The authors extend the framework of B. Peterson and the reviewer [Aequationes Math. 20, 1-17 (1980; Zbl 0434.16008)]. There we studied \(k[x]\), \(k\) a field, as a Hopf algebra with \(x\) primitive, and the coalgebra dual \(k[x]^ 0\) was identified as linearly recursive sequences. Each such sequence lies in a finite-dimensional subcoalgebra (subcomodule of \(k[x]^ 0\)), and its recursive polynomial determines a cofinite ideal (submodule) of \(k[x]\). The authors generalize this to a finite-dimensional vector space \(V\) over \(k\), the free \(k[x]\)-module \(V[x] = V \otimes k[x]\), and the cofree \(k[x]^ 0\)-comodule \(V[x]^ 0 = V^*\otimes k[x]^ 0\). \(V[x]^ 0\) consists of those functionals in \(V[x]^*\) whose kernel contains a submodule \(M\) of finite codimension. \(M\) is of the form \(PV[x]\), \(P\) an operator polynomial in \(\text{End}(V[x])\), i.e., \(P = \sum^ d_{i=0} A_ ix^ i\), \(A_ i \in \text{End }V\). Thus an element \(g\) of \(V[x]^ 0\) is of the form \(\sum^ \infty_{i=0} g_ i\otimes z_ i\), where the \(\{z_ i\}\) are the dual basis to the \(\{x^ i\}\), and the \(\{g_ i\}\) form a linearly recursive sequence in \(V^*\), i.e., \(\sum^ d_{i=0} A_ i^*g_{i+j}\) for every \(j \geq 0\). \(k[x]^ 0\)-subcomodules of \(V[x]^ 0\) are the same as \(k[x]\)-submodules of \(V[x]^ 0\), where the action is given by multiplication in \(k[x]\). In this way, there is a 1-1 correspondence between cofinite dimensional submodules of \(V[x]\) and finite dimensional subcomodules of \(V[x]^ 0\). This gives a 1-1 correspondence between equivalence classes (under \(GL_ n(k[x])\)) of regular polynomial operators \(P\) (i.e., \(\text{det }P\neq 0\) in \(k[x]\)), with cofinite dimensional submodules of \(V[x]\), and finite-dimensional subcomodules of \(V[x]^ 0\). This leads to a characterization of monic operator polynomials (the leading coefficient is the identity matrix) and comonic operator polynomials (the constant term is the identity) in terms of the corresponding subcomodules. This gives a coordinate-free approach to operator polynomials, where, roughly speaking, the coordinate free version of a Jordan matrix pair is the same as a finite-dimensional subcomodule of \(V[x]^ 0\). While the authors’ general framework is for any field \(k\), some of the technical results require \(k\) to be perfect.