Geodetic observations of interseismic deformation provide constraints on the partitioning of fault slip across plate boundary zones, the spatial distribution of both elastic and inelastic strain accumulation, and the nature of the fault system evolution. Here we describe linear block theory, which decomposes surface velocity fields into four components: (1) plate rotations, (2) elastic deformation from faults with kinematically consistent slip rates, (3) elastic deformation from faults with spatially variable coupling, and (4) homogeneous intrablock strain. Elastic deformation rates are computed for each fault segment in a homogeneous elastic half-space using multiple optimal planar Cartesian coordinate systems to minimize areal distortion and triangular dislocation elements to accurately represent complex fault system geometry. Block motions, fault-slip rates, elastic coupling, and internal block strain rates are determined simultaneously using a linear estimator with constraints from both geodetically determined velocity fields and geologic fault-slip rate estimates. We also introduce algorithms for efficiently implementing alternative fault-network geometries to quantify parameter sensitivity to nonlinear perturbations in model geometry.