I have to prove that the eigenspace, E_{lambda }is an invariant subspace.

I took some eigenvector, v in the eigenspace, then T(v) =lambda v

I know that T(v) should belong to the eigenspace , but i don't really understand why ?

E_{lambda}is the set of all eigenvectors of T, then why would (lambda) (v) be in the eigenspace ?

I am confused.