la_final - A . (b) Find a matrix Q such that Q-1 AQ is...

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Linear Algebra, Fall 2002 Final Exam Jan 10 2003, 13:10 PM - 15:00 PM. SHOW ALL YOUR WORK (1) (10 pts) Let S = span { (1 , 2 , 3) T } , find a basis for S . (2) (20 pts) Given a set of data ( x,y ) = { ( - 1 , 0) , (0 , 1) , (1 , 3) , (2 , 5) } (a) Find the best least square fit by a linear function. (b) Find the best least square fit by a quadratic function. (3) (25 pts) A = 1 1 1 0 1 0 1 0 1 , b = 1 1 1 (a) Find a basis for N ( A ), R ( A ), N ( A T ) and R ( A T ) respectively. (b) Is b R ( A )? Explain. (c) Find y 0 R ( A ) such that k b - y 0 k ≤ k b - y k for all y R ( A ). (4) (25 pts) A = 2 0 2 0 1 0 2 0 3 (a) Find the eigenvalues and eigenvectors of
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Unformatted text preview: A . (b) Find a matrix Q such that Q-1 AQ is diagonal. (c) Find a matrix B such that B 2 = A . (d) Compute A 10 . (5) (20 pts) True of False? Explain. (a) If A is diagonalizable and B is similar to A , then B is also diagonalizable. (b) If A have distinct eigenvalues then A is diagonalizable. (c) If the eigenvalues of A are not distinct, then A is not diagonalizable. (d) If A and B are both diagonalizable with the same diagonalizing matrix, then AB = BA . 1...
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