= Solution
Part b shows that all solutions have the same fitted vector, hence the same score and the same set $E$. The KKT conditions show that every nonzero coordinate of any solution belongs to $E$, since a nonzero coefficient forces the corresponding score to have absolute value $\lambda$. Thus every solution is supported on $E$ and satisfies
$$
X_E\beta_E=X\widehat\beta.
$$
If $\operatorname{rank}(X_E)=|E|$, the linear map $u\mapsto X_Eu$ is injective. Consequently $\beta_E$ and hence $\beta$ are unique.
Back to article page
