problemset11 - 12. Page 280, Problem 4 Extra Problem The...

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MAS 3106 — Linear Algebra Problem Set 11 — Due 4/7/11 1. Page 256, Problem 1 2. Page 257, Problem 3b 3. Page 257, Problem 4f 4. Page 257, Problem 8 (For c, just state the result; don’t prove it) 5. Page 258, Problem 11 (for part c, use parts a and b, not the method used in class) 6. Page 259, Problem 14 7. Page 259, Problem 15 (For b, just state the result; don’t prove it) 8. Page 260, Problem 19 9. Page 260, Problem 22 (For b, just state the result; don’t prove it) 10. Page 279, Problem 2eg 11. Page 280, Problem 3d
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Unformatted text preview: 12. Page 280, Problem 4 Extra Problem The minimal polynomial plays an important role in more advanced subjects of linear algebra. Definition: Let T be a linear operator on a finite-dimensional vector space. Polynomial p ( t ) is a minimal polynomial of T if (i) p ( t ) is monic and (ii) p ( T ) T and (iii) p ( t ) has smallest degree among all polynomials satisfying (i) and (ii). Prove: (a) If g ( t ) is a polynomial for which g ( T ) T , then p ( t ) divides g ( t ) . (b) The minimal polynomial of T is unique....
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