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Final Summer 2003 done

# Final Summer 2003 done - 110.202 Linear Algebra 2003 Summer...

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110.202. Linear Algebra 2003 Summer Final 6/16/2003 12:00Noon 1. (20pts) Let A = 011 101 110 . (a) Find an orthogonal matrix S and a diagonal matrix D such that S 1 AS = D . (b) Find a formula for the entries of A t ,wh e r e t is a positive integer. Also f nd the vector lim t →∞ A t 1 0 1 . 2. (20pts) Let A = 01 11 10 . (a) Find a singular value decomposition for A . (b) Describe the image of the unit circle under the linear trans- formation T ( ±x )= A±x . 3. (10pts) Let q be a quadratic form q ( x 1 ,x 2 )=9 x 2 1 4 x 1 x 2 +6 x 2 2 . (a) Determine the de f niteness of q . (b) Sketch the curve de f ned by q ( x 1 2 )=1 . Draw and label the principal axes, label the intercepts of the curve with the prin- cipal axes, and give the formula of the curve in the coordinate system de f ned by the principal axes. 1

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2 4. (10pts) Decide whether the matrix A = 111 010 is diagonalizable. If possible, f nd an invertible S and a diagonal D such that S 1 AS = D . 5. (10pts) Find the trigonometric function of the form f ( t )= c 0 + c 1 sin ( t )+ c 2 cos ( t ) that best f ts the data points (0 , 1) , ¡ π 2 , 2 ¢ , ( π, 2) and ¡ 3 π 2 , 1 ¢ , using lease squares. 6. (10pts) Given a matrix A = 102
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Final Summer 2003 done - 110.202 Linear Algebra 2003 Summer...

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