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.
Re:Linear algebra done right
by abecedarius on Monday August 17, 2026 @04:05PM
(#49336795)
This might come down to details of how you explain it: iirc Apostol took those basic moves (axioms) and calculated what the formula would have to be, rather than starting with a formula and checking that it has the properties of a signed volume.
But I don't know, it's been a very long time for me. It's good to have a variety of approaches to the subject.
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.
Re:Linear algebra done right
by impossiblefork on Tuesday August 18, 2026 @05:31AM
(#49343335)
Yes, but that doesn't prove that the determinant is actually the real signed volume multiplier.
I think this is what led me to feel unhappy about determinants when I was a first-year university student. You need to actually prove that the determinant is the volume of the N-dimensional parallelepiped, and the axiomatic proof doesn't do that.
So you need basically two extra lines after proving those things so that people can say "okay, the determinant eats ignores all input vector non-orthogonality so that it gives volume".
Agreed that it should help if you got to learn wedge products first (I didn't).