Let U be a linear operator on a finite dimensional vector space V, prove that:

a) $\displaystyle N(U) \subseteq N(U^2) \subseteq N(U^3) \subseteq ... \subseteq N(U^k) \subseteq N(U^k+1) \subseteq . . .$

b) If $\displaystyle rank (U^m) = rank (U^{m+1}) $ , then $\displaystyle N(U^m) = N(U^k) \ \ \ \forall k \geq m $

c) Let T be linear operator on V whose characteristic polynomial splits, and let $\displaystyle \lambda _1 \lambda _2 , . . . , \lambda _k $ be distinct eigenvalues of T. Then T is diagonalizable if and only if $\displaystyle rank (T- \lambda _i I ) = rank ((T- \lambda _i I )^2 ) $ for $\displaystyle 1 \leq i \leq k $