Prob1. Let V be a complex n -dimensional space and let T ( V ) be such that null T n− 3 = null T n− 2 . How many distinct eigenvalues can T have?
Prob2. Let V be acomplexfinite-dimensionalvectorspaceandlet T ( V )haveeigenvalues 1, 0, 1. Giventhedimensionsofthecorrespondingnullspacesbelow,determinetheJordannormalformof T
Prob 3. Let T ∈ L ( P 3 ©) be the operator
T : f ( x ) ›→ f ( x − 1) + x 3 f jjj ( x ) / 3 .
Find the Jordan normal form and a Jordan basis for T .
Prob4. Let V be a complex (finite-dimensional) vector space and let T ∈ L ( V ). Prove that there exist operators D and N in L ( V )suchthat T = D + N , D isdiagonalizable, N isnilpotent,and DN = ND .
denotethecharacteristicpolynomialof T andlet q denotethecharacteristicpolynomialof T − 1 . Prove that
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