Let $T$ be a linear transformation on $\mathbb{R}^{4}$ whose standard matrix is $\left(\begin{array}{rrrr} 1 & -1 & -1 & -1\\ 1 & 1 & 1 & -1\\ 1 & 1 & -1 & 1\\ 0 & 0 & 0 & 0 \end{array}\right).$ Does there exist a $3$-dimensional subspace of $\mathbb{R}^{4}$, $V$, and a linear transformation $S$ on $\mathbb{R}^{4}$ such that $S(\mathbf{v})=T(\mathbf{v})$ for all $\mathbf{v}\in V$ and $\left\Vert S(\mathbf{x})\right\Vert _{1}\le2$ for all $\mathbf{x}\in\mathbb{R}^{4}$ with $\left\Vert \mathbf{x}\right\Vert _{1}=1$?
Thanks in advance for any helpful answers.