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Unformatted text preview: Notice also that we have assumed t≠0 and t≠1 in order to carry out the row operations. This leads to the cases below. so the rank of this matrix is 3. so the rank of this matix is 1. Summarizing, 4) Let and be two different bases for . a. Find and . and b. Let and calculate and 5) Find the eigenvalues for . . The eigenvelues are = 3 and = 4. 6) Let u and v be eigenvectors of the matrix A corresponding to eigenvalues 1 and 2 . Prove that 1 ≠ 2 implies u and v are linearly independent. Assume that u and v are linearly dependent. It follows that u = k v (1) for some nonzero scalar k. Since u and v are eigenvectors we have (2) and (3). Combining (1) and (2) we have And It follows that or . Factoring we have Since k≠0 and v≠0, it follows that λ 1 = λ 2 . But the problem states that λ 1 ≠ λ 2 it follows that u and v are linearly independent....
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 Spring '11
 Burns
 Linear Algebra, Vector Space, standard row reductions

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