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Unformatted text preview: Homework assignment 6 pp. 95 Exercise 1. Let T be the linear operator on C 2 defined by T ( x 1 ,x 2 ) = ( x 1 , 0) . Let B be the standard ordered basis for C 2 and let B = { α 1 ,α 2 } be the ordered basis defined by α 1 = (1 ,i ) , α 2 = ( i, 2) . a What is the matrix of T relative to the pair B , B ? b What is the matrix of T relative to the pair B , B ? c What is the matrix of T in the ordered basis B ? d What is the matrix of T in the ordered basis { α 2 ,α 1 } ? Solution: a M = 2 i ¶ b M = 1 i ¶ c M = 2 2 i i 1 ¶ d 1 i 2 i 2 ¶ Exercise 4. Let V be a twodimensional vector space over the field F , and let B be an ordered basis for V . If T is a linear operator on V and [ T ] B = M = a b c d ¶ prove that T 2 ( a + d ) T + ( ad bc ) I = 0 . Solution: Notice that the zero operator and the identity operator have the same matrix with respect to any basis, namely, the zero matrix and the identity matrix. Therefore we only need to check the equality with respect to the basis B . M 2 = a 2 + bc ab + bd ca + dc bc + d 2 ¶ , ( a + d ) M = a 2 + da ba + bd ac + cd ad + d 2 ¶ , ( ad bc ) I = ad bc ad bc ¶ 1 Therefore M 2 ( a + d ) M + ( ad bc ) I = 0 ....
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This note was uploaded on 11/14/2010 for the course PHYCS 498 taught by Professor Aa during the Spring '10 term at University of Illinois, Urbana Champaign.
 Spring '10
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