Inspired by the results about projection-valued dilation for operator-valued measures or more generally bounded homomorphism dilation for bounded linear maps on Banach algebras in previous chapters, we now explore a pure algebraic structure of the dilation theory. We first consider the general linear systems acting on unital algebras and vector spaces. By introducing two natural dilation structures, namely the canonical and the universal dilation systems, we get that every linearly minimal dilation is equivalent to a reduced homomorphism dilation of the universal dilation, and all the linearly minimal homomorphism dilations can be classified by the associated reduced subspaces.