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Can there be underdetermined linear systems whose set of minimal support solutions is infinite?
If the minimal $\|x\|_0$ for a solution is $k$, any solution with $\|x\|_0 = k$ has support one of the ${n \choose k}$ subsets of $n$ with cardinality $k$.
The $m \times k$ submatrix $A_S$ of $A$ con …