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MATH 205 FINAL EXAM April 10, 2007
Name:
Your grade is based on correctness, completeness, and clarity on each exercise. You may use a calculator and the agreed-upon page of notes, but no books or other students. Good luck!
1.) (10 pts.) Given the quadratic form 8x^21 + 6x 1 x 2 ,
a.) (2 pts.) find the symmetric matrix of the quadratic form;
b.) (2 pts.) classify the quadratic form as positive definite, negative definite, or indefinite, and explain your reasoning;
c.) (6 pts.) make a change of variable, x = P y, that transforms the quadratic form into one with no cross-product term.
a.) (5 pts.) True or False: A least-squares solution of Ax = b is a vector ˆx such that ‖b − Ax‖ ≤ ‖b − Aˆx‖ for all x in Rn. If this is true, explain why. If it is false, correct the statement to make it true.
b.) (10 pts.) While boiling a pot of water, you take the temperature every two minutes. This generates the data points (0, 15), (2, 37), (4, 68), (6, 89), where the first coordinate is time, in minutes, and the second coordinate is temperature, in degrees Celsius. Using linear algebra techniques, find the equation y = β 0 + β 1 x of the least-squares line that best fits these data points.
a.) (5 pts.) Verify that v = (2, 1 , − 1 , 2) is an eigenvector of A, given below. What is the corresponding eigenvalue of v?
A =
b.) (5 pts.) Construct a 4 × 4 matrix with eigenvalues −3, 2, and 5 (with multiplicity 2). Your matrix should not be strictly diagonal - that is, there must be some nonzero entries in non-diagonal positions within the matrix.
c.) (5 pts.) Use the factorization A = P DP −^1 to compute Ak, where k represents an arbitrary positive integer.
A =
[
]
[
] [
] [
]
= P DP −^1
a.) (5 pts.) What are the three properties of a subspace H of Rn?
b.) (5 pts.) The shaded region in the image below is a set in R^2. (Include the bounding lines as part of the set.) Give a specific reason why this set is not a subspace of R^2.
c.) (5 pts.) Let A be an m × n matrix. Explain why Nul A is a subspace of Rm.
a.) (5 pts.) Suppose the vectors below are linearly independent. What can you say about the numbers a, b, c, d, e, and f?
a 0 0
b c 0
d e f
b.) (5 pts.) Write the coefficient matrix of the system of equations below.
3 x 2 − 6 x 3 + 8 x 4 = − 5 3 x 1 + x 3 − 2 x 4 = 7 4 x 1 + x 2 + 5 x 3 = 8
c.) (5 pts.) Write the augmented matrix of the system of equations in part (b). Does the system have a solution? How do you know?
