Science:Math Exam Resources/Courses/MATH152/April 2012/Question 03 (a)
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Question 03 (a) 

Let A be the matrix for the linear transformation T. Suppose we know that , , are eigenvectors of A associated to the eigenvalues , , and respectively. Express , , and as linear combinations of , and . 
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 

Expressing a vector as a linear combination of three vectors means finding coefficients a, b, c such that 
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. Let's start with . We could solve these problems by inspection but let's, at least for this case, show how one could go about solving this schematically. We want to find a, b, c such that Writing the above in matrix form we get In general we will always get a matrix with columns that are the vectors we are trying to combine. If we rowreduce this matrix (try it for yourself!) we will get Hence a = 1, b = 1, c = 0, and so finally we can write We want to emphasize that it is perfectly fine to inspect the solution visually, however this technique will always work if you are stuck. For the other vectors we will write down the answers by inspection. For , we have and for we have 