MATH221 April 2013
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Question 07 (a)

Let $T:\mathbb {R} ^{2}\rightarrow \mathbb {R} ^{2}$ be the linear transformation that reflects points through the line 3x = 4y.
a) Find the eigenvalues and eigenvectors of the standard matrix A of T. (Note: In order to do this, you do not need to evaluate A.)

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

Draw a picture which illustrates what the reflection does to a few example vectors.

Hint 2

 What does the reflection do to a vector that is on the line $\ 4y=3x$?
 What does the reflection do to a vector that is orthogonal to the line $\ 4y=3x$?

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Solution

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 For any nontrivial reflection, the eigenvalues are always 1 and 1.
 For the eigenvalue 1, we can pick any vector on the line itself. Such a vector will not be moved by the linear transformation. So, for us, we pick $\left[{\begin{array}{c}4\\3\end{array}}\right]$ as an eigenvector to the eigenvalue 1. Note that (x,y) = (4,3) is a point on the line 3x=4y since 3(4)=4(3).
 The eigenvalue of 1 comes from picking a vector in the line that is perpendicular (i.e. orthogonal) to the line $4y=3x\$. Any such vector will simply change its sign, thus will correspond to the eigenvalue 1. We choose $\left[{\begin{array}{c}3\\4\end{array}}\right]$ as an eigenvector to the eigenvalue 1.

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