\sum_{k=0}^{mn}\alpha_k(a+b)^k=\sum_{k=0}^{mn}\alpha_k\sum_{i=1}^{mn}\lambda_{k,i}c_i=\sum_{i=1}^{mn}c_i\sum_{k=0}^{mn}\alpha_k\lambda_{k,i}=\sum_{i=1}^{mn}0\cdot c_i=0