Suppose we have two linear operators $X,\,Y$ on $\mathbb{C}^n$. Now let $a,b\in\mathbb{C}$ and consider $Z_{a,b}:=aX+bY$. When we look at their images we clearly have $$im\,Z_{a,b}\subseteq \,im\,X+im\,Y,$$ but in general this inclusion is strict.
Is it true that the equality holds whenever we take $a,b\in\mathbb{C}$ sufficiently general?
What if we know that $X,Y$ satisfy $XY=YX$, that is, if $X,Y$ commute?
Here sufficiently general means that $\frac{a}{b}\in\mathbb{C}$ belongs to the complement of a certain finite set (depending on $X,Y$).
The answer to 1 is no. For instance, consider $$ X = \pmatrix{1&0\\0&0}, \quad Y = \pmatrix{0&0\\1&0}. $$ I suspect that the answer to 2 is yes. It is clear that the answer must be yes when both $X,Y$ are diagonalizable, because in this case $X$ and $Y$ commute and are therefore simultaneously diagonalizable.