Unformatted text preview: 406 8. MODULES is an ordered basis of V=V . Suppose, moreover, that the matrix of T j V with respect to .v 1 ;:::;v k / is A 1 and the matrix of T with respect to .v k C 1 C V ;:::;v n C V / is A 2 . Show that .v 1 ;:::;v k ;v k C 1 ;:::;v n / is an orderd basis of V and that the matrix of T with respect to this basis has the form A 1 B A 2 ; where B is some k –by– .n k/ matrix. 8.6.9. Use the previous two exercises, and induction on n to conclude that V has some basis with respect to which the matrix of T is upper triangular ; that means that all the entries below the main diagonal of the matrix are zero. 8.6.10. Suppose that A is the upper triangular matrix of T with respect to some basis of V . Denote the diagonal entries of A by . 1 ;:::; n / ; this sequence may have repetitions. Show that T .x/ D Q i .x i / . 8.6.11. Let .v 1 ;:::;v n / be a basis of V with respect to which the matrix A of T is upper triangular, with diagonal entries ....
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 Fall '08
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 Algebra, Determinant, Characteristic polynomial, Diagonal matrix, Triangular matrix, 1 k

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