Given $T$ be a linear operator on the the finite dimensional space $V$, let $R$ be the range of $T$. Then, I have to show that
(a) $R$ has a complementary $T$-invariant subspace if and only if $R$ is independent of the null space $N$ of $T$.
(b) If $R$ and $N$ are independent, I need to prove that $N$ is the unique $T$-invariant subspace complementary to $R$.
Please suggest how to proceed.
I supposed $R$ has a complementary $T$-invariant subspace, say, $W$. Then $R$ should be $T$-admissible. I assumed to the contrary, that $R$ intersection $T$ is not equal to $\{\mathbf{0}\}$. I took a point in the intersection but could not proceed further. Please suggest what to do.