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.