Regular expressions are a great example of bundling up some really neat and complex mathematical theory into a valuable interface. Linear algebra feels similar to me.
It always amazes me how given the appropriate field, so much math can be transformed into linear algebra. Even Möbius transformations on the complex plane w=(az+b)/(cz+d) can be turned into linear algebra.
Linear transformations preserve the structure of the space so you can keep applying them. It's not surprising that you can always find some "space-preserving" part of a problem and fold the rest (the "non-linear" structure) into transformations or the definition of the space itself.
That usually means the representation is getting close to the truth. Good interfaces have intrinsic value, which many result-focused people do not appreciate.