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