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We propose a simple model of the neutrino mass matrix which can explain thesolar and atmospheric neutrino problems in a 3($\nu_L + \nu_R$) framework.Assuming that only two right-handed neutrinos are heavy and a Dirac mass matrixhas a special texture, we construct a model with four light neutrinos. Thefavorable structure of flavor mixings and mass eigenvalues required by thoseneutrino deficits is realized as a result of the seesaw mechanism. Bi-maximalmixing structure might be obtainable in this scheme. Since it contains a lightsterile neutrino, it has a chance to explain the LSND result successfully. Weconsider an embedding of this scenario for the neutrino mass matrix into theSU(5) grand unification scheme using the Froggatt-Nielsen mechanism based on$U(1)_{F_1} \times U(1)_{F_2}$. Both a small mixing angle MSW solution and alarge mixing angle MSW solution are obtained for the solar neutrino problemdepending on the charged lepton mass matrix.Comment: 22 pages, latex, 3 eps-figure

A Light Sterile Neutrino Based on the Seesaw Mechanism