The tools, ideas, and insights from linear algebra, abstract algebra, and functional analysis can be extremely useful to signal processing and system theory in various areas of engineering, science, and social science including approximation, optimization, parameter identification, big data, etc. Indeed, many important ideas can be developed from the simple operator equation A x = b by considering it in a variety of ways. If x and if b are vectors from the same or, perhaps, different vector spaces and A is an operator, there are three interesting questions that can be asked which provide a setting for a broad study.