D0TheMath's Shortform

Garrett Baker

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There is a mystery which many applied mathematicians have asked themselves: Why is linear algebra so over-powered?

An answer I like was given in Lloyd Trefethen's book An Applied Mathematician's Apology, in which he writes (my summary):

Everything in the real world is described fully by non-linear analysis. In order to make such systems simpler, we can linearize (differentiate) them, and use a first or second order approximation, and in order to represent them on a computer, we can discretize them, which turns analytic techniques into algebraic ones. Therefore we've turned our non-linear analysis into linear algebra.

Compare also the central conceit of QM /Koopmania. Take a classical nonlinear finite-dimensional system X described by a say a PDE. This is a dynamical system with evolution operator X -> X. Now look at the space H(X) of C/R-valued functions on the phase space of X. After completion we obtain an Hilbert space H. Now the evolution operator on X induces a map on H= H(X). We have now turned a finite-dimensional nonlinear problem into an infinite-dimensional linear problem.

3Rana Dexsin2y

That description is distinctly reminiscent of the rise of containerization in software.

3lemonhope2y

This is cool I never heard of this. There are many other exceptions of course. Particularly with "turning things on" (car starting, computer starting, etc)