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601hw-ii - MATH 601 AUTUMN 2007 Additional homework II...

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MATH 601, AUTUMN 2007 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 j = e j j e a a T a j under the basis changes e j = e j j e j , e a = e a a e a . 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 (1) - 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 ) = x 0 f ( ξ ) ξ defines a linear endomorphism T : W W , and evaluate Trace T . 4. Find the dimension of the kernel of the linear operator T : C 1 ((0 , 1) , R ) C ((0 , 1) , R ) given by ( Tf )( x ) = xf ( x ) + f ( x ) , where
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