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.
Best Answer
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.