Science:Math Exam Resources/Courses/MATH152/April 2015/Question B 2 (c)
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Question B 2 (c) 

Let be the matrix

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 

Don't try to find other eigenvectors. Note that we have three distinct eigenvalues; what does that mean? 
Hint 2 

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. In fact, eigenvectors corresponding to distinct eigenvalues are linearly independent. Let’s prove it. Suppose that are eigenvectors corresponding to distinct eigenvalues , respectively. Let be the largest integer for which are linearly independent. If we had , we could write for some not all zero. Now , whence . As the are distinct, this implies (by linear independence) that , so , contradicting the assumption that was an eigenvector. We conclude that , so all the eigenvectors are linearly independent. Each of the 3 eigenvalues found in part (b) has a corresponding eigenvector, and by the above, they are linearly independent. (Clearly, there cannot be more linearly independent eigenvectors, since the dimension of the matrix is 3.) Thus, the answer is 
Solution 2 

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