For which sets $\{A_i \}_{i=1}^n \subseteq M_d$ of linearly independent $d\times d$ complex matrices does there exist a vector $v\in \mathbb{C}^d$ such that $$\text{dim} [\text{span}\{A_1v, A_2v, \ldots A_nv\}] = \begin{cases} n, \quad \text{if } n\leq d \\ d, \quad \text{if } n> d \end{cases} \quad ?$$
If $n\geq d$ (say), then linear independence of the given matrices alone does not guarantee the existence of the required vector, see the answer provided here. Hence, it is natural to ask for additional constraints on the given matrices which allow the required vector to exist. For instance, if we demand that the matrices are diagonalizable and mutually commuting, then there exists a common eigenbasis $\{w_i\}_{i=1}^d$ of $\mathbb{C}^d$ in which all the matrices are diagonal. By choosing $v=\sum_{i=1}^d w_i$, it is easy to see that the linear independence of $\{A_i\}_{i=1}^n \subseteq M_d$ is equivalent to the linear independence of $\{A_i v \}_{i=1}^n \subseteq \mathbb{C}^d$, which implies that $n\leq d$ and $\text{dim} [\text{span}\{A_1v, A_2v, \ldots A_nv\}] = n$.
It would be wonderful to get hold of other such simple and easily verifiable conditions on the given matrix set.
