solution6 - Let B be the standard ordered basis for C2 and...

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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 0 = { 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 0 ? b What is the matrix of T relative to the pair B 0 , B ? c What is the matrix of T in the ordered basis B 0 ? d What is the matrix of T in the ordered basis { α 2 , α 1 } ? a M = 2 0 - i 0 b M = 1 - i 0 0 c M = 2 - 2 i - i - 1 d - 1 - i - 2 i 2 Let V be a two-dimensional 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 . 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 0 0 ad - bc 1
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Therefore M 2 - ( a + d ) M + ( ad - bc ) I = 0 . Let θ be a real number. Prove that the following two matrices are similar over the field of complex numbers: cos( t ) - sin( t ) sin( t ) t cos( t ) , e it 0 0 e - it .
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