ass4 - ASSIGNMENT 4 MATH 270 WINTER 2010 DUE 5PM...

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ASSIGNMENT 4 MATH 270, WINTER 2010. DUE: 5PM, MARCH 25, 2010. Assignments must have your name, student number, course number and TA’s name on the first page. Please submit them by placing them in the box marked “Assignments” outside the Maths Office (Burn- side Hall, 10th floor). Show all working and justifications! (1) Maclaurin series are all about approximating functions around x = 0 by polynomials. For these ap- proximations, it is natural to use the basis 1 , x , x 2 , x 3 ,... . However, linear algebra thinks that this is a lousy basis because it fails miserably at being orthogonal. (a) Perform Gram-Schmidt on the set E = 1 , x , x 2 , x 3 to get an orthonormal basis B for the vector space P of real polynomials of degree 6 3. Use the inner-product h p , q i = 1 2 Z 1 - 1 p ( x ) q ( x ) d x . These orthonormal polynomials are known as Legendre polynomials and crop up in all sorts of applications. [ 5 marks ] (b) Consider the linear differential operator L = ( x 2 - 1 ) d 2 d x 2 + 2 x d d x acting on the space P . Compute its matrix representatives [ L ] E and [ L ] B with respect to the bases E and B . [ 3 marks ] (c) Conclude that the first four Legendre polynomials are eigenvectors of L (acting on P ). What are the corresponding eigenvalues? [ 2 marks ] [ Total: 10 marks ] (2) (a) Diagonalise the permutation matrix A = 0 0 0 1 1 0 0 0 0 1 0 0 0 0 1 0 , giving the entries of the diagonal matrix D in order of increasing argument
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