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 $\displaystyle V=R(T) \oplus W $ and W is T-invariant. Prove that $\displaystyle W \subseteq N(T) $

where R(T) is the range of linear transformation $\displaystyle T:V \rightarrow V $ 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 $\displaystyle w \in W, w \in N(T)$ and w is in N(T) if and only if T(w)={0}, so what i need to show is for all $\displaystyle w \in W, T(w) = 0 $.

Since V is a direct sum of R(T) and W, then $\displaystyle V=R(T)+W $ and $\displaystyle R(T) \cap W = \{0\} $ . Since V=R(T)+W, i know that for all $\displaystyle v \in V $, there exists $\displaystyle y \in R(T) $ and there exists $\displaystyle w \in W $ such that v=y+w. Since there exists y in R(T), then there exists an $\displaystyle x \in V $ such that T(x)=y. And from W being T-invariant, i know $\displaystyle T(w) \in W $.

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

Thank you for any help!!