This preview shows pages 1–3. Sign up to view the full content.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
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 = { 1 , 2 } be the ordered basis defined by 1 = (1 ,i ) , 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 { 2 , 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 ....
View Full
Document
 Spring '10
 aa

Click to edit the document details