T-invariant

• September 17th 2009, 10:18 PM
Last_Singularity
T-invariant
Question: Let $W$ be a subspace of vector space $V$ and that $T: V \rightarrow V$ is linear. Prove that subspaces $\{0\},V,R(T),N(T)$ are all T-invariant.

I feel that the trick is really simple but I just cannot grasp it. Any pointers would be appreciated - thanks.
• September 17th 2009, 10:32 PM
Jose27
Quote:

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

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

$\{ 0 \}$ and $V$ are obvious since $T:V \longrightarrow V$ is linear. Since $T(v) \in V$ for all $v \in V$ we have $T(T(v)) \in R(T)$ and so $T^n(v) \in R(T)$ by definition, if $v \in N(T)$ then $T(v)=0 \in N(T)$ and $T^n(v) \in N(T)$