601-hw-ii - MATH 601, AUTUMN 2008 Additional homework II,...

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MATH 601, AUTUMN 2008 Additional homework II, October 2 PROBLEMS 1. Given a linear operator T : V W between finite-dimensional vector spaces V, W , along with bases e j in V and e a in W , we characterize the components T a j of T relative to the e j and e a by Te j = T a j e a . Verify the transformation rule T a 0 j 0 = e j j 0 e a 0 a T a j under the basis changes e j 0 = e j j 0 e j , e a = e a 0 a e a 0 . 2. Let V denote the real vector space of polynomials in the real variable x whose degree does not exceed 5, and let F V * be given by = ϕ 0 (1) - R 1 0 ϕ ( x ) dx . Find the coefficients F j of the expansion F = F j e j , where e j is the dual basis for the basis e j of V such that for each j = 1 , . . . , 6 , e j equals x to the power j - 1. 3. Denote W the real vector space of all polynomials f = f ( x ) in the real variable x for which deg f 3 and f (0) = 0. Verify that the formula ( Tf )( x ) = Z x 0 f ( ξ ) ξ defines a linear endomorphism T : W W , and evaluate Trace T . 4.
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601-hw-ii - MATH 601, AUTUMN 2008 Additional homework II,...

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