In this chapter, we consider the central issue of minimality of the state-space system representation, as well as equivalences of representations. The question introduces important new basic operators and spaces related to the state-space description. In our time-variant context, what we call the Hankel operator plays the central role, via a minimal composition (i.e., product), of a reachability operator and an observability operator. Corresponding results for LTI systems (a special case) follow readily from the LTV case. In a later starred section and for deeper insights, the theory is extended to infinitely indexed systems, but this entails some extra complications, which are not essential for the main, finite-dimensional treatment offered, and can be skipped by students only interested in finite-dimensional cases.