[Math] How to prove that every linear operator on a finite dimensional vector space is a sum of invertible linear operators

Question

Let $V$ be a finite dimensional vector space , then how do we prove that for every linear operator $T$ on $V$ , there exist invertible linear operators $S_T' , S_T'',…$ such that $T(\vec v)=S_T' (\vec v) + S_T''(\vec v)+… \forall \vec v \in V$ ?

Answer 1

Hint: if $t \ne 0$ is not an eigenvalue of $T$, then ...

EDIT: In the case of dimension $1$ over the field $GF(2)$ with two elements, there is only one invertible linear operator ($1$), and it is not the sum of two invertible linear operators (though it is the sum of three).