In this exercise, we work through the proof of the converse of Theorem 10.1, in particular showing that if the $\ell_1$-relaxation has a unique optimal solution, equal to the $\ell_0$-solution, for all $S$-sparse vectors, then the set $\operatorname{null}(\mathbf{X}) \backslash\{0\}$ has no intersection with $\mathbb{C}(S)$.
(a) For a given vector $\beta^* \in \operatorname{null}(\mathbf{X}) \backslash\{0\}$, consider the basis-pursuit problem
$$
\underset{\beta \in \mathbb{R} P}{\operatorname{minimize}}\|\beta\|_1 \quad \text { such that } \mathbf{X} \beta=\mathbf{X}\left[\begin{array}{c}
\beta_S^* \\
0
\end{array}\right] .
$$
What is the link between its unique optimal solution $\widehat{\beta}$ and the vector
$$
\left[\begin{array}{c}
0 \\
-\beta_{S^c}^*
\end{array}\right] ?
$$
(b) Use part (a) to show that $\beta^* \notin \mathbb{C}(S)$.