An unidentified quantum fluid designated the pseudogap (PG) phase is produced by electron-density depletion in the CuO$_2$ antiferromagnetic insulator. Current theories suggest that the PG phase may be a pair density wave (PDW) state characterized by a spatially modulating density of electron pairs. Such a state should exhibit a periodically modulating energy gap $Delta_P(pmb r)$ in real-space, and a characteristic quasiparticle scattering interference (QPI) signature $Lambda_P(pmb q)$ in wavevector space. By studying strongly underdoped Bi$_2$Sr$_2$CaDyCu$_2$O$_8$ at hole-density ~0.08 in the superconductive phase, we detect the $8a_0$-periodic $Delta_P(pmb r)$ modulations signifying a PDW coexisting with superconductivity. Then, by visualizing the temperature dependence of this electronic structure from the superconducting into the pseudogap phase, we find evolution of the scattering interference signature $Lambda(pmb q)$ that is predicted specifically for the temperature dependence of an $8a_0$-periodic PDW. These observations are consistent with theory for the transition from a PDW state coexisting with d-wave superconductivity to a pure PDW state in the Bi$_2$Sr$_2$CaDyCu$_2$O$_8$ pseudogap phase.
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