Let $T \in \mathcal{L}(\mathbb{R}^4, \mathbb{R}^3)$ and suppose its matrix representation with respect to the standard bases for $\mathbb{R}^4$ and $\mathbb{R}^3$ is of the form
$\mathcal{M}(T) = \begin{bmatrix} 1 & 2 & 2 & * \\ 3 & 4 & * & * \\ * & 5 & 6 & * \end{bmatrix}$,
Determine $T$, that is, find the remaining entries of $\mathcal{M}(T)$, so that dim null $T = 2$ and so that $u = \begin{bmatrix} 1 & -1 & 2 & 0 \end{bmatrix}^T$ and $v = \begin{bmatrix} 1 & 0 & 2 & -1 \end{bmatrix}^T$ are both in range $T^*$, or prove that no such $T$ exists.
