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A lot of MSE posts asking for introductory books on Linear Algebra, linked here and here mention Sheldon Axler’s Linear Algebra Done Right.

I got my hands on a copy of the Second Edition of the text but noticed that the Preface to the Instructor and Preface to the Student start with “ You are probably about to teach a course that will give students their second exposure to linear algebra. During their first brush with the subject, your students probably worked with Euclidean spaces and matrices. In contrast, this course will emphasize abstract vector spaces and linear maps.“ (emphasis mine) and “You are probably about to begin your second exposure to linear al- gebra. Unlike your first brush with the subject, which probably empha- sized Euclidean spaces and matrices, we will focus on abstract vector spaces and linear maps... This book starts from the be- ginning of the subject, assuming no knowledge of linear algebra.” (emphasis mine) respectively.

The thing is, I have no idea of what a Euclidean space is. In our high school curriculum, we were taught vectors and matrices as separate concepts, though vectors did involve some determinants.
So in matrices we were taught things like matrix multiplication, adjoint, inverse, (involutory, idempotent, nilpotent, orthogonal) matrices, determinant evaluation with row and column operations, and how to solve systems of linear equations in three variables using adjoint method/determinant method on AX=C. (No eigenvectors)
While in vectors we were taught basically the geometrical aspect, and scalar and vector triple products. (No span/basis etc.)

So, my question is, will Axler’s book be well-suited to me? Or should I read up something more basic?
I also noted a few mentions of Hoffman and Kunze’s book, which, in this link is supposed to be for people of the level of completing Spivak. I have just started with Spivak (Basic Properties of Numbers), so should I stay away from it ?

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  • $\begingroup$ Read something else. Axler is trying to avoid matrices too artificially (in my opinion). $\endgroup$
    – Qi Zhu
    Oct 19, 2022 at 4:59
  • $\begingroup$ "Euclidean space" just means $\mathbb{R}^n$ (roughly). Why don't you try reading the beginning of Axler to see if you like his style? $\endgroup$
    – Qiaochu Yuan
    Oct 19, 2022 at 5:03
  • $\begingroup$ I suggest first studying chapter $(1)$ from Hoffman Kunze's book or similar chapter from Artin's book (this will help you in exercises in Axler's as some exercises will ask you to create some examples/counter-examples and that can be done with more ease if you understand matrix properties well) and then studying Axler. $\endgroup$
    – Koro
    Oct 19, 2022 at 5:51
  • $\begingroup$ The readership of linear algebra is too diverse. I don't think there is a perfect or near-perfect fit for everyone. If you have access to a book, just give it a cursory read and see if it suits your needs. $\endgroup$
    – user1551
    Oct 19, 2022 at 6:47

4 Answers 4

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I don't have any specific book to recommend but there are two levels to learning linear algebra.

In a first course, the book will start with systems of linear equations and end with an introduction to vector spaces.

Since linear algebra is the study of vector spaces, a second course, usually taken by math majors in college, starts with the definition of a vector space and goes from there.

In practice, the first course is usually taken along with calculus and is sometimes included in a multivariable calculus course. The second course is usually taken after the calculus sequence and for many math majors is the first course they take that requires proof writing.

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  • $\begingroup$ I am sorry that I left it out of my original post. We have been taught systems of linear equations in three variables, but only that. $\endgroup$
    – insipidintegrator
    Oct 19, 2022 at 5:11
  • $\begingroup$ @insipidintegrator Have you used matrices and row reduction to solve these systems? A first course usually introduces things like that. These courses tend to be more computational than rigorous. $\endgroup$
    – John Douma
    Oct 19, 2022 at 6:16
  • $\begingroup$ Yes I have, like depends on the matrix A, we take adj(A) if A is singular etc. $\endgroup$
    – insipidintegrator
    Oct 19, 2022 at 6:40
  • $\begingroup$ Have you been exposed to $\mathbb R^n$ for general $n$? Have you worked with spaces whose points are tuples of the form $(x_1,...x_n)$ where $n$ is not specified? If so, then you may be able to handle that book. A good rigorous introduction to linear algebra is worth owning even if you must defer its usage until you have had the prerequisites. $\endgroup$
    – John Douma
    Oct 19, 2022 at 8:04
  • $\begingroup$ No. I don’t even know what “space” is supposed to mean. $\mathbb R^3$ (3d geometry) is the best I have been taught. $\endgroup$
    – insipidintegrator
    Oct 19, 2022 at 8:23
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I am learning Linear Algebra from Axler's book at the moment and I think that it is a great book. The book is completely self contained and doesn't expect any prior knowledge of linear algebra. This is actually my first time learning linear algebra and I have only learnt it from Linear Algebra Done Right. I do go back and interpret the results in terms of matrices like what an introductory course might teach. Before picking up this book, all I knew was row reduction and nothing else.

The book does require that you be familiar with writing proofs and basic introductory logic(as would any other pure math book).

What I am trying to get at is that you will be fine. I have been learning from that book without any knowledge of introductory linear algebra. All I knew was row reduction. I have however always tried to interpret the results in terms of matrices. For example, a linear map being diagonaliazable and a matrix being diagonalizable have two different definitions but the underlying idea is the same for both of them.

If you have any further questions, feel free to ask!

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The thing is, I have no idea of what a Euclidean space is.

A Euclidean space is a multidimensional space with Euclidean geometry. Basically if you know what a dot product is you have done this.

So, my question is, will Axler’s book be well-suited to me? Or should I read up something more basic?

It's not possible for anyone except you yourself to answer that. If you are doing a formal course then you should consult the instructor about what text to use. If you are not doing a formal course you can do whatever you like, so decide what your goals are and then pick the book you want to study. Axler is making the fourth edition of his book free to download, so from next year you can legally download a copy of the new edition, and try it, and see if it suits you.

I also noted a few mentions of Hoffman and Kunze’s book, which, in this link is supposed to be for people of the level of completing Spivak. I have just started with Spivak (Basic Properties of Numbers), so should I stay away from it ?

I have no opinion about whether you should "stay away" from it, but as far as I know Hoffman and Kunze's book is roughly the same level as Axler's book, but Hoffman and Kunze goes slightly further.

I can't imagine why anyone would warn anyone else to "stay away" from a mathematics book. I never heard instructors at my old university say things like "Stay away from 'Functional Analysis' by Rudin". It's not like you will end up joining a drug-dealing motorcycle gang on the basis of using the wrong maths textbook.

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  • $\begingroup$ Thanks for your answer (and the last line :))! $\endgroup$
    – insipidintegrator
    Oct 19, 2022 at 9:27
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Here are some (GOOD) linear algebra books:

⋆ "Linear Algebra" by "Friedberg, Insel and Spence".

⋆ "Linear Algebra" by "Hoffman and Kunze".

⋆ "Linear Algebra Done Wrong" by "Sergei Treil".

⋆ "Linear Algebra Done Right" by "Sheldon Axler".

⋆ "Introduction to Linear Algebra" by "Serge Lang".

⋆ "Linear Algebra" by "Serge Lang". (different from the previous one).

For me, as a beginner, [self-study], I took a look at all of the books mentioned above. (I listed them in order, starting from the best (in my opinion)).

However, there is a better book than these $6$ books, which is:

$\color{blue}{\text{Linear algebra and its applications}}$ by $\color{blue}{\text{David C. Lay}}$

For me, the best introductory book, amazing for beginners.

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    $\begingroup$ I know a linear algebra text co-authored by Hoffman and Kunze, but who is Kunze Hoffman? $\endgroup$
    – user1551
    Oct 19, 2022 at 6:42
  • $\begingroup$ @user1551 Corrected. Thanks! $\endgroup$
    – Hussain-Alqatari
    Oct 19, 2022 at 8:59
  • $\begingroup$ Hi, are all these books the same level, or is there an order of difficulty in which I should study all of them? $\endgroup$
    – insipidintegrator
    Oct 19, 2022 at 9:29

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