Definitions. Let V be a vector space, and let T: V rightarrow V be linear. A subspace W V is said to be T - invariant if T(x) W for every x W, that is, T(W) W. If W is T - invariant, we define the restriction of T on W to be the function TW: W rightarrow W defined by Tw(x) = T(x) for all x W. Prove that the subspace {0}, V, R(T), and N(T) are all T - invariant.
Show transcribed image text Definitions. Let V be a vector space, and let T: V rightarrow V be linear. A subspace W V is said to be T - invariant if T(x) W for every x W, that is, T(W) W. If W is T - invariant, we define the restriction of T on W to be the function TW: W rightarrow W defined by Tw(x) = T(x) for all x W. Prove that the subspace {0}, V, R(T), and N(T) are all T - invariant.