solnsexamII - 1. SOLUTIONS (1)(20 points) ( a ) If the...

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Unformatted text preview: 1. SOLUTIONS (1)(20 points) ( a ) If the sequence a n satisfies lim n a n +1- a n = 0 then lim n a n exists and is finite. F Take a n = n. ( b ) If q is a polynomial of degree 5 and q (0) = q 00 (0) = q (4) (0) = 0 and q (0) = 1 , q 000 (0) =- 1 6 , q (5) (0) = 1 120 then q is the fifth degree Taylor polynomial for f ( x ) = sin x at c = 0 . T Since f (0) = f 00 (0) = f (4) (0) = 0 and f (0) = 1 , f 000 (0) =- 1 6 , f (5) (0) = 1 120 . ( c ) The matrix A = 0 1 0 0 0 1 0 0 0 is diagonalizable. F One way to see this is that the only eigenvalue is 0 , it has multiplicity 3 , and the eigenspace N ( A- I ) = { ( t, , 0) T : t R } so there are not three linearly independent eigenvectors. Another way to see this is that if A were diagonalizable, then for some invertible matrix X we would have A = XDX- 1 and D is the diagonal matrix with the eigenvalues of A on its diagonal. But this would be the 0 matrix and as a result A is also the 0 matrix, but it is not, so A can not be diagonalizable. 1 2 ( d ) If an eigenvalue of the matrix A has multiplicity greater than one, then A is defective....
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solnsexamII - 1. SOLUTIONS (1)(20 points) ( a ) If the...

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