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> I think this is because we introduce a complicated way to calculate determinants and then we use determinants to calculate the eigenvalues? Yes, the determin
by mydogcanpurr 3y ago
> I think this is because we introduce a complicated way to calculate determinants and then we use determinants to calculate the eigenvalues?
Yes, the determinant should be taught and defined as the volume of the parallelepiped in n-dimensions defined by the columns of the given square matrix. This perspective makes it immediately obvious that the eigenvalues scale the parallelepiped in each of its dimensions (a basis of eigenvectors makes it even simpler). Of course the volume (determinant) must be the product of these scaling factors (eigenvalues)! Since algebra is too convenient for solving problems, this geometric intuition is often an afterthought if it's even taught at all.
- lupire 3y agoWhat trash math classes were you all in that didn't teach all of this?
- programjames 3y agoI think you first need to define "volume" as a bunch of simplices put together, or the (-1)^{...} term is unmotivated.
- defrost 3y agoAs anecdata this was taught in first year university mathematics for math, engineering, physics, chemistry, etc. students in 1981 in all three universities in Perth Western Australia aka "the most isolated city in the world" [1] It never occurred to me that geometric parallels would not be given in linear algebra courses. [1] https://about.soar.earth/blog-pages/how-the-worlds-most-isolated-city-became-the-city-of-light https://about.soar.earth/blog-pages/how-the-worlds-most-isol...