Find a linear transformation $ T: \mathbb{R^4} \to \mathbb{R^3}$ such that $\ker T$ and $\operatorname{Range}T$ are spanned by given vectors

I have got the following entrance exam question.

Find a linear transformation $ T: \mathbb{R^4} \to \mathbb{R^3}$ such that $\ker T$ and $\operatorname{Range}T$ are respectively spanned by $$\{(1,1,1,1), (1, 0, 0, 1)\} \text{ and } \{(1,1,0), (1, 0, 1)\}$$

My approach: $\dim \ker T = 2$ and $\dim\operatorname{Range}T = 2$. $T(1,1,1,1) = (0,0,0)$, $T (1, 0, 0, 1)= (0,0,0)$. But with this information I am not able to proceed further. Kindly help me with this.