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  • 14
    $\begingroup$ The problem with Axler is that he tries to avoid algebra at all cost - when it makes things easier and when it makes thing harder. Sometimes much harder, actually. Even if one goes into physics and never has to work over a ring different from R and C, one will realize one day that in order to compute the characteristic polynomial one doesn't have to bring the matrix in upper triangular form (no joke, this is how Axler defines the characteristic polynomial), and that often, the characteristic polynomial matters and upper diagonalization doesn't. $\endgroup$ Commented Mar 10, 2010 at 18:50
  • 12
    $\begingroup$ For someone who plans to work in algebra or algebraic geometry, linear algebra learnt from Axler is mostly wasted time. I don't understand what he has against the notion of determinant; this notion (with the sum-over-permutations formula that he seems to hate) comes out straightforwardly if one tries to apply Gaussian elimination to a general systems where the coefficients of the system are variables. $\endgroup$ Commented Mar 10, 2010 at 18:53
  • 4
    $\begingroup$ Maybe not everyone is going into algebraic geometry? I think that a lot of the proofs in Axler suggest the right way to think about things in functional analysis. Maybe it was just that my first linear algebra course was totally computational and had very little motivation, and my second exposure to linear algebra out of Axler showed me that there were actually some beautiful ideas there. $\endgroup$ Commented Mar 10, 2010 at 19:21
  • 12
    $\begingroup$ From the description given (starts with a $\textit{review}$ of matrices and fields, uses modules and exterior algebra, etc), it is patently obvious that this book (HK) is unsuitable to people without abstract algebra under their belt. Quite a bit of mathematical maturity that you cannot reasonably expect from non-math majors with or without prior proof experience is required as well. $\endgroup$ Commented May 23, 2010 at 1:31
  • 6
    $\begingroup$ @Harry: Bear in mind that what works for $\textit{you}$ (and other math majors at UM), doesn't necessarily work for others. And you have just confirmed that you were comfortable with abstract algebra, at the level higher than most non-majors ever see, before starting out. Learning composition of morphisms without being able to multiply matrices is $\textit{truly}$ pointless. One imperfect analogy: it's possible to learn AG from EGA or Harstshorne ("it has been done"), but as first books they are nowhere near Shafarevich, Cox-Little-O'Shea, Reid, Mumford, and any number of other texts. $\endgroup$ Commented May 23, 2010 at 6:49