If the matrix of a linear map is independent from the basis, then the map is a multiple of the identity map. Let $V$ be a finite dimensional vector space over $F$, and let
$$T:V\to V$$
be a linear map.
Suppose that given any two bases $B$ and $C$ for $V$, we have that the matrix of $T$ with respect $B$ is equal to that with respect to $C$.
How can we show that this implies that there exists some $\lambda\in F$ such that $T(v)=\lambda v$, $\forall v\in V$?

Hint: This is equivalent to say that any nonzero vector is an eigenvector of $T$
