Based on the range space property (RSP), the equivalent conditions between nonnegative solutions to the partial sparse and the corresponding weighted $l_1$-norm minimization problem are studied in this paper. Different from other conditions based on the spark property, the mutual coherence, the null space property (NSP) and the restricted isometry property (RIP), the RSP-based conditions are easier to be verified. Moreover, the proposed conditions guarantee not only the strong equivalence, but also the equivalence between the two problems. First, according to the foundation of the strict complementarity theorem of linear programming, a sufficient and necessary condition, satisfying the RSP of the sensing matrix and the full column rank property of the corresponding sub-matrix, is presented for the unique nonnegative solution to the weighted $l_1$-norm minimization problem. Then, based on this condition, the equivalence conditions between the two problems are proposed. Finally, this paper shows that the matrix with the RSP of order $k$ can guarantee the strong equivalence of the two problems.