# Math Homework

*∈L ƒ*

**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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