Let $V$ be a finite dimensional vector space over $\mathbb{Z}_2$ with a linear map $f_i : V \to V$ for each $i$ in some finite index set $I$.

Then one can always find some subset $G \subseteq V$ of minimal cardinality such that the set of all elements:

$f_{i_1} \circ \dots \circ f_{i_n}(g)$ where $g \in G$, $i_j \in I$ and $n \geq 0$

spans the vector space $V$. Likewise in the dual situation $(V^*,(f_i^*)_{i \in I})$ there is some respective minimal generating set $G' \subseteq V^*$.

Here are my questions:

How does the cardinality of $G'$ compare to that of $G$?

What if each $f_i$ is assumed to be idempotent?

Any help much appreciated.