  Inspired by some recent development on the theory about projection valued
dilations for operator valued measures or more generally bounded homomorphism
dilations for bounded linear maps on Banach algebras, we explore a pure
algebraic version of the dilation theory for linear systems acting on unital
algebras and vector spaces. By introducing two natural dilation structures,
namely the canonical and the universal dilation systems, we prove 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 contained in the kernel of
synthesis operator for the universal dilation.
