Hey #mathsky — What are your favorite motivations for change of basis? I'm already planning to parametrize an ellipse, but I'd love to hear about other quick and illuminating examples 😎
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Even in classical mechanics, the fact that physics is the same no matter what direction you look leads to orthogonal change of bases (and the fact that physics should be invariant under them).
Representing a recurrence relation as a companion matrix and finding the closed form solution for the nth element in a sequence is pretty damn cool. z transforms are eigendecompositions in disguise.
Converting to orthogonal basis polynomials is a favorite of mine. Basically just QR decomposition for polynomials, but it's also nice to manipulate the coefficients themselves, which won't typically be orthogonal (but they are in full SVD, which is very cool too)
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