Science:Math Exam Resources/Courses/MATH221/December 2007/Question 08 (d)
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Question 08 (d) 

Let be the linear map whose matrix is given by (a shear in the ydirection). Find a basis of consisting of eigenvalues of T, or explain why this is not possible. 
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Hint 

Science:Math Exam Resources/Courses/MATH221/December 2007/Question 08 (d)/Hint 1 
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Solution 

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Please rate my easiness! It's quick and helps everyone guide their studies. No such basis exists. Note that 1 is the only eigenvalue for T, as the characteristic polynomial is . To determine the eigenvectors corresponding to the eigenvalue 1, we solve the corresponding equation
Thus is an eigenvector for T if and only if x = 0. This means that there is only one linearly independent eigenvector, namely . As a basis of consists of two linearly independent vectors, there exists no basis consisting of eigenvectors of T. 