MAT 272 Assignment 6 - solutions - Notice also that we have...

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MAT 272 Assignment #6 - solutiuons Due: Dec. 10, 2009 1) Let V be a vector space with basis . Let x,y V with coordinates relative to B of {c 1 ,c 2 , …, c n } and {d 1 , d 2 , …d n } respectively. a. Find coordinates for x + y and k x relative to B. and so and Similarily b. Are your coordinates unique? Why? The coordinates are unique because the coordinates for x and y are unique. 2) Let V be a vector space and a basis for V. For any x V define the coordinate mapping for B as the function where and . Prove that T is a linear transformation. Must show that 1) T(x+y) = T(x) + t(y) and 2) T(kx) = kT(x). 1) In problem 1 we showed that so 2) In problem 1 we learned that so .
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3) Let . Describe how the rank of A depends on the value of t. You should be able to tell the values of t necessary for A to have rank 1, 2 or 3. Row reduce the matrix using standard row reductions. This last matrix has rank 3 unless t = -2 in which case the rank is 2.
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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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MAT 272 Assignment 6 - solutions - Notice also that we have...

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