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It's more that in linear algebra, we have "the nicest theorems we can hope for", so there's no "nasty surprises" lurking around. This is qualitatively different
by bollu 6y ago
It's more that in linear algebra, we have "the nicest theorems we can hope for", so there's no "nasty surprises" lurking around. This is qualitatively different from pretty much all other algebraic structures one studies, where there is some amount of nastiness which we attempt to control.
- Koshkin 6y agoIndeed, “linear” and “nice” are almost synonymous in mathematics and, especially, in its applications. (For example, the entire differential calculus is based on the idea of approximating arbitrary “nice” functions with linear maps.)
- eru 6y agoThere's also lots of clever tricks to make things more linear. There's things like linear approximations. Or, my favourite, picking a clever semi-ring in which your problem is linear. See eg the analysis of Google's BBR TCP variant. Eg https://labs.xjtudlc.com/labs/wldmt1/papers/Traffic%20management%20congestion%20control,%20router%20mechanisms%20and%20pricing/BH00.pdf https://labs.xjtudlc.com/labs/wldmt1/papers/Traffic%20manage... Or 'Linear regression over the max-plus semiring:algorithms and applications' (https://arxiv.org/pdf/1712.03499.pdf https://arxiv.org/pdf/1712.03499.pdf)