Does the preimage of finite codimension spaces have finite codimension?

Question

Let $X$ a real or complex vector space and $F$ a subspace of $X$ with finite codimension (that is, there exists a finite dimensional subspace $E$ of $X$ such that $X=E+F$). Is it true that, for any linear map $T:X\to X$, the space $T^{-1}(F)$ has finite codimension?

I think the answer is positive, so I tried to show it by contradiction but haven't had success... Have you any tips?

Answer 1

Yes, it is true the you have an injective map $X/T^{-1}(F)\rightarrow X/F$ induced by $T$, since $X/F$ is finite dimensional so is $X/T^{-1}(F)$.