i think the first part, we can say that assume A represents a linear map $\displaystyle \alpha : V\longrightarrow W$ and dimV is greater than dimW. So alpha is surjective which means n(A) is nonzero.

And k is the least number makes the linear map surjective, which means, dimV=dimW+1, and n(A)=1

but i got no idea with (b) and etc.