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Unformatted text preview: MATH 405 Take Home Final Exam W. Adams Due: December 17, 2003 Show all work. Justify all answers. Use an examination book and sign the pledge. That is, ALL work on this exam must be done alone. No books should be consulted except the text and your text from the first course. NOTE: All vector spaces on this examination should be assumed to be finite dimensional and over the field F unless otherwise stated. 1. (20 points) Consider the vector space V = P 3 ( F ) with basis 1 , t, t 2 , t 3 . Define the operator T on V by T ( f ( t )) = f ( t- 1). (You may assume everything stated so far.) (a) Find the matrix of T with respect to the given basis. (b) Using the previous part of the problem find the matrix of T 2 with respect to the given basis. 2. (25 points) Let H be a bilinear from on a finite dimensional vector space V . For each x V define L x V * by L x ( y ) = H ( x, y ) for all y V . (It has already been noted in the book that L x is a linear transformation, i.e. is in V *...
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