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I am a self-learner majoring in computer science, and I have learnt an introductory linear algebra course which is not rigorous. However, I really love pure math. So, I start to self learn fundamental courses of pure math such as advanced calculus and linear algebra. I find that many books and lecture notes I find through googling "linear algebra lecture notes/books pdf" are not rigorous. I find sergei treil's $\it{Linear}$ $\it{algebra}$ $\it{done}$ $\it{wrong}$ and axler's $\it{Linear}$ $\it{algebra}$ $\it{done}$ $\it{right}$ are rigorous, however I find that the former is not complete, and he didn't mention some standard contents in linear algebra such as quotient space, invariant space and so on, and the latter I find the complex numbers' definition have some problems which made it not rigorous. So, I give up the two books.

After googling more things about linear algebra, I find that artin's $\it{Algebra}$ is very good, however, the linear algebra relative contents are not complete, he focus more on absract algebra.

So, now, I want someone can recommend books or lecture notes which have complete linear algebra contents and natrually combined it with abstract algebra. I find that knapp's $\it{Basic}$ $\it{Algebra}$ is very suitable for my expectation, but I think it's too hard for me at current stage. Hope that someone can recommend books like knapp's but simpler and more complete than it.

Plus:Answer to @Chubby Chef: My reply is a little long, so I place it here. Axler's book's chapter 1 Example 1.2 make me confused. In fact, he didn't define multiplication operation on real numbers between complex numbers, though he states that $a \in \mathbb{R}$ should be identify with $a+0i$, however, I think that he should also prove that this kind of "equality" can keep operaters(add,multiply) in complex numbers. Additionally, in his definition 1.16, he states that $-x=(-x_1,-x_2,...,-x_n)$, however, he should prove that the additive inverse of $x$ is unique, then we can say $-x$ is like that formula. Though I admit that this book is really great, the inaccuracy at beginning chapter make me a little uncomfortable.

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    $\begingroup$ Can you elaborate a little bit on what your problem with Axler’s book is? It’s perfectly rigorous (and, I would argue, pretty good!) $\endgroup$ Apr 1 at 10:31
  • $\begingroup$ I recommend Paolo Aluffi 'Algebra Chapter 0'. It's very complete regarding the topics you want to learn about fairly detailed. $\endgroup$
    – S.Farr
    Apr 1 at 11:03
  • $\begingroup$ What coverage is considered complete? I think most linear algebra texts more or less omit some topics that are covered in other textbooks. Textbooks written for mathematics majors tend to omit matrix decompositions such as QR or SVD, while textbooks written for engineers and scientists tend to omit topics about elementary divisors, rational canonical form and smith normal form. $\endgroup$
    – user1551
    Apr 1 at 11:18
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    $\begingroup$ These nitpicks with Axler are a silly reason to give up on an excellent book. Authors don’t always prove things that are obvious. Fill these infinitesimal gaps yourself and carry on! $\endgroup$ Apr 1 at 13:05
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    $\begingroup$ @KevinArlin I think you are right. Maybe I have to go on with Axler's book. In his preface, he said that readers should read a single page for at least an hour, maybe it is due to these gaps, and we have to spend at least an hour to find the gaps and try to fill them up. $\endgroup$
    – UESTCfresh
    Apr 1 at 14:13
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For an abstract algebra textbook that covers also typical (non-numerical) linear algebra topics, you may try Cohn's Classic Algebra (Mathematical Gazette review). I haven't read Knapp or Cohn carefully, but I think Cohn goes deeper in linear algebra than Knapp does. However, Cohn doesn't discuss any matrix decomposition, if I remember correctly.

For a linear algebra textbook whose treatment is more algebraic, you may try Berberian's Linear Algebra (MAA review). Someone recommended this book to me on this site before and I have read it once from cover to cover. I remember that I quite liked it, but I don't remember why I liked it. Most introductory texts discuss matrices over some fields, but Berberian also discusses matrices over principal ideal rings. It has a brief discussion of multilinear algebra, but the coverage is not strong. Also, although it includes some abstract algebra topics (such as factorisations over integral domains), this is a linear algebra text. So, don't expect yourself to learn abstract algebra from it.

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Just a thought, I noticed that in the Dover series of paperbacks, there's one entitled Linear Algebra and Group Theory. So this may be the sort of thing you are looking for. But again all I have is the title (it caught my eye): I haven't looked at it.

They're two of my favorite subjects.

Later when I get a chance I will look up the author for you.

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