Science:Math Exam Resources/Courses/MATH152/April 2013/Question B 01 (d)
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Question B 01 (d) 

Let be the line . Find an eigenvector of the matrix A associated to the eigenvalue 1 where A was computed in part (b) and represents the reflection across the line L. 
Make sure you understand the problem fully: What is the question asking you to do? Are there specific conditions or constraints that you should take note of? How will you know if your answer is correct from your work only? Can you rephrase the question in your own words in a way that makes sense to you? 
If you are stuck, check the hints below. Read the first one and consider it for a while. Does it give you a new idea on how to approach the problem? If so, try it! If after a while you are still stuck, go for the next hint. 
Hint 1 

Remember that the eigenvector(s), corresponding to a particular eigenvalue of a matrix satisfy where I is the identity matrix. 
Hint 2 

For an alternative solution, think about which vector flips it's direction when reflected along a line? 
Checking a solution serves two purposes: helping you if, after having used all the hints, you still are stuck on the problem; or if you have solved the problem and would like to check your work.

Solution 1 

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Please rate my easiness! It's quick and helps everyone guide their studies. From the statement of the problem we know that is an eigenvalue of the matrix A. The corresponding eigenvector satisfies the following equation where I is the identity matrix. Solving for gives: Any nonzero constant multiple of is also an acceptable answer. 
Solution 2 

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Please rate my easiness! It's quick and helps everyone guide their studies. Any vector perpendicular to the line of reflection will flip it's direction, and hence be an eigenvector with eigenvalue 1. To find the vector that is perpendicular to the line we rewrite 3x=4y in normal form In other words, all point on the line 3x=4y are perpendicular to . Hence must be an eigenvector with eigenvalue 1. 