Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 205 / 1 / c / 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 satisfiesIf $\mathrm{rank}(X_E)=|E|$, the linear map $u\mapsto X_{E}u$ is injective. Consequently ${\beta }_E$ and hence $\beta$ are unique.