I know this might be trivial, but when practicing for exam, I usually write the "inverse" values of the cheat sheet, and want to make sure I'm not making a mistake. Is the eigenspace 1,-1 the same as -1,1?
I think your difficulty is your "notation" which is so terse as to be confusing. 'The eigenspace 1, -1' makes no sense to me. If you mean "the subspace spanned by eigenvectors of linear operator A corresponding to eigenvalues 1 and -1" then it should be clear that the order in which you mention the eigenvalues is irrelevant. (I started to write "the subspace of all eigenvectors of linear operator A corresponding to eigenvalues 1 and -1" but you understand that that set is NOT a subspace, right?)
To put it a bit differently, an eigenspace is a (vector ) subspace , and {1,-1} is not, at least not in any way I' familiar with. Did you mean the eigenspaces associated to each of these eigenvectors?
If your notation 1, -1 is intended to mean the vector <1, -1> in R^{2}, then yes, the space spanned by the eigenvector <1, -1> is the same as that spanned by the vector <-1, 1>. Both vectors lie along the line y = -x but point in opposite directions.