4 ms·

My two favorites are probably the hedgehog theorem (= number of discontinuities if you were to comb a hedgehog) https://clever-geek.imtqy.com/articles/784617/i

My two favorites are probably the hedgehog theorem (= number of discontinuities if you were to comb a hedgehog) https://clever-geek.imtqy.com/articles/784617/index.html https://clever-geek.imtqy.com/articles/784617/index.html And that the determinant is the volume of the parallelepiped spanned by the vectors: https://math.libretexts.org/Bookshelves/Linear_Algebra/Interactive_Linear_Algebra_%28Margalit_and_Rabinoff%29/04%3A_Determinants/4.03%3A_Determinants_and_Volumes https://math.libretexts.org/Bookshelves/Linear_Algebra/Inter... It's quite stunning how far one can get with visualizations. I think it's an indirect discovery of Euler though (who did many visual proofs) that visualizations are really just that and to do proofs textual reasoning is necessary.

roenxi 5y ago
> And that the determinant is the volume of the parallelepiped spanned by the vectors: It always bugs me when people throw that out because it doesn't underscore why that fact is useful. I saw this in my linear algebrea textbook as a practice problem and never got it until 3Blue1Brown spelled it out. It is quite likely that the determinant was originally defined as the volume of the parallelepiped spanned by the vectors, when the mathematicians were working out linear systems. Then they threw out the intuition and kept the formula for the textbooks. The point is that because the transform is linear, the volume change of the unit square as it is stretched into a parallelepiped is the same transform as will be experienced by any volume in the space, which is quite possibly how the determinant was first identified. So the perspective where this is an important fact is that the determinant is the volume change of unit volume -> parallelepiped (or higher dimensional equivalent) when a matrix is viewed as a spatial transform, then that turns out to be a remarkably fundamental measure that characterises a lot of useful properties about the entire matrix.
- kirsebaer 5y ago
  Yeah, this should be emphasized in any linear algebra course as a primary way to think about determinants. Otherwise the determinant seems like just some math formula to memorize that keeps appearing for some reason in other math formulas.

roenxi 5y ago

> And that the determinant is the volume of the parallelepiped spanned by the vectors: It always bugs me when people throw that out because it doesn't underscore why that fact is useful. I saw this in my linear algebrea textbook as a practice problem and never got it until 3Blue1Brown spelled it out. It is quite likely that the determinant was originally defined as the volume of the parallelepiped spanned by the vectors, when the mathematicians were working out linear systems. Then they threw out the intuition and kept the formula for the textbooks. The point is that because the transform is linear, the volume change of the unit square as it is stretched into a parallelepiped is the same transform as will be experienced by any volume in the space, which is quite possibly how the determinant was first identified. So the perspective where this is an important fact is that the determinant is the volume change of unit volume -> parallelepiped (or higher dimensional equivalent) when a matrix is viewed as a spatial transform, then that turns out to be a remarkably fundamental measure that characterises a lot of useful properties about the entire matrix.

kirsebaer 5y ago
Yeah, this should be emphasized in any linear algebra course as a primary way to think about determinants. Otherwise the determinant seems like just some math formula to memorize that keeps appearing for some reason in other math formulas.