Please enlighten me on the following issue. Let's say I'm supposed to numerically solve a linear system Ax = b. But since the space is too large I can only work on a truncated version of b, call it b'. b' contains the kets with big weights and in fact the weight of kets in b decays exponentially, under appropriate reordering of basis kets. One consequence of the truncation is that A itself should also be truncated and become A'. Than, I'm contemplating on how does the solution of A'*y=b' differ from x. How to analyze the overlap between $x$ and $y$? I made some attempts but failed. One possible way might be to decompose the whole space in to the direct sum of two subspaces.
Note: in my case matrix element can be calculated when it's being accessed so no need to save them.