Let $V,W$ be finite dimensional vector spaces over an infinite field $k$. Fix a positive integer $n \leq \dim(W), \dim(V)$.
Given $n$ injective linear maps $f_i:V\rightarrow W$, such that the $f_i$ are linearly independent as elements of $\operatorname{Hom}(V,W)$, is there necessarily a vector $v\in V$ with $f_i(v)$ linearly independent in $W$?
This holds for $n=1,2$ and if the $f_i$ commute and are diagonalisable, since then one may simultaneously diagonalise.
In terms of the conditions, we clearly require linear independence of the maps for the conclusion, and injectivity is required to exclude the case of $\dim(V) > \dim(W)$, where the images will never be linearly independent.
Best Answer
This fails for $n=3.$ Consider $f_i$ with matrix representations
$$ \begin{pmatrix}1&0&0\\0&1&0\\0&0&1\end{pmatrix}, \begin{pmatrix}1&1&0\\0&1&0\\0&0&1\end{pmatrix}, \begin{pmatrix}1&0&1\\0&1&0\\0&0&1\end{pmatrix}. $$
Or in other words: $\mathrm{id}, \mathrm{id}+e_1e_2^*, \mathrm{id}+e_1e_3^*.$ Then for any $v,$ all three vectors $f_i(v)$ lies in the space spanned by $\{v,e_1\}.$