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Any sources that discuss this viewpoint wrt the determinant? Seems I'm still at the "seemingly-arbitrary number" stage.

If you like textbooks, try Section 1.3 of Artin's Algebra (find it at 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.

The 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=PLZHQObOWTQ...

[1] https://www.youtube.com/watch?v=Ip3X9LOh2dk&list=PLZHQObOWTQ...

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