Science:Math Exam Resources/Courses/MATH152/April 2010/Question B 02 (b)
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Question B 02 (b) 

The matrix is known to have eigenvalues 1 and 4. (b) Find two linearly independent eigenvectors that correspond to eigenvalue 1. 
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 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 

Recall that eigenvectors with eigenvalue satisfy 
Checking a solution serves two purposes: helping you if, after having used the hint, you still are stuck on the problem; or if you have solved the problem and would like to check your work.

Solution 

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Please rate my easiness! It's quick and helps everyone guide their studies. We want to find two eigenvectors of
First Eigenvector: a=1, b=0 To find the first eigenvector we will use set the first free parameter to 1 and the second to 0 (a=1,b=0). Note that you can pick any values you want (except for a=0,b=0 because that will lead to the eigenvector being zero). With a=1 and b=0 we have
To find the second eigenvector we will use set the first free parameter to 0 and the second to 1 (a=0,b=1). Note that you can pick any values you want (except for a=0,b=0 because that will lead to the eigenvector being zero) as long as they are different from the first values you chose, otherwise you will get the same eigenvector. With a=0 and b=1 we have
