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math34600spring2005

# math34600spring2005 - THE CITY COLLEGE MATHEMATICS 34600...

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THE CITY COLLEGE DEPARTMENT OF MATHEMATICS MATHEMATICS 34600 FINAL EXAMINATION SPRING 2005 PART I: ANSWER ALL THREE QUESTIONS (40%) 1. (15) Let A = . 0 1 1 1 2 1 2 2 1 a. Find the eigenvalues and bases for the eigenspaces of A. b. Find a diagonal matrix D and an invertible matrix P such that P AP = D. 1 c. Use the result of part b. to compute A . 5 2. (10) Let T: V W be a linear transformation. a. Define the kernel of T and prove that it is subspace of V . b. Define the range of T and prove that it is subspace of W . 3. (15) Suppose L: P P is given by L (p(t)) = 2p(t) – 3p'(t). Given the basis B = {1, 3t + 1, 2t } for P , 2 2 2 2 a. Find the B -coordinates of p(t) = 5 – 6t +4t . 2 b. Find [L] , the B -matrix for L. B c. Using the results of parts a. and b., compute [L(5 – 6t +4t )] . 2 B PART II: ANSWER FIVE OUT OF SEVEN QUESTIONS (OMIT TWO) (60%) 4. (12) Suppose that S = { v 1 , v } is a linear independent set and that

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math34600spring2005 - THE CITY COLLEGE MATHEMATICS 34600...

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