Linear Transformations · Challenge problem

Consider the linear maps $T:\mathbb{R}^4\to\mathbb{R}^3$ such that $T(1,1,1,1) = \mathbf 0$ and every output $T(\mathbf v)$ lies in the plane $x + y + z = 0$. These maps form a subspace of the $12$-dimensional space of all linear maps $\mathbb{R}^4\to\mathbb{R}^3$. What is its dimension?