2 questions in Statement of Primary Decomposition Theorem and it’s Corollary in Linear Algebra

Question

I am self studying Linear Algebra from Hoffman Kunze and I have 2 questions in section 6.8 whose image I am adding->

Questions : (1) In the last paragraph how does one can deduce that $W_{i}'s $ are invariant under T?

(2) In last line of corollary : How is each subspace $W_{i} $ invariant under U ?

Answer 1

Well, (2) implies (1) as for $T$ commutes with itself, so about (2): For $i = 1, …, k$, as $W_i = \ker p_i^{r_i}(T)$, $$(p_i^{r_i}(T)∘U)(W_i) = (U∘p_i^{r_i}(T))(W_i) = U(p_i^{r_i}(W_i)) = U(0) = 0,$$ so $U(W_i) ⊆ \ker p_i^{r_i}(T) = W_i$.