MATH 110 - Spring 2005 - Ribet - Final (solution)

# MATH 110 - Spring 2005 - Ribet - Final (solution) - Math...

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Math 110 Professor K. A. Ribet Final Exam May 18, 2005 This exam was a 180-minute exam. It began at 5:00PM. There were 7 problems, for which the point counts were 8, 9, 8, 7, 8, 7, and 7. The maximum possible score was 54. Please put away all books, calculators, electronic games, cell phones, pagers, .mp3 players, PDAs, and other electronic devices. You may refer to a single 2-sided sheet of notes. Explain your answers in full English sentences as is customary and appropriate. Your paper is your ambassador when it is graded. At the conclusion of the exam, please hand in your paper to your GSI. 1. Let T be a linear operator on a vector space V . Suppose that v 1 , . . . , v k are vectors in V such that T ( v i ) = λ i v i for each i , where the numbers λ 1 , . . . , λ k are distinct elements of F . If W is a T -invariant subspace of V that contains v 1 + ··· + v k , show that W contains v i for each i = 1 , . . . , k . See my solutions for homework set #11. 2. Assume that T : V W is a linear transformation between ﬁnite-dimensional vector spaces over F . Show that T is 1-1 if and only if there is a linear transfor- mation U : W V such that UT is the identity map on V . One direction is obvious; if UT = 1 V and T ( v ) = 0, then v = U ( T ( v )) = 0, so that T must be injective. The harder direction is to construct U when T is given as 1-1. Choose a basis v 1

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## This note was uploaded on 07/03/2011 for the course MATH 110 taught by Professor Gurevitch during the Spring '08 term at Berkeley.

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MATH 110 - Spring 2005 - Ribet - Final (solution) - Math...

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