In this paper, we provide, for the first time in the literature, rigorous and complete proofs to all the key properties of the special coordinate basis of linear time-invariant systems. The special coordinate basis decomposition or technique developed by Sannuti and Saberi in 1987 has a distinct feature of explicitly displaying the finite and infinite zero structures, the invertibility structures, as well as the invariant and almost invariant geometric subspaces of a given system. The technique has been extensively used in the literature to solve many system and control problems. We believe that the result of this paper is a complement of the seminal work of Sannuti and Saberi. It makes the theory of the special coordinate basis more complete.