Science:Math Exam Resources/Courses/MATH221/April 2013/Question 07 (b)
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Question 07 (b) 

Let be the linear transformation that reflects points through the line 3x = 4y. b) Using the result of a), find A (the standard matrix of T). 
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 

Using the information from part a), the matrix of T can be described by using the eigenvectors and eigenvalues. 
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Solution 

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Please rate my easiness! It's quick and helps everyone guide their studies. What we know from part a) Using the eigenvectors , and the eigenvalues , we know that and .
Then, we can write the information from above in matrix notation.
The first equality follows from this: when we product two 2 by 2 matrices, the first column of the resulting matrix is the product of the first matrix with the first column of the second matrix. Similarly, the second column of the resulting matrix is the product of the first matrix with the second column of the second matrix. In the last step we use, in reverse, that the product of two matrices is the sum of (columns of the first matrix) times (rows of the second matrix). We define the matrix as the matrix of eigenvectors of and write the equation above as
This leads us to .
First the inverse of B is given by the usual method for 2x2 matrices (where ):
We can now compute A as claimed: . 