
Linear Transformations
Plese, help me to solve this:
Let T is in L(V). Put R=Im(T) and N=null(T). Note that both R and T are Tinvariant. Show that R has a complementary Tinvariant subspace W (i.e. V=R (direct sum) W and T(W) is included in W) if and only if R intersect N = {0}, in which case N is the unique Tinvariant subspace complementary to R.
Thank you in advance!

First, it follows from the ranknullity theorem that any complementary subspace for R must have the same dimension as N.
So if $\displaystyle R\cap N = {0}$ then it's obvious that N is a Tinvariant complementary subspace for R.
Conversely, suppose that S is a Tinvariant complementary subspace for R. If $\displaystyle x\in S$ then $\displaystyle Tx\in R$ (obviously, since Tx is in the range of T) and $\displaystyle Tx\in S$ (because S is Tinvariant). Therefore $\displaystyle Tx\in R\cap S=\{0\}$. In other words, $\displaystyle x\in N$. Hence $\displaystyle S\subseteq N$. But these subspaces have the same dimension, and so S = N.