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Material Type: Exam; Class: Matrices & Linear Algebra; Subject: Mathematics; University: University of North Carolina - Charlotte; Term: Fall 2008;
Typology: Exams
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EXAM STRATEGY The exam has 225 total points (75 points on the mandatory part, 150 points on the optional part). As a reminder, if you circled “No” above, I will substitute the average of your midterm exam scores in its place. If you circled “Yes” above, no such substitution will take place. Your final exam score will replace the lower of your two midterm scores, provided the final exam score is not the lowest of the three. You have 3 hours to finish this exam. It is roughly 1/3 new material and 2/3 old material. So if you’re writing both parts, you should spend 1 hour on the mandatory part and 2 hours on the optional part.
MANDATORY PART OF EXAM (75 points)
MANDATORY CONCEPTUAL QUESTIONS (4 points each - 20 points total).
MANDATORY FREE RESPONSE QUESTIONS (55 points total).
(a) (3 points) Write down the change of coordinates matrix PB (no work required).
(b) (3 points) Find [x]B if x =
. (Hint: PB[x]B = x. Go to echelon form and use back substi-
tution.)
(c) (2 points) Is B linearly independent? Why or why not? (Hint: Cite your work from part (b).)
(d) (2 points) Is B a basis for R^3? Why or why not? (Hint: Cite your answer from part (c).)
(a) (1 points) Reduce A to an echelon form (do not go to reduced echelon form). Circle the pivots.
(b) (3 points) What is rank A? (Hint: rank is the dimension of the column space – no work required).
(b) (3 points) What is nullity A? (Hint: nullity is the dimension of the null space – no work required).
(c) (3 points) What is rank A + nullity A? Does Rank + Nullity Theorem hold? Why or why not?
(a) (3 points) What are the eigenvalues of A^5? (Hint: Since A is triangular, so is A^5. Think about the diagonal of A^5 , but do not calculate A^5 .)
(b) (4 points) Obviously λ = 2 is an eigenvalue of A with multiplicity 2. Determine E 2 , the eigenspace corresponding to λ = 2. (Hint: Recall E 2 = Nul (A − 2 I).)
(c) (3 points) Explain why A is not diagonalizable. (Hint: Cite what you found out in part (b).)
. Diagonalize A as A = P DP −^1 by following these steps:
(a) (3 points) Determine the eigenvalues of A and their multiplicities.
(b) (3 points) Find a basis for each eigenspace.
OPTIONAL PART OF EXAM (150 points)
CONCEPTUAL QUESTIONS (4 points each - 40 points total).
FREE RESPONSE QUESTIONS (110 points total).
(a) (2 points) A =
. One-to-One: YES or NO. Onto: YES or NO.
(b) (2 points) A =
. One-to-One: YES or NO. Onto: YES or NO.
(c) (2 points) A =
. One-to-One: YES or NO. Onto: YES or NO.
(d) (2 points) A =
. One-to-One: YES or NO. Onto: YES or NO.
(a) (4 points) Express Nul A as the span of a set of vectors.
(b) (4 points) Express Col A as the span of a set of vectors. (Hint: Reuse your work from (a).)
(c) (4 points) Express Row A as the span of a set of vectors. (Hint: Reuse your work from (a).)
, v 2 =
, v 3 =
h
, B = {v 1 , v 2 , v 3 }, and u =
(a) (5 points) For what values of h is B linearly independent? (One row reduction step is enough!)
(b) (5 points) Assuming h = 1, express u as a linear combination of the vectors in B (i.e. find x 1 , x 2 , x 3 such that u = x 1 v 1 + x 2 v 2 + x 3 v 3 ). (Hint: Go to echelon form and use back-substitution.)
(a) (5 points) Find the LU factorization of A (Hint: One row reduction step gives you L and U ).
(b) (5 points) Use the LU factorization found in part (a) to solve Ax =
(a) (5 points) Find the standard matrix A of T.
(b) (5 points) Find a vector x such that T (x) = (1, 3).
(a) (6 points) Calculate A−^1.
(b) (4 points) Using A−^1 you calculated in part (a), solve Ax =
. (Hint: Use a certain matrix-
vector multiplication and not row reduction).