3 ms·

Determinants always fascinated me, partly (completely?) because of the cool sounding name. I always wanted to get intuition for them because their formula is so

• ndriscoll 4y ago
  It's unfortunate that more theory makes determinants really intuitive, but most people don't get there. The upshot is that there's a thing called the exterior algebra which is relatively simple to define and calculate with, and essentially encodes the notions of areas, volumes, 4-volumes, etc. over a space. Every linear map on a space uniquely specifies a map on the exterior algebra (in fact this is a functor, which makes calculations easier), and the determinant of a linear map is then just the corresponding exterior algebra map evaluated on the unit n-volume. The minors are the exterior algebra map evaluated on the unit volume for various subspaces. The rules for the exterior algebra and the fact that this is a functor let you learn a couple simple rules to simplify expressions into a standard form, and then you just apply those rules mechanically and don't have to remember minus signs or what multiplies with what. It becomes a process that requires no thought. It's sort of like how once you learn how to deal with complex exponentials, you can forget pretty much every rule from trigonometry, making it entirely pointless to memorize those rules.
  • green_on_black 4y ago
    It's unfortunate that lower level "math" is so banally taught (i.e. calc and below), at least in the US.
• lambdaxymox 4y ago
  One of the great parts about linear algebra is that there is almost always a simple geometric idea underneath. The geometric picture underneath is one of the things that keeps me in awe of the subject despite its seeming simplicity, and I keep getting something out of it every time I come back to it. It's a bummer since finite-dimensional linear algebra is one of a handful of mathematics topics where one can answer all the questions posed at the beginning of a course in it by the end of a course in it, so it is a pretty self-contained topic. After learning e.g. exterior algebra (differential forms), Clifford algebra (geometric algebra in these parts), and so on, the geometric picture of the determinant as the size of an oriented volume makes deriving the algebraic formula super duper slick. Like in Clifford algebra, the formula can be proven in two or three lines. It's unfortunate that it seems like e.g. exterior algebra never get introduced sooner in the pedagogy of linear algebra or multivariable calculus because when used right they make the underlying ideas shine through beautifully. It's a bummer since exterior algebra is much simpler than it looks, though like many things in mathematics, it's takes a lot of work to make that simple idea rigorous. But unfortunately algebra in general given it's abstract nature can absolutely lobotomize the real deal geometric ideas underneath a lot of this stuff when used poorly.
  • flebron 4y ago
    The trifecta of: * There's almost always a simple geometric intuition, and low-dimensional intuition can get you quite far even in high dimensional cases. * You can surprisingly often get by with closing your eyes and saying "my problem is linear" three times. See: All of neural networks. * Linear problems have practically all nice properties you could ever ask of any function. Has made linear algebra by far the most bang/buck mathematics topic I've studied in my life. Close behind is asymptotic analysis.
• edflsafoiewq 4y ago
  A shocking number of people are apparently taught the determinant without learning the geometric interpretation. It's educational malpractice.
• IIAOPSW 4y ago
  watch his fourier trick. you're welcome