Suppose $X$, $Y$, and $Z$ are finite sets. If we have a function $f : X \longrightarrow Y$ and another $g : Y \longrightarrow Z$ then the composite function $g \circ f$ has the property that $ |\mbox{Im} \, g \circ f | \leq |Y|. $
If we replace $f$ by a formal linear combination of functions $X \longrightarrow Y$, and $g$ by a formal linear combination of functions $Y \longrightarrow Z$, then we may "compose" these formal combinations (enforcing the distributive law) to obtain a linear combination of functions $X \longrightarrow Z$, none of which have image exceeding $|Y|$ in cardinality.
My question is the converse: can any such linear combination be obtained?