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Note that "done right" means done with Axler's completely subjective and unusual hatred of determinants, chronicled here [0]. It is in no way "done right" in so

• renyicircle 1mo ago
  Yeah I feel like this "done right" part is responsible for most of the popularity of this book. Makes the reader think they've been learning it wrong. Kind of like these clickbait videos "you've been folding your laundry wrong your whole life!" or whatever
  • inigyou 1mo ago
    Or "this one simple trick makes Big Linear Algebra hate you"
  • fn-mote 1mo ago
    The popularity of Linear Algebra Done Right (IMO) comes from two sources: 1. Excellent exercises. Challenging. Really make you put the concepts together. 2. Good, opinionated pedagogy. If you agree with the philosophy (among other things, determinants are not a beginner tool), the explanations are good. LADR is hardly the only book to eschew determinants for a long time. IIRC Lang takes a similar approach, but is not as digestible.
• ak_111 1mo ago
  Determinants are easy to use but very hardly to grasp intuitively, this is not a minority point of view. See countless of StackOverflow questions begging for a conceptual exposition of determinants. The easiest conceptual handle is geometric: volume expansion, but seeing how this is related to the combinatorial sum over all permutations, or how those two point of views are related to the algebraic one (that a set of equations having a solution or not), is not easy to see even in the 2D case. I won't be surprised if math professors don't have this issue like you said (especially if someone is comfortable with wedge products), but the vast majority of newcomers who are interested in understanding why something works rather than just how to use it struggle all the time with determinants.
  • traes 1mo ago
    I think the volume explanation is one of the most intuitive pieces of math in existence, personally! Uninvertibility of a tranformation corresponds to a volume of zero because the transformation must squish two dimensions together, leaving them impossible to differentiate, det(AB) = det(A)det(B) because applying two transformations applies their scaling successively, det(A^-1) = 1/det(A) because you have to undo the scaling to invert a transformation etc. I don't think the permutation definition is even strictly necessary; if I recall correctly Linear Algebra Done Wrong defines the determinant in terms of its geometric definition and develops its formula from the properties it must have. I think that the concept is well worth the investment of initial confusion. Axler disagrees, however.
    • ak_111 1mo ago
      But if you see the geometric view then the permutation sum, for certain obsessive learners they want to see why they are equivalent and thats the hard part.
  • abecedarius 1mo ago
    It seems to me more like determinants are often badly motivated. The way I remember it from Apostol's Calculus was like "a volume multiplier would be very useful; it needs to be a signed volume for linearity, which implies antisymmetry. Here are axioms collecting these requirements. They're uniquely satisfied by the determinant. Proof: ..." Agreed that it should help if you got to learn wedge products first (I didn't).
    • ak_111 1mo ago
      This is indeed a good concise description of how to connect the two, but even making peace with this, there is something still magical in how the permutations in the sum cancel neatly (in an inclusion-exclusion kind of a way) to get the volume.
    • fn-mote 1mo ago
      I found that “properties of the determinant uniquely determine this formula that I guessed” approach to determinants to be extremely unconvincing when I was learning linear algebra.
• tanderson92 1mo ago
  Maybe you need to speak to better math professors.