Direct sum of a range and T-invariant subspace problem

Hi!

I've been working for this problem for quite a while, but am seriously stuck now, so I would appreciate any help!

Here's the problem:

Suppose and W is T-invariant. Prove that

where R(T) is the range of linear transformation and N(T) is a null space of T, and W is a subspace of V.

Here's what I got:

I need to show that for all and w is in N(T) if and only if T(w)={0}, so what i need to show is for all .

Since V is a direct sum of R(T) and W, then and . Since V=R(T)+W, i know that for all , there exists and there exists such that v=y+w. Since there exists y in R(T), then there exists an such that T(x)=y. And from W being T-invariant, i know .

Ok, knowing it is good, but i cannot relate that to N(T). What do I do next??(Worried)

Thank you for any help!!