Here, we review the theory of the Koopman operator, an in nite-dimensional linear operator that represents the dynamics of nonlinear systems in terms of its action on observables. We highlight the key role of linear algebra in Koopman operator analysis, spectral theory, and nite-dimensional approxima- tions. Further, we demonstrate how machine learning approaches utilize these linear algebraic principles to generate data-driven, nonlinear system models in the Koopman framework.
