I think we need a specific example. But I cannot find one so I am trying to work through the logic. So to my understanding, we must map vectors of the form (x,y,0) to vectors that are a linear combination of (1,1,1) and (1,2,3).
So we can say v $\in(x,y,0)$ and let v = $\sum a_ie_i$ where $e_i$ is the standard basis vectors of R^3. Now it follows by the property of a linear map that it is is completely determined by T($e_i$). So it makes sense to write v such that T(v) => $\sum a_i$T($e_i$). We are saying this must equal some vector that is span of (1,1,1) and (1,2,3).
But at this point I am having trouble because I am not able to define T($e_i$). Well, we know this is equal to some vector in the set spanned by (1,1,1) and (1,2,3). We know (1,1,1) and (1,2,3) forms the rest of the vectors so would T($e_1$) = (1,1,1) and T($e_2$) = (1,2,3)?
Now if this is the case, how do we even answer the question?