> I object on moral grounds to inverting a matrix with determinants!

The determinant of a linear map is the induced effect it has on volumes. So it makes sense that it appears when inverting a map: if the forward map scales volumes by detA the the inverse needs to scale them by 1/detA. It also makes sense as an invertibility criterion: you can invert a map iff it didn't collapse the space down to a lower dimension iff it doesn't reduce volumes to 0.

Of course this is presented completely opaquely at a low level with the even more opaque cofactor matrix stuff. So the trouble is that we really need to incorporate wedge products and some of the underlying geometry better at the lower level.

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tptacek 6 days ago | parent [-]

That first paragraph is more valuable than a unit of determinant matrix inversion problems.

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ndriscoll 6 days ago | parent [-]

I generally agree, though I think that's kind of a rejection of approaches like MA entirely (with the caveat that I've never used MA, so I'm assuming what it is based on discussions I've read): mathematics naturally gives you spaced repetition from new concepts building upon previous concepts, so calculation problems are mostly needed as a "check whether you are truly following along", and doing an entire unit of any type of calculation is somewhat pointless once you reach a certain maturity level unless there's some specific insight you're supposed to get from each example calculation.

	▲	tptacek 6 days ago \| parent [-]
		I see exactly what you're saying and I agree: the kind of understanding your exposition gives is different than the kind of upskilling Math Academy can promise. My only thing here is (a) there's value in both approaches and (b) everything I've read about self-teaching math is that you can't get fluency without grinding problem sets. I buy that you need both (and: I supplement MA with ChatGPT, which is pretty great at Socratically working through concepts), but also I cringe whenever people say "oh, you want to understand eigenvectors, just watch the 3blue1brown videos", because, those videos are great, and watching them in isolation gets you approximately 0% closer to being able to do (or appreciate) a PCA. I think a set of videos like Strang's 18.06 and Math Academy (but for arbitrary subjects, like Calc II or undergrad Stat) would be a pretty killer combination. Are a killer combination, is what I mean to say. But I've only been at it a few months!