The algebraic force-field Hamiltonian expansion recently proposed for vibrations of bent triatomic molecules [T. Sako, K. Yamanouchi, and F. Iachello, Chem. Phys. Lett. 299, 35 (1999)] is extended to linear polyatomic molecules by introducing operators constructed from the unitary algebra U(3). A comparison with the conventional force-field method is made by fitting the experimental vibrational-level energies of CO2. The results show that the algebraic Hamiltonian in local coordinates reproduces the level-energies with a smaller root-mean-square deviation and much faster convergence.