 7, 2012
A much, much better coverage of linear algebra than the book my undergrad class used. As many others have noted, it largely avoids the use of matrices and determinants to focus on the purely algebraic aspects of the material. It is also nice that vector spaces other than R^n or C^n are frequently considered, such as the space of all polynomial functions of a single variable. I was able to read through almost the entire book in one long sitting since it was a lot of review for me was well-presented, and in the course of it I solidified some ideas and formed some new ones. My motivation for picking it up was to get a formal introduction to direct products and a few other scattered ideas that were not covered in my lin-alg course but it was so nice to read such clear expositions that I just kept reading.
February 12, 2022
Beautiful. One of the books where the reading process doesn't feel like learning more material, but rather like individual components falling into one big picture of the world of (finite-dimensional) linear spaces.
That being said, don't pick this up without also studying additional material. LADR is beautiful, but by no means comprehensive:
1) It lacks a lot of material, either in the topics presented, or additional topics (I would love to see the author's treatment of bilinear forms)
2) Nearly no material or exercises on computations. I don't consider this a weakness, it's not what this book is about and such material would clutter the otherwise clean and beautiful presentation.
3) No matter how many times the author makes a remark of how superior this approach is, knowing both is what everyone should do. Another text leaning more on the determinant, computations, applications, operations with matrices and geometrical approach text would be a great complement to this book.
That being said, don't pick this up without also studying additional material. LADR is beautiful, but by no means comprehensive:
1) It lacks a lot of material, either in the topics presented, or additional topics (I would love to see the author's treatment of bilinear forms)
2) Nearly no material or exercises on computations. I don't consider this a weakness, it's not what this book is about and such material would clutter the otherwise clean and beautiful presentation.
3) No matter how many times the author makes a remark of how superior this approach is, knowing both is what everyone should do. Another text leaning more on the determinant, computations, applications, operations with matrices and geometrical approach text would be a great complement to this book.
December 29, 2020
I enjoyed teaching out of this book this year, and expect to keep doing so for years to come.
April 19, 2020
Oh man, this book is SO GOOD! SO. GOOD. I learned linear algebra from /book/show/3..., but this book is SO MUCH better in terms of motivating any number of abstractions. I'm not sure if it's because of my having becoming more familiar with abstraction since having read that book or if it's actually this book, but I thought this was far more approachable, without being any less rigorous. It certainly went through all the same material, so all the material that I didn't understand the reasoning for (i.e. considering the dual space and annihilators in that space or the permutation formulation of determinants or generalized eigenspaces) made so much more sense in reading this book. There's also an ungodly number of exercises here, of which I probably did around 1/3. They had a good range of difficulty, in the space of completely elementary if you understood the material to requiring a decent bit of thought to piece together some of the concepts.
Anyway, that was a long enough rant. But this book is GREAT
Anyway, that was a long enough rant. But this book is GREAT
January 3, 2021
I really liked this book, but I think there is some room for improvement. I will describe my reasoning below. Nevertheless, among the arid and dry landscape of mathematics, this book is an oasis.
The premise of the book is simple and outrageous at first sight. Let us teach linear algebra without the use of determinants. "Anathema!" I hear you all shout. "It is impossible" (I heard myself think). However, Sheldon Axler proved me wrong. And honestly, it makes things much more intuitive and easy.
But wait, isn't this like some sort of new math that would disable people to actually deal with determinants when encountering them elsewhere? Not at all. The very last chapter of the book is discussing determinants. This goes all the way after discussing finite dimensional vector spaces, linear maps, kernels, images, inner product spaces, the spectral theorem, the Cayley Hamilton theorem, and everything else you would expect in a typical undergraduate level math course.
Each chapter is having plenty of exercises to give the student a relatively good grasp on the concepts. The book itself doesn't have the solutions, but they are available on a website.
All this make this book a favorite of mine in regards to this subject. Nevertheless, it appears to me that there is some room for improvement. I do understand the need for proof for every theorem. The author has a point to make (i.e. that linear algebra can be taught in a more efficient way by delaying the study of the determinants). What I would have improved though is the intuitive aspect of this branch of mathematics.
I believe that the books is a step in the right direction, and of course, I know that exercises help. However, I believe that due to the very abstract nature of this area of mathematics, some geometric approach of some kind, would have helped. When I first learned linear algebra, a lot of the concepts seem simply introduced as definitions, without a reasoning behind the why and how. As such, these courses were for me more of an exercise in memorization, and developed little practical understanding. I was not able to understand why people were coming up with the concepts and names of linear algebra.
Mathematics is notorious for its abstractness and that is the main reason behind it. Consider the following (rather stupid) example that occurred to me when I first heard of the term "linear transforms" (linear mappings):
A linear mapping is a kind of function. Of course, it has certain properties. For example, if we have a linear mapping "f" on a vector space, then f(a)+f(b) = f(a+b). Also, m*f(a) = f(m*a) . The question that occurred to me, was this: "Why is a translation on a line not a linear transformation?". Of course, the simple answer is this: "It doesn't respect the properties above". (A translation is a function like this T(x) = x + c where c is a constant)
But really, a translation is simply moving something in a straight line, from point A to point B. It is the most obvious use of the word linear. Of course, there are explanations and since then I came to know better.
Similar questions arise time and time again in the study of mathematics (why are eigenvalues so important? What about the adjoint of an operator? What is the idea behind them? What is the intuition behind the dual spaces? How come someone cared enough to name them? etc.)
To me it seems that a lot of these issues can and should be explained better. I honestly don't believe these are stupid questions. Of course, I don't believe that simply giving a simplistic reasoning would increase your capacity of problem solving. However, this type of translation and understanding can certainly be done to great profit, and I believe it shortens the curve. Also, I do know that many of the questions do not have a satisfying answer. For example, I wasn't able to really understand why the mathematicians chose the term spectral for the spectral theorem. I do know when it was used first, and who used it, but apparently there is no clear understanding as to why he used it or whether he saw any connections to physics.
I believe that Sheldon Axler's book is a huge step in the right direction, but I feel that further strides can be easily done, and I believe that there is enough room for improvement. At the moment at least, I think it is light years above my meager undergraduate course in explanatory power. I believe he could have added more examples taken from geometry and physics.
The premise of the book is simple and outrageous at first sight. Let us teach linear algebra without the use of determinants. "Anathema!" I hear you all shout. "It is impossible" (I heard myself think). However, Sheldon Axler proved me wrong. And honestly, it makes things much more intuitive and easy.
But wait, isn't this like some sort of new math that would disable people to actually deal with determinants when encountering them elsewhere? Not at all. The very last chapter of the book is discussing determinants. This goes all the way after discussing finite dimensional vector spaces, linear maps, kernels, images, inner product spaces, the spectral theorem, the Cayley Hamilton theorem, and everything else you would expect in a typical undergraduate level math course.
Each chapter is having plenty of exercises to give the student a relatively good grasp on the concepts. The book itself doesn't have the solutions, but they are available on a website.
All this make this book a favorite of mine in regards to this subject. Nevertheless, it appears to me that there is some room for improvement. I do understand the need for proof for every theorem. The author has a point to make (i.e. that linear algebra can be taught in a more efficient way by delaying the study of the determinants). What I would have improved though is the intuitive aspect of this branch of mathematics.
I believe that the books is a step in the right direction, and of course, I know that exercises help. However, I believe that due to the very abstract nature of this area of mathematics, some geometric approach of some kind, would have helped. When I first learned linear algebra, a lot of the concepts seem simply introduced as definitions, without a reasoning behind the why and how. As such, these courses were for me more of an exercise in memorization, and developed little practical understanding. I was not able to understand why people were coming up with the concepts and names of linear algebra.
Mathematics is notorious for its abstractness and that is the main reason behind it. Consider the following (rather stupid) example that occurred to me when I first heard of the term "linear transforms" (linear mappings):
A linear mapping is a kind of function. Of course, it has certain properties. For example, if we have a linear mapping "f" on a vector space, then f(a)+f(b) = f(a+b). Also, m*f(a) = f(m*a) . The question that occurred to me, was this: "Why is a translation on a line not a linear transformation?". Of course, the simple answer is this: "It doesn't respect the properties above". (A translation is a function like this T(x) = x + c where c is a constant)
But really, a translation is simply moving something in a straight line, from point A to point B. It is the most obvious use of the word linear. Of course, there are explanations and since then I came to know better.
Similar questions arise time and time again in the study of mathematics (why are eigenvalues so important? What about the adjoint of an operator? What is the idea behind them? What is the intuition behind the dual spaces? How come someone cared enough to name them? etc.)
To me it seems that a lot of these issues can and should be explained better. I honestly don't believe these are stupid questions. Of course, I don't believe that simply giving a simplistic reasoning would increase your capacity of problem solving. However, this type of translation and understanding can certainly be done to great profit, and I believe it shortens the curve. Also, I do know that many of the questions do not have a satisfying answer. For example, I wasn't able to really understand why the mathematicians chose the term spectral for the spectral theorem. I do know when it was used first, and who used it, but apparently there is no clear understanding as to why he used it or whether he saw any connections to physics.
I believe that Sheldon Axler's book is a huge step in the right direction, but I feel that further strides can be easily done, and I believe that there is enough room for improvement. At the moment at least, I think it is light years above my meager undergraduate course in explanatory power. I believe he could have added more examples taken from geometry and physics.
May 12, 2011
This is how linear algebra should always be presented. The unfortunate presentation one first encounters that emphasises matrices and determinants without appeal to the algebra and geometry that motivates them is absent from this text. It is a full presentation that will give you an intuitive grasp of linear algebra from both the geometric and algebraic points of view.
February 24, 2025
bu adam kadar determinantlardan nefret eden biri yok, the biggest hater there ever was adam o kadar nefret ediyo ki determinantlardan DETERMİNANT KULLANILMAYACAK bir matematik subgenre’sı bulmuş eline sağlık reis
June 5, 2021
Best book on Linear Algebra that I found from a whole bunch that were recommended to me. The author's determination to ignore determinants until the fag end of the textbook and just explaining concepts in more intuitive terms and not the rote, god-awful pedantic tradition of drawing up lines and boxes and putting numbers around them. It's extremely imaginative and makes an effort to get you to truly "see" linear spaces in a new light beyond all the greeks.
If for some reason, you prefer the determinant-centric(?) approach and like more of a traditional approach to the subject, there's Gilbert Strang's legendary book too. But I have invited people to read this book and give linear algebra another shot (coz it's dope) and they have loved it just as much as I have. So this one's definitely my go-to comfort math book because it's just so much fun to read!
P.S: This book is known to be extremely polarising among teachers, but I have so far only heard of the good experiences. Btw, there's also Linear Algebra Done Wrong by Sergei Treil that I think you should check out. Despite the name, another solid book.
If for some reason, you prefer the determinant-centric(?) approach and like more of a traditional approach to the subject, there's Gilbert Strang's legendary book too. But I have invited people to read this book and give linear algebra another shot (coz it's dope) and they have loved it just as much as I have. So this one's definitely my go-to comfort math book because it's just so much fun to read!
P.S: This book is known to be extremely polarising among teachers, but I have so far only heard of the good experiences. Btw, there's also Linear Algebra Done Wrong by Sergei Treil that I think you should check out. Despite the name, another solid book.
April 3, 2008
This was the textbook used in my undergraduate theory of linear algebra course (taught by Seymour Goldberg, so you know the relevant view). It does have the nice property, basically shared with Lax's book, that it's clearly a book about linear algebra in general (finite-dimensional) linear spaces, and not a mix of some linear algebra, some matrix analysis, and some elementary modeling concepts. Also, Axler is in the "short and sweet" school of mathematical writing, so it doesn't loom large on the shelf.
May 13, 2024
Now that I’ve fallen back in love with everything I learned in college, textbooks are becoming a cover-to-cover affair, which makes them fair game.
Axler’s approach to linear algebra was completely foreign to me, especially with the largely application-oriented focus my previous classes had; the subject existed more as a means to an end than an abstract, theoretical path of study in its own right. LINEAR ALGEBRA DONE RIGHT works well for what it is, which is a non-comprehensive but valuable tool in understanding more of the intuition surrounding linear algebra, but certain computational tie-ins could have been useful. I appreciate Axler cutting out the bloat, but I don’t think everything that was omitted necessarily should have been.
There's also a bit of inconsistency in terms of assumptions Axler makes about the level of understanding the reader has. It does read as an undergraduate text, but when approaching Chapters 6-8, which weren’t covered by my linear algebra course, there were points where I felt out of my depth. The basics of inner product spaces and singular value decomposition would've passed me by if not for the cursory understanding I had from numerical analysis, and it took me a few tries to grasp the spectral theorem, although the real/complex split and respective explanations were well done.
Overall, though, Axler’s focus on operators and algebra rather than computation and matrix manipulation has ultimately been worthwhile for me, especially as one of my first forays into the sphere of math that is linearly independent to all my engineering curriculum (haha). His notation and wording, although unconventional (I’m sure “nice� is a very credible mathematical term) and with some exceptions (the way he expresses bilinear symmetric/alternating forms looks awful), make topics both intuitive and engaging.
just don’t ask me to explain a tensor product
Axler’s approach to linear algebra was completely foreign to me, especially with the largely application-oriented focus my previous classes had; the subject existed more as a means to an end than an abstract, theoretical path of study in its own right. LINEAR ALGEBRA DONE RIGHT works well for what it is, which is a non-comprehensive but valuable tool in understanding more of the intuition surrounding linear algebra, but certain computational tie-ins could have been useful. I appreciate Axler cutting out the bloat, but I don’t think everything that was omitted necessarily should have been.
There's also a bit of inconsistency in terms of assumptions Axler makes about the level of understanding the reader has. It does read as an undergraduate text, but when approaching Chapters 6-8, which weren’t covered by my linear algebra course, there were points where I felt out of my depth. The basics of inner product spaces and singular value decomposition would've passed me by if not for the cursory understanding I had from numerical analysis, and it took me a few tries to grasp the spectral theorem, although the real/complex split and respective explanations were well done.
Overall, though, Axler’s focus on operators and algebra rather than computation and matrix manipulation has ultimately been worthwhile for me, especially as one of my first forays into the sphere of math that is linearly independent to all my engineering curriculum (haha). His notation and wording, although unconventional (I’m sure “nice� is a very credible mathematical term) and with some exceptions (the way he expresses bilinear symmetric/alternating forms looks awful), make topics both intuitive and engaging.
just don’t ask me to explain a tensor product
April 21, 2022
It's a really good book overall... but it's biggest triumph--putting the determinant off until the end--is also its biggest downfall. Axler is right that linear algebra can be more elegantly developed without determinants than it is classically. This book is his proof that a coherent and clean theory can be developed according to his program.
There are many areas where leaving out determinants leads to a more beautiful theory. But it isn't categorically true. For example, Axler was right that it is possible to develop the theory of eigenvalues/vectors without determinants. But the chapter on them feels like it's missing a good answer to computing eigenvalues. By far the most natural way, of course, is to take the determinant of A-lambda*I. It feels unnatural to talk about eigenvalues without mentioning this natural fact.
Moreover, as a geometer, this book didn't entirely prepare me for what I needed to use linear algebra for. Most importantly, Axler was able to dance around using the determinant in linear algebra, but it occupies a much more unavoidable role in geometry, a fact owing ultimately to its place in tensor calculus.
I've spent this review trying to justify giving only four stars to this book, because it really is excellent other than these problems.
There are many areas where leaving out determinants leads to a more beautiful theory. But it isn't categorically true. For example, Axler was right that it is possible to develop the theory of eigenvalues/vectors without determinants. But the chapter on them feels like it's missing a good answer to computing eigenvalues. By far the most natural way, of course, is to take the determinant of A-lambda*I. It feels unnatural to talk about eigenvalues without mentioning this natural fact.
Moreover, as a geometer, this book didn't entirely prepare me for what I needed to use linear algebra for. Most importantly, Axler was able to dance around using the determinant in linear algebra, but it occupies a much more unavoidable role in geometry, a fact owing ultimately to its place in tensor calculus.
I've spent this review trying to justify giving only four stars to this book, because it really is excellent other than these problems.
August 2, 2022
Finally finished, using this textbook for my second course in Linear Algebra, and I definitely agree this is one of the better organized LA textbooks. However, I still think it struggles especially in the middle chapters to instill a full understanding. Additionally, many of the exercises were either too easy, or completely irrelevant. That being said still a solid book (may go up to 4 stars if I do well on my final 😂).
January 9, 2025
Definitely the go-to textbook for your first linear algebra course. Good examples and explanations. After I found this book, the class got way easier.
December 20, 2019
Get ready to write a lot of proofs.
December 2, 2024
The book has great pedagogy and some parts are simply amazing. However, there are a handful proofs that had me bored to death. This book's audience is very much for students already familiar with elementary linear algebra, who want to increase their intuition of the subject. Despite being a book on linear algebra, matrices are introduced far into the book and are rarely used. This unusual way of teaching linear algebra does indeed appear to be the right way.
July 13, 2024
This book is a freaking gem! In my case it was a perfect second textbook in linear algebra. I was already familiar with basic concepts (jordan form, eigenvectors, determinants etc.) from the matrix perspective. However it was a mystery to me why the hell a trace (which as far as I knew was just a sum of the diagonal matrix elements) is important at all, or why is determinant defined the way it is, or what does matrix transpose mean except a step in the computational procedures. Not to mention eigenvalues and eigenvectors - I knew the definitions but the intuition behind them eluded me for years. And then I read this book and all became clear - the clouds went away and the sun shines brightly over adjoints and invariant subspaces!
Seriously though, thinking about linear algebra from the linear transformations perspective helped me to develop the critical intuition and understanding lacking in the matrix-oriented courses. Highly recommend doing the exersices as well - nothing helps to internalize the abovementioned understanding better than using it.
Seriously though, thinking about linear algebra from the linear transformations perspective helped me to develop the critical intuition and understanding lacking in the matrix-oriented courses. Highly recommend doing the exersices as well - nothing helps to internalize the abovementioned understanding better than using it.
May 9, 2020
This is the first math book I couldn't stop reading after the first few chapters. I was hooked after the author's proof of the Rank-Nullity theorem - the elegance of the proof was superb.
The book also is the only book that gives a satisfactory theoretical explanation/justification of determinants. (Hint: It's not pretty that's why it's eschewed.)
The book also is the only book that gives a satisfactory theoretical explanation/justification of determinants. (Hint: It's not pretty that's why it's eschewed.)
September 14, 2024
Well, it wasn't easy. But the learning curve was just right. If you work through it sequentially, there isn't point where it becomes impossible to follow.
It's definitely not an applied mathematics, or a particularly *practical* book on linear algebra. When Axler calls it "Done Right", what he means is that it is mathematically rigorous and he provides proofs for just about anything and everything.
This is a book for a pure mathematician, and there's nothing wrong with that. By learning more about the mathematical structures of linear algebra and the deep explanations for why things are as they should be, it is also an immense aid for any physicist or engineer who wants to do better than just apply theorems blindly.
I highly recommend this book for... well, people who would be willing to read it. Good luck.
It's definitely not an applied mathematics, or a particularly *practical* book on linear algebra. When Axler calls it "Done Right", what he means is that it is mathematically rigorous and he provides proofs for just about anything and everything.
This is a book for a pure mathematician, and there's nothing wrong with that. By learning more about the mathematical structures of linear algebra and the deep explanations for why things are as they should be, it is also an immense aid for any physicist or engineer who wants to do better than just apply theorems blindly.
I highly recommend this book for... well, people who would be willing to read it. Good luck.
April 5, 2016
This is a great second book on linear algebra - something you might want to read once you read a foundation text on linear algebra first. It is a full color book which makes the book rather expensive. It's not a large book, but it's packed with just the right explanations and amount of theory to get you to understand the more mathematical aspects of linear algebra - the topics of spaces, linear maps and related theorems and concepts. The book is very accessible, has plenty of exercises (no solutions though!) and is quite well designed.
December 31, 2018
lives up to its name. And as the subhead says, this should not be your first introduction to linear algebra.
All quants should read it eventually. (I define a quant as anyone who uses matrices at work.)
All quants should read it eventually. (I define a quant as anyone who uses matrices at work.)
September 1, 2021
I really like this text. It's a nice complement to Halmos, which is curious as Halmos was Axler's advisor.
Read
May 21, 2015jhjhljkhjlkjkj.k
March 29, 2020
Extremely lucid explanations for the abstract aspects of Linear Algebra including vector space, etc. Read this for machine learning:D
March 26, 2022
Picked up to review linear algebra for a tutoring assignment. It’s extremely good
July 6, 2021
First of all, a big disclaimer,
JUST DON'T READ THIS BOOK AS YOUR INTRODUCTION TO LINEAR ALGEBRA !!!
I still can't find out why we were forced to have this book for our first course in linear algebra. This book is totally abstract, you have to prove things, there aren't many numerical problems or numerical stuff. So after finishing this book you'll probably end up with an Existential crisis if you want to enter machine learning by the time you've finished this book.
Sheldon Axler wants to be different. Ok, that's fine but, why on earth I have to burden this attitude? He introduced determinants as the wrap-up for his book. why? cause he just wants to be different.
He introduced SVD in some kinda weird and ugly way. why? cause he just wants to be different.
literally not in the formal notation that you would saw in other books and real-life and you won't probably understand why this thing matters and instead of that you can have Jordan form (Such a blessing thanks god!!)
Ah this book totally reminds me of the Babadook meme
Sheldon Axler why can't you be normal :(
It was a fun but abstract experience and clearly, don't recommend it if you're an application-based guy rather than proof-based. I feel I've wasted my time with something that I didn't mean to learn.
Instead of this go with Gilbert Strang's books for god sake.
JUST DON'T READ THIS BOOK AS YOUR INTRODUCTION TO LINEAR ALGEBRA !!!
I still can't find out why we were forced to have this book for our first course in linear algebra. This book is totally abstract, you have to prove things, there aren't many numerical problems or numerical stuff. So after finishing this book you'll probably end up with an Existential crisis if you want to enter machine learning by the time you've finished this book.
Sheldon Axler wants to be different. Ok, that's fine but, why on earth I have to burden this attitude? He introduced determinants as the wrap-up for his book. why? cause he just wants to be different.
He introduced SVD in some kinda weird and ugly way. why? cause he just wants to be different.
literally not in the formal notation that you would saw in other books and real-life and you won't probably understand why this thing matters and instead of that you can have Jordan form (Such a blessing thanks god!!)
Ah this book totally reminds me of the Babadook meme
Sheldon Axler why can't you be normal :(
It was a fun but abstract experience and clearly, don't recommend it if you're an application-based guy rather than proof-based. I feel I've wasted my time with something that I didn't mean to learn.
Instead of this go with Gilbert Strang's books for god sake.
September 8, 2021
I had this book checked out of the library for nearly a year and kept renewing it because of how well it guides the reader through the theoretical topics. I have also watched every video that the author has posted on YouTube. This is the absolute best textbook I have read because it most significantly changed the way I think about not only math, but all technical topics in general.
I have not finished all of the exercises, and it may take me a few more years to fully understand all of the content, but learning the concepts of Linear Algebra will save you time and energy in the long run. If you learn the concepts of vector spaces correctly and fully, then you won't be confused by mediocre explanations of vectors that you might get from a standard engineering textbook. The only downside is that by reading this book, you will expect more rigor from the next proofs you read, and that is a high bar to live up to.
If you are new to higher level mathematics, it may take you all day to get through a few pages, especially since many new terms and notations are introduced. Also, the problems at the end of each chapter can be extremely difficult, but you can find worked-out solutions online since the book has become so popular among students lately. But all of the pieces are laid out, so you will know both why and how the solution is correct when you see it.
There is a twist ending, where unlike other textbooks, don't show up until the last chapter.
I have not finished all of the exercises, and it may take me a few more years to fully understand all of the content, but learning the concepts of Linear Algebra will save you time and energy in the long run. If you learn the concepts of vector spaces correctly and fully, then you won't be confused by mediocre explanations of vectors that you might get from a standard engineering textbook. The only downside is that by reading this book, you will expect more rigor from the next proofs you read, and that is a high bar to live up to.
If you are new to higher level mathematics, it may take you all day to get through a few pages, especially since many new terms and notations are introduced. Also, the problems at the end of each chapter can be extremely difficult, but you can find worked-out solutions online since the book has become so popular among students lately. But all of the pieces are laid out, so you will know both why and how the solution is correct when you see it.
There is a twist ending, where unlike other textbooks, don't show up until the last chapter.
October 23, 2022
I thought this was too easy for a second course in Linear Algebra, but for someone taking this for the first time, it might be slightly too difficult. Not because the content is necessarily difficult, but Linear Algebra is usually taught with lots of numbers, and this book is clearly geared more for a pure math audience. There is a lot of computation skills that are not taught in this book, and I think it would be imperative for an undergraduate student to learn those computation skills in an introductory Linear Algebra course. I wrote solutions to this book at , and I did include some computation skills.
I thought it was a good refresher for Linear Algebra though, but a bit unsure where this book fits in terms of best use -- maybe as a concise book for instructors to plan out their lessons and guide a curriculum from?
I thought it was a good refresher for Linear Algebra though, but a bit unsure where this book fits in terms of best use -- maybe as a concise book for instructors to plan out their lessons and guide a curriculum from?
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