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**Unformatted text preview: **MATH 601, AUTUMN 2008 Additional homework VIII, November 19 PROBLEMS 1. Let h , i be the symmetric bilinear form on R 2 given by h ( x,y ) , ( a,b ) i = 4 xa + 2 yb- xb- ya . Verify that h , i is an inner product and find the minimum distance d between the vector v = (1 , 0) and all vectors forming the line L given by y = x . (The distance corresponds here to the norm determined by h , i , and not the standard Euclidean norm.) 2. For a fixed integer n ≥ 2, let V be the space of all polynomials f of the real variable x with deg f ≤ n and f (0) = f (1) = 0, endowed with the L 2 inner product h f,h i = Z 1 f ( x ) h ( x ) dx, and let T : V → V be the differential operator given by ( Tf )( x ) = ( x 2- x ) f 00 ( x ). Find an explicit description of the adjoint T * . 3. Find the least-squares solution ( x,y,z ) of the problem 1 1 1 1 1 1 1 1 1 1 1 · x y z = 2 1 , where R 5 is endowed with the standard Euclidean inner product. 4. Find the function f of the form f ( x ) = ax + b the graph of which provides the best fit, in the least-squares sense, to the data (- 1 , 1) , (0 , 1) , (1 ,- 1) , (2 , 5) in the xy-plane. 5. Find the function f of the form f ( x ) = ax 2 + bx + c the graph of which provides the best fit, in the least-squares sense, to the data (- 1 , 0) , (0 ,- 1) , (1 ,- 1 / 2) , (2 , 0) in the xy-plane. 6. In the space C ([0 , 1] , R ) of all real-valued continuous functions f : [0 , 1] → R on the closed interval [0 , 1], with the L 2 inner product defined as in Problem 2, find the system h 1 ,h 2 ,h 3 obtained by the standard orthonormalization of the system f 1 ,f 2 ,f 3 with f 1 ( x ) = 1 , f 2 ( x ) = √ x , f 3 ( x ) = x . 7. For a fixed integer n ≥ 2, let V be the space of all polynomials f of the real variable x with deg f ≤ n , endowed with the L 2 inner product (see Problem 2), and let W be the subspace of V consisting of all such polynomials that are L 2-orthogonal to x and x 2 . Find the minimum distance d between the constant polynomial 1 and all vectors in W . 8. Let T : R 3 → R 3 be the operator defined by T x y z = 16 25 + 9 √ 3 50 3 10 12- 6 √ 3 25- 3 10 √ 3 2 2 5 12- 6 √ 3 25- 2 5 9+8 √ 3 25 · x y z . Is T a rotation by some angle θ about an axis (i.e., a 1-dimensional subspace) L ⊂ R 3 ?...

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