(i) Find a linear transformation $f: \mathbb R^4 \to \mathbb R^4$ for which $$\ker(f) = \langle (1,0,1,0) \rangle \text{ and Im }(f^2) = \langle (1,0,1,0),(1,1,1,1) \rangle.$$ (ii) Find all the possible values of $\dim(\ker (f) \cap \text{Im}(f^2))$ where $f: \mathbb R^4 \to \mathbb R^4$ is a linear transformation.
Here is my idea for (i): we know that the $4 \times 4$ Jordan block $J$ with zeroes as diagonals and three ones satisfy the condition when $(1,0,1,0)$ and $(1,1,1,1)$ are replaced by $e_1$ and $e_2$, respectively. Thus, if we can somehow manipulate $J$ with suitable invertible linear transformation $g$ such that $g(e_1) = (1,0,1,0)$ and such that Im($g \circ J)^2 = \langle (1,0,1,0),(1,1,1,1) \rangle$, I think that would solve the problem. The question is, I am (1) unsure if this approach will work, and (2) how to go about finding the specific $g$.
For (ii), I think this is just looking at all the possible Jordan forms of $4 \times 4$ matrix, and simply computing the possible dimensions. However, I am wondering if there is any other simpler way to solve (ii), since it seems slightly lengthy to do the rote calculations (the above questions were given on a time-constrained test).