# Math Help - Linear Transformations

1. ## 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 T-invariant. Show that R has a complementary T-invariant 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 T-invariant subspace complementary to R.
So if $R\cap N = {0}$ then it's obvious that N is a T-invariant complementary subspace for R.
Conversely, suppose that S is a T-invariant complementary subspace for R. If $x\in S$ then $Tx\in R$ (obviously, since Tx is in the range of T) and $Tx\in S$ (because S is T-invariant). Therefore $Tx\in R\cap S=\{0\}$. In other words, $x\in N$. Hence $S\subseteq N$. But these subspaces have the same dimension, and so S = N.