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

Question

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 $$\{(1,1,1,1), (1, 0, 0, 1)\} \text{ and } \{(1,1,0), (1, 0, 1)\}$$

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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.

Answer 1

Let $v_1=(1,1,1,1)$ and $v_2=(1,0,0,0)$. Now take two other vectors $v_3$ and $v_4$ such that $\{v_1,v_2,v_3,v_4\}$ is linearly independent, such as $v_3=(1,0,0,0)$ and $v_4=(0,1,0,0)$. And then take the only linear map $T:\mathbb{R}^4\longrightarrow\mathbb{R}^3$ such that $T(v_1)=T(v_2)=(0,0,0)$, $T(v_3)=(1,1,0)$, and $T(v_4)=(1,0,0)$.