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.