Projection methods for oversized linear algebra problems

Everybody who has some experience in doing mathematics knows, that dimensional reduction and projection are useful tools to confront problems that are too complicated to solve without any simplification. Who hasn’t, occasionally, but notwithstanding timidly, suggested that perhaps it would be a good idea to study the simple one-dimensional case first before trying to understand the real-world three-dimensional problem? Apparently, it is a wide-spread faith
that such simplifications will not damage the essential mathematical or physical truth that is hidden in the original problem. But is this faith founded? Regardless of the answer, one should realize that in many applications there is no plausible alternative, so it would be unfair to judge too harshly on those who solve reduced problems and, with due mathematical care, formulate interesting and strong theorems and hypotheses on the full problem. Among them are the people from the field of numerical linear algebra.