Science:Math Exam Resources/Courses/MATH152/April 2015/Question A 28
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Question A 28 

Let be a matrix which represents a reflection across a line in . Suppose that is an eigenvector with eigenvalue and is an eigenvector with eigenvalue . What are the values of and ? 
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If you are stuck, check the hint below. Consider it for a while. Does it give you a new idea on how to approach the problem? If so, try it! 
Hint 

Observe that the points fixed by a reflection across a line are exactly the points on the line of reflection. What happens to vectors perpendicular to this line, and what does this tell you about the eigenvalues and eigenvectors of ? 
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Solution 

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Please rate my easiness! It's quick and helps everyone guide their studies. Recall that the fixed points (i.e., eigenvectors of eigenvalue 1) of a reflection are the points on the line of reflection. Since we are given that the vector must lie on the line of reflection. Any vector perpendicular to must therefore point in the opposite direction after reflection (i.e., must be an eigenvector of eigenvalue 1). For instance, We have thus found two eigenvalues of : 1 and 1. These are all of its eigenvalues, since a 2by2 matrix can have at most 2 distinct eigenvalues. Since the eigenvector we seek does not correspond to the eigenvalue 1, it must correspond to the eigenvalue Now, the eigenspace of each of these eigenvalues is onedimensional (since each has dimension at least 1, and the sum of their dimensions is at most 2), so it follows that for some whence Hence 