Let be the set of vectors in of the form where . Find a subspace such that .

Attempt at a solution:

Now it is easy to prove that is of dimension 2 with a basis as . All we need is a 2 dimensional subspace such that the basis of both have no element in common.

By trial, with a basis as satisfies the given condition. My question is, is there a systematic method to solve such questions? What do I do if the basis of the involved vector spaces are so high that trial and error don't work?

Thanks!