3 - Assignment for Applied Linear Algebra - Math 270...

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Assignment for Applied Linear Algebra - Math 270 February 11, 2011 Problem 1 (3 pts.) Find the coefficients b 1 ,b 2 ,b 3 R such that sin( t ) cos(2 t ) = b 1 sin( t ) + b 2 sin(2 t ) + b 3 sin(3 t ) . SOLUTION: sin( t ) cos(2 t ) = ± e it - e - it 2 i ²± e i 2 t + e - i 2 t 2 ² = 1 4 i ( e i 3 t + e - it - e it - e - i 3 t ) = - 1 2 e it - e - it 2 i + 1 2 e i 3 t - e - i 3 t 2 i . Therefore: b 1 = - 1 2 , b 2 = 0, b 3 = 1 2 . Problem 2 (9 pts.) Let P 1 be the vector space of real polynomials of degree less or equal than 1. Define the (nonlinear) function E : P 1 R as E ( p ) = Z 1 0 ± 2 π cos ³ πx 2 ´ - p ( x ) ² 2 dx, (where p = p ( x ) is a polynomial in P 1 ). Find the point of minimum for E , i.e., find the polynomial q P 1 such that E ( q ) E ( p ) for all p P 1 . SOLUTION: For brevity, let f ( x ) := 2 π cos( πx 2 ), then, using the length k·k defined in the hint, min p P 1 E ( p ) = min p P 1 k f - p k 2 is given by the orthogonal projection of f on P 1 . The projection can be computed in the following
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3 - Assignment for Applied Linear Algebra - Math 270...

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