Perhaps my question is naive, but let me try.

Take a (real or complex) vector space $V$ and consider an ideal $\mathcal{I}$ of subsets of $V$ with the following property (call it (*)): for each linear map $A\colon V\to V$ and each $S\in \mathcal{I}$ we have $A(S)\in \mathcal{I}$.

The ideal of finite sets enjoys this property. Is there a maximal ideal (that is, such that the set $\{E\setminus S\colon S\in \mathcal{I}\}$ is an ultrafilter) with (*)?

What if we replace vector spaces and linear maps by Banach spaces and bounded linear operators?