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I think the key is that linear algebra textbooks do not tell students straight away that matrices are encodings of linear maps. Hence a square matrix is an enco
by nextos 6y ago
I think the key is that linear algebra textbooks do not tell students straight away that matrices are encodings of linear maps. Hence a square matrix is an encoding of a map that preserves dimensions.
Axler, and a few others, say this straight away in the intro. This makes things much simpler to understand instead of developing linear algebra historically from systems of linear equations.
- dnautics 6y agoI happened to be the other way, I learned first that matrices were linear transformations and, being thick-headed, it took me a really long time to understand that matrices could also be used as "dumb excel spreadsheets". there's a fantastic historiography of linear algebra and why we're stuck in this situation here: https://www.youtube.com/watch?v=C2RO34b_oPM https://www.youtube.com/watch?v=C2RO34b_oPM
- nextos 6y agoA worse situation is that of statistics, where many textbooks insist on using approximations to the normal which obscure concepts and are not needed in the age of computer-based inference [1]. Calculus is also a bit confusing as for historical reasons we use a lot of notation from Newton-Leibniz informal infinitesimal approach, but we try to teach Bolzano-Weirstrass formal analysis, ending up with a confusing potpurri. [1] https://escholarship.org/uc/item/6hb3k0nz https://escholarship.org/uc/item/6hb3k0nz
- dnautics 6y agoI didn't mind the issue in calculus. The first thing we did in analysis after being introduced the Bolzano weierstrass form was to prove that it was equivalent to epsilon delta proofs.