In this chapter, the author aims to present the ideas in a manner that is accessible, providing insights into the construction of frames and wavelets from a differential geometric standpoint. Treating the frames constructed (by discretizing the ax+b and Heisenberg Lie groups) as prototypes of a much more general phenomenon, the author aims to develop a far-reaching and unified theory capable of handling a large class of induced representations. As a byproduct, we produces an algorithmic procedure that outputs frames for function spaces over any linear Lie group. Additionally, the frames constructed with these methods satisfy several desired regular conditions that make them suitable for applications in signal analysis, time-frequency, generalized wavelet, shearlets, and operator theories.