College Algebra Exam Review 392

College Algebra Exam Review 392 - S 1 AS is the rational...

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402 8. MODULES last two columns of the polynomial matrix P ± 1 , which are 2 6 6 6 6 4 0 0 0 0 0 0 ± 1 C 3x ± x 2 ± 3 4 . ± 1 C x/ 4 3 . ± 2 C x/ 1 3 7 7 7 7 5 : The cyclic vector v 1 for V 1 is . ± 1 C 3x ± x 2 / O e 4 C 4 3 .x ± 2/ O e 5 D . ± 1 C 3A ± A 2 / O e 4 C 4 3 .A ± 2/ O e 5 D 2 6 6 6 6 4 4 0 0 1 8=3 3 7 7 7 7 5 : The cyclic vector v 2 for V 2 is ± 3 4 . ± 1 C x/ O e 4 C O e 5 D ± 3 4 . ± 1 C A/ O e 4 C O e 5 D 2 6 6 6 6 4 0 3 3 0 1 3 7 7 7 7 5 : The remaining vectors in the basis of V 2 are Av 2 D 2 6 6 6 6 4 3 6 6 0 4 3 7 7 7 7 5 ; A 2 v 2 D 2 6 6 6 6 4 9 15 15 0 10 3 7 7 7 7 5 ; A 3 v 2 D 2 6 6 6 6 4 21 39 42 0 22 3 7 7 7 7 5 : The matrix S has columns v 1 ;v 2 ;v 3 ;v 4 ;v 5 . Thus S D 2 6 6 6 6 4 4 0 3 9 21 0 3 6 15 39 0 3 6 15 42 1 0 0 0 0 8=3 1 4 10 22 3 7 7 7 7 5 Finally, to check our work, we can compute that, indeed,
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Unformatted text preview: S 1 AS is the rational canonical form of A . The characteristic polynomial and minimal polynomial Let A 2 Mat n .K/ . Write x A for xE n A . We dene the charac-teristic polynomial of A by A .x/ D det .x A/ . The reader can check...
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