Let $k$ be an infinite field and $V,W$ vector spaces over $k$. Let $\varphi, \psi:V \longrightarrow W$ be $k$-linear morphisms. Consider the linear combinations $\varphi_\lambda := \varphi + \lambda \psi$. I want to know under what conditions we obtain infinitely many different images $\text{im}\,\varphi_{\lambda}$ from these morphisms? For example, this always happens if $\varphi$ and $\psi$ are injective and the intersection of their images is zero. However, this condition does not include all case. Is there a more general nice condition when this happens?
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