Comment by throwawaymath

Because in fairness, it does do so many things right. It's kind of controversial[1] because of Axler's notation and de-emphasis on determinants. But it's an ideal book to give a student who has had a first, computational course in linear algebra (i.e. a course entirely devoted to solving systems of linear equations and drilling matrices).

I think Hoffman & Kunze is really the best textbook for the arrangement of material I'm talking about, but it's outdated at this point and its exposition is strictly less pedagogical than Axler. My vision for the "perfect" linear algebra textbook would be loosely based on Hoffman & Kunze, but replacing a lot of the theoretical exposition with Axler's exposition from Linear Algebra Done Right, then keeping (and even expanding) the mterial on Gaussian elimination and matrices. You could get a solid three-semester sequence of computational and proof-based linear algebra out of such a textbook, and you'd be all set to go to something grad-level like Roman.

Unfortunately it's also hard to judge a lot of linear algebra textbooks because the bifurcation between proof-based and computation-based linear algebra isn't as clean as it is in calculus. Calculus has its own separate course sequence in real/complex analysis, whereas linear algebra doesn't have a distinct name for the more rigorous coverage of the subject. So you have a lot of textbooks which choose one or the other thing, which then results in an over-emphasis on computational stuff in the first course and a complete de-emphasis on the motivation in the second course. When you are learning these back to back in distinct courses that's usually fine, but many first courses actually jump right to Axler despite the words of caution he writes in his own preface.

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1. Noam Elkies uses it when he teaches Harvard's Math 55a. See his list of comments and errata for the book: http://www.math.harvard.edu/~elkies/M55a.17/index.html