intersection of point stabilisers is trivial

Question

Let $G=\operatorname{GL}_{n}(2)$. Let $v_{i}$ be the basis elements of the natural module of $G$. I observed by computing with Magma that the intersection of all Stabiliser($G, v_{i}$) is trivial for small $n$. Is it true in general? I suppose so since the point-stablisers are all conjugate to each other... It could be generalized to a general finite field if it is true, I think?

Answer 1

If a linear map $f : V \rightarrow V$ fixes every basis vector, it fixes every $v \in V$. So $f$ is the identity map on $V$.