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But I had another question. What about all the possible vectors x for a given b, such that Ax=b ? I get that this wouldn't always work because say, some systems are inconsistent and have no solutions so it would be empty in this case. It seems like this wouldn't necessarily be a subspace because it wouldn't necessarily be closed under addition (just because x is a solution to Ax=b doesn't mean A(2x)=b is a solution). But sometimes it would be a subspace. Because for example, the nullspace would just be the special case where b = 0, the 0 vector, and we know the nullspace is a subspace. Is this "solution space" given special importance? If so what's it called? (I keep calling it a solution space and I just don't know what the proper name of it is)