1. ## T-invariant

Question: Let $\displaystyle W$ be a subspace of vector space $\displaystyle V$ and that $\displaystyle T: V \rightarrow V$ is linear. Prove that subspaces $\displaystyle \{0\},V,R(T),N(T)$ are all T-invariant.

I feel that the trick is really simple but I just cannot grasp it. Any pointers would be appreciated - thanks.

2. Originally Posted by Last_Singularity
Question: Let $\displaystyle W$ be a subspace of vector space $\displaystyle V$ and that $\displaystyle T: V \rightarrow V$ is linear. Prove that subspaces $\displaystyle \{0\},V,R(T),N(T)$ are all T-invariant.

I feel that the trick is really simple but I just cannot grasp it. Any pointers would be appreciated - thanks.
$\displaystyle \{ 0 \}$ and $\displaystyle V$ are obvious since $\displaystyle T:V \longrightarrow V$ is linear. Since $\displaystyle T(v) \in V$ for all $\displaystyle v \in V$ we have $\displaystyle T(T(v)) \in R(T)$ and so $\displaystyle T^n(v) \in R(T)$ by definition, if $\displaystyle v \in N(T)$ then $\displaystyle T(v)=0 \in N(T)$ and $\displaystyle T^n(v) \in N(T)$