finweb2003 - M346 Final Exam 1 In the vector space R 2 t...

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Unformatted text preview: M346 Final Exam, December 10, 2003 1. In the vector space R 2 [ t ], consider the basis B = { 1 , 1 + t, 1 + t + t 2 } , the basis D = { 1 + t + t 2 , t + t 2 , t 2 } and the vector v = 2 t 2- 1. a) Find [ v ] B and [ v ] D . (That is, find the coordinates of v in the B basis, and the coordinates of v in the D basis.) b) Find the change-of-basis matrices P BD and P DB . 2. Consider the operator L : R 2 [ t ] → R 2 [ t ] defined by ( L p )( t ) = p ( t +1)- p ( t ). a) Find the matrix of L in the standard basis { 1 , t, t 2 } . b) Find the matrix of L in the basis B = { 1 , 1 + t, 1 + t + t 2 } (this is the same basis B you saw in problem 1). 3. Matrices and eigenvalues: a) Find the eigenvalues and eigenvectors of the matrix 2 1 4 2 1 1 2 . b) Find a matrix whose eigenvalues are 1 , 3 , 4 and eigenvectors are 1 3 1 2 2 , 1 3 2 1- 2 and 1 3 - 2 2- 1 . [Hint: there is an easy way to compute P- 1 ] 4. Let A =- 1 2 2- 4 . (You may find useful the fact that A is Hermitian, but you don’t need this fact to solve the problem)....
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finweb2003 - M346 Final Exam 1 In the vector space R 2 t...

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