that is not correct! the dimesnion of the eigenspace in this example is 1 not 0. the dimesnion of the eigenspace corresponding to an eigenvalue is never 0. it's always at least 1.

the eigenspace corresponding to the eigenvalue of a transformation is the solution set of for example, in the above example, you need to solve

because so we'll have which gives us and can be anything. so the eigenspace corresponding to is the set which is

obviously generated by so {(1,0)} is a basis for the eigenspace and thus the dimension of the eigenspace is 1. i think you can now answer your second question yourself.