Let $M \in \mathcal{M}_{k \times d}(\mathbb{R})$ with $\operatorname{rank}(M) = k$, $b \in \mathbb{R}^k \setminus \{0\}$, $\bar{x} \in C$ (where $C := \{x \in \mathbb{R}^d : Mx = b, \|x\|_1 = r\}$), and $q \in \operatorname{Ker}(M)^\perp \setminus \{0\}$ as in question 23. Let $K$ be the set of $y \in \mathbb{R}^d$ such that
$$My = b, \quad y_i = 0 \quad \forall i \in I_0(\bar{x}), \quad q_i y_i \geq 0 \quad \forall i \in \{1, \ldots, d\}.$$
Show that $K$ is non-empty and included in $C$.