A lot of the discordance in this thread can be boil down to "different people having different goals when reading". As for Linear Algebra, due to its wide relevance, it is hard for one book to accommodate to all needs of people from different fields and levels of mathematical maturity. I would say there are 2 main reasons for one to study LA:
- To understand and use its toolbox in practice: in fields like computer graphics, engineering, statistics, data analysis, modeling, etc.
- To understand and use its toolbox in other fields of (pure) math: single, multivariable, and functional analysis, abstract algebra, etc.
(Sometimes the latter goal is coupled with a secondary goal of introducing the student to mathematical proof. However, for the self-learners, I don't think using a LA book is good for learning how to write proof for the first time.)
Unlike in calculus where Spivak serves as the one-size-fits-all, readers should identify their reasons before picking an LA book. I have not much experience with the first goal. Strang's book seems quite popular, and given the title, I would say it fits this goal. Axler's book falls squarely into the second goal, and I wager it's a good book for this purpose if not for two things: his avoidance of determinants and the dearth of any computational aspects or applications (yes, even theoretical LA books ought to be grounded in reality). There are three books that, in my opinion, do better than Axler here:
- "Linear Algebra" by Stephen H. Friedberg, Arnold J. Insel, and Lawrence E. Spence: dry, does not instill in me the sense of excitement that Axler does. However, it is comprehensive and rigorous, with a decent amount of explanation and a good selection of exercises. Application aspects are present but quite disparate and incoherent. This is the old and long-standing recommendation.
- "Linear Algebra" by Elizabeth S. Meckes and Mark W. Meckes: a new book that somehow manages to combine the material of Friedberg and the eloquence of Axler. The rigor and exercises are comparable to Friedberg, but the applications are presented much better here.
- "Linear Algebra" by Sterling K. Berberian: a little bit more abstract than the other two, but the exposition is still very clear. It connects LA to other fields of math and gives plenty of examples and motivations (shocking I know). The exercises department is good.
Honorable mentions to the classics of Serge Lang and Paul Halmos. They are concise and, as with the rest of their works, wonderfully written. For "Linear Algebra Done Wrong", well, I feel like it's a stripped down Friedberg: shorter but not better. (The proof of the multiplicativity of determinant is just some hand-waving at "a lucky coincidence".)
- To understand and use its toolbox in practice: in fields like computer graphics, engineering, statistics, data analysis, modeling, etc.
- To understand and use its toolbox in other fields of (pure) math: single, multivariable, and functional analysis, abstract algebra, etc.
(Sometimes the latter goal is coupled with a secondary goal of introducing the student to mathematical proof. However, for the self-learners, I don't think using a LA book is good for learning how to write proof for the first time.)
Unlike in calculus where Spivak serves as the one-size-fits-all, readers should identify their reasons before picking an LA book. I have not much experience with the first goal. Strang's book seems quite popular, and given the title, I would say it fits this goal. Axler's book falls squarely into the second goal, and I wager it's a good book for this purpose if not for two things: his avoidance of determinants and the dearth of any computational aspects or applications (yes, even theoretical LA books ought to be grounded in reality). There are three books that, in my opinion, do better than Axler here:
- "Linear Algebra" by Stephen H. Friedberg, Arnold J. Insel, and Lawrence E. Spence: dry, does not instill in me the sense of excitement that Axler does. However, it is comprehensive and rigorous, with a decent amount of explanation and a good selection of exercises. Application aspects are present but quite disparate and incoherent. This is the old and long-standing recommendation.
- "Linear Algebra" by Elizabeth S. Meckes and Mark W. Meckes: a new book that somehow manages to combine the material of Friedberg and the eloquence of Axler. The rigor and exercises are comparable to Friedberg, but the applications are presented much better here.
- "Linear Algebra" by Sterling K. Berberian: a little bit more abstract than the other two, but the exposition is still very clear. It connects LA to other fields of math and gives plenty of examples and motivations (shocking I know). The exercises department is good.
Honorable mentions to the classics of Serge Lang and Paul Halmos. They are concise and, as with the rest of their works, wonderfully written. For "Linear Algebra Done Wrong", well, I feel like it's a stripped down Friedberg: shorter but not better. (The proof of the multiplicativity of determinant is just some hand-waving at "a lucky coincidence".)