A simple four configuration model that quantitatively reproduces all of the 1(1)B(u) and 2(1)A(g) state 0-0 energies that have been measured in high resolution spectroscopic experiments has been described previously. This model has been useful for explaining trends in the electronic properties of series of unsubstituted and substituted linear polyenes. While this model led to analytical expressions for the excitation energies, there were no closed form expressions for other quantities, such as the transition dipoles, that depend explicitly on the coefficients of the one-electron molecular orbitals. This paper derives exact expressions for the one-electron molecular orbital coefficients for an alternating chain as well as exact expressions for the transition dipole moments. This facilitates a detailed examination of the dependence of the transition dipole moments on the polyene length and alternation parameter which leads to an expression that accurately describes the dependence of the transition dipole moments on these chain parameters. The application of these expressions to an analytical analysis of nonlinear response in Linear polyenes will be the subject of a subsequent paper.