&#62; Many practical problems - optimization, differential equations, signal processing, etc. - boil down to solving linear systems, even when the original problems are non-linear. Finite element software, for example, spends nearly all its time solving linear equations.<p>If I remember correctly, it's not that all these problems are actually sets of linear sub-problems. We just are a lot more apt to think in terms of linear equations, making it so that we tend to break down any problem we have to a linear problem. If done incorrectly of course this has dire consequences to your results.<p>This is similar to how in physics we try to reduce or approximate every problem to a harmonic oscillator, because everything else is just too damn complicated to actually solve.