Science:Math Exam Resources/Courses/MATH152/April 2015/Question B 6 (b)
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Question B 6 (b) 

The matrix represents a rotation in 3D relative to some axis.

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 

If a vector is unchanged by a linear transformation, that means it is an eigenvector with eigenvalue 1. 
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 are looking for the eigenvector associated with eigenvalue . This means that solves the equation It remains to solve for . We do this by row reduction, beginning with dividing the second row by : Adding the first row to the second and third rows: Lastly we add times the second row to the first row and multiply the first row by and the second row by (to get leading zeros): Translating this back into a system of equations, this matrix is saying and . Thus, this operation rotates vectors around the axis spanned by the vector 