1Introduction 2Organization of This Paper 3Introduction to SVD and Waves—Sets of Point Sources and Receivers 4Introductory Example—Three Sources and Three Receivers 5Scalar Wave Examples with Point Sources and Receivers 6Mathematics of Continuous Functions, Operators, and Vector Spaces 7Communications Modes and Common Families of Functions 8Extending to Electromagnetic Waves 9Quantizing the Electromagnetic Field Using the M-Gauge 10Linear Scatterers and Optical Devices 11Mode-Converter Basis Sets as Fundamental Optical Descriptions 12Conclusions