There are actually two eigenvectors 1/sqrt(2) {1,1} and 1/sqrt(2) {1,-1}, corresonding to eigenvalues 1.5 and 4.5.

{1,1} doesn't span R^{2}. It spans the line y = x. Likewise {1,-1} spans the line y = -x

together these two vectors span R^{2}

I'm not sure I've answered your question.