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> I think if I were in charge of presenting this material to students I'd do it by introducing the concept of memorylessness and by showing how good memorylessn
by exmadscientist 2y ago
> I think if I were in charge of presenting this material to students I'd do it by introducing the concept of memorylessness and by showing how good memorylessness is, how many wonderful things you can do with it. And then one day I'd be like, "well, it sure would be nice if we had any distributions like that!" and then whirl around with my piece of chalk to deliver the exciting news that we do. Exactly one, in fact.
Incidentally, this also goes for the determinant of a matrix. It's got a lot of neat and desirable properties, and it turns out to be the only thing that does. When it was finally taught to me this way, those weird algorithms we use to compute this seemingly-arbitrary number finally made sense. (And, in fact, this is the easiest way to prove that all those algorithms have to be computing the same seemingly-arbitrary number. Because the algorithms preserve the properties that define The Determinant, and The Determinant is the unique thing that preserves all of those properties, so must those algorithms all be computing The Determinant, no matter how different they might look.)
So I can vouch that this style of explanation really does work, at least for people like me.
- AgentMatt 2y agoCan you give some examples for algorithms which aren't obviously logically connected but use the determinant for its nice properties?
- mturmon 2y agoTwo ways to compute the determinant: 1) product of eigenvalues 2) cofactor expansion (e.g., https://textbooks.math.gatech.edu/ila/determinants-cofactors.html https://textbooks.math.gatech.edu/ila/determinants-cofactors...) The second is the one I was taught first, in the context of linear system solution, but it does not seem obvious that it would be related to the product of eigenvalues. It’s kind of a weird pile of arithmetic manipulations.
- jackthetab 2y agoAny sources that discuss this viewpoint wrt the determinant? Seems I'm still at the "seemingly-arbitrary number" stage.
- traes 2y agoThe entire 3blue1brown series[0] on linear algebra is well worth watching, it has really intuitive graphical explanations of a bunch of concepts. Here's the one on determinants in particular[1]. TL;DW the determinant represents how much you scale the area/volume/hypervolume (depending on dimension) of a shape by applying a matrix transformation to each point. [0] https://www.youtube.com/watch?v=fNk_zzaMoSs&list=PLZHQObOWTQDPD3MizzM2xVFitgF8hE_ab&index=2 https://www.youtube.com/watch?v=fNk_zzaMoSs&list=PLZHQObOWTQ... [1] https://www.youtube.com/watch?v=Ip3X9LOh2dk&list=PLZHQObOWTQDPD3MizzM2xVFitgF8hE_ab&index=6 https://www.youtube.com/watch?v=Ip3X9LOh2dk&list=PLZHQObOWTQ...
- exmadscientist 2y agoIf you like textbooks, try Section 1.3 of Artin's Algebra (find it at https://media.githubusercontent.com/media/storagelfs/books/main/Applied%20Mathematics/Algebra/Artin.%20Algebra.%201991.pdf https://media.githubusercontent.com/media/storagelfs/books/m... among others). Do be warned that Algebra is a... high-octane... text written for serious math students and can be... powerful.
- nerdponx 2y agoMatrix multiplication and the Gaussian distribution are also like that. A lot of things are like that. I really dislike that this approach is not a core tool for teaching in math.