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quiz2_S07

# quiz2_S07 - (a Find the 3-volume of the parallelepiped...

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Princeton University Department of Mathematics Name: Instructor: A. Cannas J. Holland N. Pavlovic P. Raphael A. Salehi Golsefidy J. Szeftel A. Yang Class time: MAT 202 – Linear Algebra with Applications QUIZ 2 – Spring 2007 This quiz is due Friday, April 20, at 2 pm at your instructor’s office. You have 60 minutes to complete your work. Please show all work , explain your answers and write neatly. Each lettered part of a problem corresponds to 10 points. Books, notes, calculators, computers are not permitted on the quiz. As part of your obligations under the Honor Code, do not discuss this quiz with anyone until after the Friday 2 pm deadline. PLEASE WRITE OUT AND SIGN THE PLEDGE: I pledge my honor that I have not violated the Honor Code during this examination. for the grading problem points score 1 20 2 20 3 20 4 20 5 20 total 100

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2 (1) Let A = 0 1 1 1 1 1 and b = 2 3 5 . (a) Find the least-squares solution of the system Ax = b .
3 (b) What is the vector in the image of A which is closest to the vector b ?

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4 (2) Let A =

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Unformatted text preview: . (a) Find the 3-volume of the parallelepiped deﬁned by the columns of A . 5 (b) Let v be a unit vector in R 4 . The augmented matrix [ A | v ] is 4 × 4 . What is the maximum value det[ A | v ] may take? 6 (3) Let A = ± 5-4 4-5 ² . (a) Compute the eigenvalues and the eigenvectors of A . 7 (b) Is the matrix A similar to B = ± 3 1 3-2 ² ? 8 (4) Let A be the 2 × 2 matrix with eigenvalues 1 2 and 2 , for which ± 2 1 ² and ± 1 2 ² are corresponding eigenvectors. Consider the discrete dynamical system ± x n +1 y n +1 ² = A ± x n y n ² . (a) For the initial value ± x y ² = ± 3 3 ² , ﬁnd the limit of the quotient y n x n as n goes to + ∞ . 9 (b) Find A . 10 (5) Determine whether the following statements are true or false, explaining why. (a) There is a nonzero skew-symmetric matrix A such that I n + A is an orthogonal matrix. (b) For any n × m matrix A , we have im ( A T ) = im ( A T A ) ....
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