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# ass2 - ASSIGNMENT 2 MATH 270 WINTER 2010 DUE 5PM FEBRUARY 9...

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ASSIGNMENT 2 MATH 270, WINTER 2010. DUE: 5PM, FEBRUARY 9, 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 (Burnside Hall, 10th floor). Show all working and justifications! (1) (a) Use Gaussian elimination to invert the matrix U = 1 3 1 - 2 - 2 - 2 1 - 2 - 2 - 2 1 and thereby compute all powers U n of U . [ 6 marks ] (b) Substitute U into the Maclaurin series expansion of cos x to show that cos U is a mul- tiple of the identity matrix. [ 4 marks ] [ Total: 10 marks ] (2) Define the degree of a polynomial p ( x ) to be the maximal power of x appearing. For example, x 2 + 3 x + 2 has degree 2 whereas the constant polynomial 5 has degree 0. Funnily enough, the degree of the zero polynomial is taken to be - . (a) Is the set of all polynomials of degree 6 a vector space? [ 1 mark ] (b) Is the set of all polynomials of degree less than 6 a vector space? [ 1 mark ] (c) How about the set of all polynomials of degree greater than 6?

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