Let T be a linear operator on V. Suppose V=W1⊕⋯⊕Wk, where each Wi is invariant under T. Let Ti be the induced (restriction) operator on Wi.
(a) Prove that det(T)=det(T1)⋯det(Tk).
(b) Prove that the characteristic polynomial for T is the product of the characteristic polynomials for T1,…,Tk.
(c) Prove that the minimal polynomial for T is the least common multiple of the minimal polynomials for T1,…,Tk. (Hint: Prove and then use the corresponding facts about direct sums of matrices.)