Actually, the problem is not that there is not enough information but that what you give is contradictory. You say T is from to but then you say T((1, 2))= (0, 0) which is a transformation from to .
Is the subspace the set of all linear transformation from to such that T((1,2))= (0,0,0)? Or did you mean that T is from to ?
Assuming you meant the former, any linear transformation from to can be written as a matrix:
and your condition is
so you must have a+ 2b= 0, c+ 2d= 0, and e+ 2f= 0. Those, of course, give a= -2b, c= -2d, and e= -2f. The matrix can be written
If you mean to just follow the same ideas.