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A linear transformation of the state variables of a linear dynamical system generates another linear system, which may have some properties in common with the original system. The transformations considered here are those which leave invariant a functional of a general algebraic form of the system state variables. It is shown that these define a subgroup of transformations within the linear group. The general concepts of group theory are reviewed, and an example is given of the application of a finite transformation to a linear electrical network.