Linear operator equations with applications in control and signal processing

The author gives several definitions, including the definition of linear vector spaces, inner products, and Hilbert spaces. He defines linear operators and the Hilbert adjoint operator, and gives several illustrative examples. He presents a diagram which he says is key to understanding linear operator equations. It is a pedagogically important tool for understanding linear operators. Its details are discussed. When attention is restricted to linear matrix equations, the singular-value decomposition completely characterizes the fundamental subspaces of the operator, as is also discussed. The author presents several applications of the theory, including least squares, minimum-norm solutions, controllability and observability of linear systems, optimal control, optimal estimation, and modeling mechanical systems. The examples were chosen to illustrate the wide variety of problems that can be solved using the theory presented in the previous sections