boutiquemathsDec 8, 2022
The difference btw "change of coordinate system" and "change of basis" has always puzzled me. There is a great subtle answer here (what? the plane is not automatically a vector space? 🤯 ). Maybe trying to equate coordinate systems and bases is a category error?
https://www.quora.com/Does-polar-coordinate-have-basis-vectors?share=1
Does polar coordinate have basis vectors?

Answer (1 of 3): Polar coordinates are not some linear combination of basis vectors. Polar coordinates (and Cartesian coordinates) are ways of mapping pairs of numbers to points on a plane. The plane, in this case, is not a vector space, so there are not necessarily any basis vectors. It is pos...

Quora
Show threadAleathaDec 9, 2022
@boutiquemaths I don’t think it’s a category error, no, it just takes you out of linear algebra. W polar you still represent a point as a combination of coefficients on basis functions, just like Cartesian spaces (linear bases)… 1/n
Show threadAleatha
@boutiquemaths but also Fourier transforms or wavelets (non-linear bases), or deep learning (which is basically finding an optimal combination of non-linear functions to represent data. Remove the nonlinear activation function & DL is a Rube Goldberg machine for finding eigenvectors.) 2/2
Dec 9, 2022 at 5:16amWeb