Name:
AMAT 220: Linear Alge
a
Final Exam
May, 2022
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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 co
esponding 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?