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## Main Question or Discussion Point

When studying linear algebra when encountering a system Ax=b, I always read of the fundamental subspaces of A: N (the null space, all solutions x of Ax=0), the column or domain space of A: (the space spanned by the columns of A, or in other words, all possible b for Ax=b), the row space (the space spanned by the rows of A. I have a harder time wrapping my head around this one).

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)

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)