A linear inverse space-mapping (LISM) optimization algorithm for designing linear and nonlinear RF and microwave circuits is described in this paper. LISM is directly applicable to microwave circuits in the frequency- or time-domain transient state. The inverse space mapping (SM) used follows a piecewise linear formulation, avoiding the use of neural networks. A rigorous comparison between Broyden-based "direct" SM, neural inverse space mapping (NISM) and LISM is realized. LISM optimization outperforms the other two methods, and represents a significant simplification over NISM optimization. LISM is applied to several linear frequency-domain classical microstrip problems. The physical design of a set of CMOS inverters driving an electrically long microstrip line on FR4 illustrates LISM for nonlinear design.