A transcriptional regulatory network (TRN) constitutes the collection of regulatory rules that link environmental cues to the transcription state of a cells genome. We recently proposed a matrix formalism that quantitatively represents a system of such rules (a transcriptional regulatory system [TRS]) and allows systemic characterization of TRS properties. The matrix formalism not only allows the computation of the transcription state of the genome but also the fundamental characterization of the input-output mapping that it represents. Furthermore, a key advantage of this pseudo-stoichiometric matrix formalism is its ability to easily integrate with existing stoichiometric matrix representations of signaling and metabolic networks. Here we demonstrate for the first time how this matrix formalism is extendable to large-scale systems by applying it to the genome-scale Escherichia coli TRS. We analyze the fundamental subspaces of the regulatory network matrix (R) to describe intrinsic properties of the TRS. We further use Monte Carlo sampling to evaluate the E. coli transcription state across a subset of all possible environments, comparing our results to published gene expression data as validation. Finally, we present novel in silico findings for the E. coli TRS, including (1) a gene expression correlation matrix delineating functional motifs, (2) sets of gene ontologies for which regulatory rules governing gene transcription are poorly understood and which may direct further experimental characterization, and (3) the appearance of a distributed TRN structure, which is in stark contrast to the more hierarchical organization of metabolic networks.