Science:Math Exam Resources/Courses/MATH221/April 2009/Question 07 (a)
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Question 07 (a) 

Let be the reflection across the line , and let A be the standard matrix of this linear transformation. Find a basis for consisting of eigenvectors of A. 
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Hint 

Science:Math Exam Resources/Courses/MATH221/April 2009/Question 07 (a)/Hint 1 
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Solution 

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Please rate my easiness! It's quick and helps everyone guide their studies. We first note that as T is a reflection mapping, it leaves the length of a vector unchanged. This is important as it allows us to determine the eigenvalues and eigenvectors without ever having to explicitly determine what T is. Since T leaves the length of a vector unchanged, the only eigenvalues are 1 and 1. The eigenvalue 1 corresponds to a vector which is invariant under the reflection; this vector must therefore lie on the line . Now the vector with coordinates lies on the line , so we conclude is an eigenvector corresponding to the eigenvalue 1. The eigenvalue 1 corresponds to vectors which, under the reflection, has a 180 degree reversal in orientation. Such vectors necessarily must lie on the line perpendicular to the line . The equation for this perpendicular line is given by , and the vector lies on this perpendicular line. Therefore is an eigenvector corresponding to the eigenvalue 1. Hence, we conclude is a basis for consisting of eigenvectors of T. 