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Unformatted text preview: Notice also that we have assumed t0 and t1 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 non-zero 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 k0 and v0, it follows that 1 = 2 . But the problem states that 1 2 it follows that u and v are linearly independent....
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- Spring '11