This chapter starts developing our central linear time-variant (LTV) prototype environment, a class that coincides perfectly with linear algebra and matrix algebra, making the correspondence between system and matrix computations a mutually productive reality. People familiar with the classical approach, in which the z-transform or other types of transforms are used, will easily recognize the notational or graphic resemblance, but there is a major difference: everything stays in the context of elementary matrix algebra, no complex function calculus is involved, and only the simplest matrix operations, namely addition and multiplication of matrices, are needed. Appealing expressions for the state-space realization of a system appear, as well as the global representation of the input–output operator in terms of four block diagonal matrices and the (now noncommutative but elementary) causal shift Z. The consequences for and relation to linear time-invariant (LTI) systems and infinitely indexed systems are fully documented in *-sections, which can be skipped by students or readers more interested in numerical linear algebra than in LTI system control or estimation.