Subspace consisting of normal operators

Question

$V$ is an $n$-dimensional inner product space. Let $L=L(V,V)$ be a vector space of all linear operators on $V$, and let $T\in V$ be a normal operator. If the char. poly of $T$ splits, show that there is an $n$-dimensional subspace of $L$ containing $T$ and consisting of normal operators. Unsure about how the characteristic polynomial of a normal operator here is relevant.

Answer 1

Hint: Consider the subspace spanned by the powers of $T$ (including the identity matrix, which is treated as the "$0$th power").