  A unitary Fermi gas in an isotropic harmonic trap is predicted to show scale
and conformal symmetry that have important consequences in its thermodynamic
and dynamical properties. By experimentally realizing a unitary Fermi gas in an
isotropic harmonic trap, we demonstrate its universal expansion dynamics along
each direction and at different temperatures. We show that as a consequence of
SO(2,1) symmetry, the measured release energy is equal to that of the trapping
energy. We further observe the breathing mode with an oscillation frequency
twice the trapping frequency and a small damping rate, providing the evidence
of SO(2,1) symmetry. In addition, away from resonance when scale invariance is
broken, we determine the effective exponent $\gamma$ that relates the chemical
potential and average density along the BEC-BCS crossover, which qualitatively
agrees with the mean field predictions. This work opens the possibility of
studying non-equilibrium dynamics in a conformal invariant system in the
future.
