For range, I would just have to find a vector v such that L(v)=w, so w is in the range of L. So a is not in kernel L but b is and c is in range L but d is not? For the range, I set up a 2x3 matrix with the vector in question as column 3 then found rref. Is this correct? d would not be in range because in the reduced matrix -1 does not equal 0.
Notice that .
Thus, to be in the range it needs to be a linear combination of .
There are of course many other ways to solve this problem, I just happen to like the approach that uses linear combinations. I looks like that approach uses least amount of computation since these are just matrices.