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We report the observation of strongly ferromagnetic F=1 spinor Bose-Einstein condensates of ^{7}Li atoms. The condensates are generated in an optical dipole trap without using magnetic Feshbach resonances, so that the condensates have internal spin degrees of freedom. Studying the nonequilibrium spin dynamics, we have measured the ferromagnetic spin interaction energy and determined the s-wave scattering length difference among total spin f channels to be a_{f=2}−a_{f=0}=−18(3) Bohr radius. This strong collision-channel dependence leads to a large variation in the condensate size with different spin composition. We were able to excite a radial monopole mode after a spin-flip transition between the |m_{F}=0〉 and |m_{F}=1〉 spin states. From the experiments, we estimated the scattering length ratio a_{f=2}/a_{f=0}=0.27(6), and determined a_{f=2}=7(2) and a_{f=0}=25(5) Bohr radii, respectively. The results indicate the spin-dependent interaction energy of our system is as large as 46% of the condensate chemical potential.