Please check attached file.

Please check attached file.


Name: AMAT 220: Linear Algebra Final Exam May, 2022 Show all work for each problem in the space provided. If you run out of room for an answer, continue on the back of the page. Question Points Bonus Points Score 1 0 0 2 0 0 3 0 0 4 0 0 Total: 0 0 1. Given the matrix A =  1 1 2 0 3 2 0 0 9  OR A =  1 1 1 0 2 −1 0 −3 0  1. Compute the characteristic polynomial of A. 2. Using 1., compute the eigenvalues of A. 3. Find bases of the eigenspaces corresponding to the eigenvalues found in 2. 2. Given the matrix A =  0 1 1 1 0 1 1 1 0  OR A =  7 0 5 0 5 0 −4 0 −2  1. Assume A = PDP−1, that is, that A is diagonalizable. Compute P . 2. Find the inverse of P in part one. 3. Compute A6. 3. Consider the transformation T : R3[x]→ R2[x] given by T (ax3 + bx2 + cx+ d) = cx+ d. OR Consider the transformation T : R2 → R2 given by T (x1e1 + x2e2) = 2x1 + 3x2 4x1 − 5x2  1. Verify that T is a linear transformation. 2. Compute the matrix of T relative to the bases B = {1, x, x2, x3} and D = {1, x, x2} OR E = {e1, e2} is the standard basis of R2 and B = {b1 = 1 2  b2 = 2 5 } 3. Is the transformation T diagonalizable? Why or why not? Justify your answer. 4. Let T (x) be a linear transformation from R3 onto itself and suppose R3 is spanned by the non-standard basis B = {b1, b2, b3} consisting of eigenvectors of T . Suppose further that T (x) = Ax, where A =  0 1 1 1 0 1 1 1 0  OR A =  7 0 5 0 5 0 −4 0 −2  1. Compute a matrix representation of T?