Pretty much anything in classical linear algebra can be abstracted to something more powerful by taking a swig of the Axiom of Choice.
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Replying to @EmilyGorcenski
A great example is PCA. It's often viewed as a dimensionality reduction technique, where we construct an orthonormal basis from data.
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Replying to @EmilyGorcenski
But orthonormalization exists in infinite dimensional spaces, too, and so with the AoC we could choose an infinite basis instead.
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Replying to @EmilyGorcenski
So I could pick, oh, a family of polynomials as the basis
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