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Is it problematic if we have many solutions since wouldn't mapping a larger space onto a smaller space still result in the same smaller space?


In general, is there an application-transparent way to deal with these problems? I would imagine either 1. the library/system provide validation checks, e.g. for non-parallel equations in the first case; 2. or provide corrections/signals after calculations.

Since it was mentioned that, sadly, the state of the common algorithms right now is to give random values when there are errors like above, why are the validation checks not used? Are they computationally expensive, e.g. I could imagine language support to simplify a batch of equations into a few... But even then, could the mathematical properties be preserved (i.e. validation checks done at the beginning can protect the entire batch)?


Would there only be one solution when we're mapping from a space to the same dimensional space, since mapping to a larger space would always result in not every point being "reached"?


What happens to our program/algorithm if the system we have yields no solutions? Do we get rid of any constrictions until the system isn't overdetermined or look for an approximation that minimizes error?