cannot be in the image of . Consider the element , for example.
The correct answer, I believe, is that . I got this just by writing out a few elements of , noting that if an element is in the image of then its dot product with each of those elements must be 0, and solving a system of equations based on this fact.
As far as finding a formula for , I'm not 100% sure what to do. I mean, we can define a lot of different linear transformations whose image is . I guess just pick one.