 Then every formula_18-sparse signal formula_29 is the unique solution to the formula_1-relaxation problem with formula_31 if and only if formula_17 satisfies the nullspace property with order formula_18.formula_34 For the forwards direction notice that formula_35 and formula_36 are distinct vectors with formula_37 by the linearity of formula_17, and hence by uniqueness we must have formula_21 as desired.For the backwards direction, let formula_40 be formula_18-sparse and formula_42 another (not necessary formula_18-sparse) vector such that formula_44 and formula_45.