Let T be a linear endomorphism on a vector space V . Now I must show that a subspace W is T-invariant, if and only if it is (T − λI)-invariant. Here we say λ is some number.
So I know with T-Invariance that T(W)⊂W, but I have to show that this is true, only if it is "(T − λI)-invariant".
Right now I thought of defining the subspace W such that $ W=Span (v_1,v_2,...,v_m) $ so by the problem (T − λI)(W)⊂W? I'm wondering if my train of thought here is correct and how to really start this problem.
Take $\;w\in W\;$ , then if $\;W\;$ is $\;T\,-$ invariant, then for any scalar$\;\lambda\;$:
$$(T-\lambda I)w=Tw-w\in W\;,\;\;\text{since}\;\;Tw\in W$$
Try now to do the reverse deduction. If you understood the above is almost trivial.