This investigation deals with the selection of test data for vector-valued sequences which can be generated by a linear first order system of difference equations with rational coefficients of limited degree. Let $\mathcal{F}(\nu,k,l,m)$ be the class of v-dimensional sequences y which satisfy an equation of the form $P(n)y(n + 1) = A(n)y(n) + R(n)$ where $A,P$, and R are polynomial matrices of degree $ \leqq k,\, l$, and m, respectively, and P is diagonal. It is shown that there is a finite sample which uniquely identifies members of $\mathcal{F}(\nu,k,l,m)$. A finite sample also exists for the case where y is considered to be a function of its initial value. If y satisfies two distinct equations of the above type, then there is a transformation under which certain coordinates must eventually be rational functions, or even polynomials if P is the identity.