Is this matrices exist, with this propertie

Question

if $\begin{bmatrix} 1\\ 1\\ 0 \end{bmatrix}$, $\begin{bmatrix} 0\\ 0\\ 1 \end{bmatrix}$ $\in R(A)$, and $\begin{bmatrix} 2\\ 5 \end{bmatrix}$, $\begin{bmatrix} 1\\ 2 \end{bmatrix}$ $\in R(A^{T})$

I think that is not exist because, from here we can see that this matrices must be $3x2$ but when I find base of $N(A)$ and $N(A^{T})$ they have dimension 1, so than we need matrice $3x3$ which is not what we expect.

Answer 1

Let consider

$$A=\begin{bmatrix}1&0\\1&0\\0&1\end{bmatrix}$$

and note that $\operatorname{rank}(A)=2$ therefore $R(A^T)$ = $\mathbb{R^2}$.