because any vector of the form (a,0) = a(1,0) satisfies x_{2}= 0, so this is an eigenvector iff a ≠ 0.

yes this is the same as being a "free variable", any choice of x_{1}(freely made) will do (but of course, the fact that an eigenvector must be non-zero means we can't choose x_{1}= 0, although (0,0) *is* in the eigenspace E_{λ1}).