Science:Math Exam Resources/Courses/MATH221/April 2009/Question 07 (b)

MATH221 April 2009

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Question 07 (b)
Let $T:\mathbb {R} ^{2}\to \mathbb {R} ^{2}$ be the reflection across the line $x_{1}+3x_{2}=0$, and let A be the standard matrix of this linear transformation. Find the matrix 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 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

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Solution

Found a typo? Is this solution unclear? Let us know here. Please rate my easiness! It's quick and helps everyone guide their studies. From part (a), we know that 1 is an eigenvalue of A with eigenvector ${\begin{pmatrix}3\\-1\end{pmatrix}}$, and -1 is an eigenvalue of A with eigenvector ${\begin{pmatrix}1\\3\end{pmatrix}}$. If we form the matrix $P={\begin{pmatrix}3&1\\-1&3\end{pmatrix}}$ whose columns are the eigenvectors, with corresponding diagonal matrix $\Lambda ={\begin{pmatrix}1&0\\0&-1\end{pmatrix}}$, we know that the matrix A decomposes via the standard matrix diagonalization $A=P\Lambda P^{-1}.$ Upon computing $P^{-1}={\frac {1}{10}}{\begin{pmatrix}3&-1\\1&3\end{pmatrix}}$, we multiply the three matrices together to find A: $A={\frac {1}{10}}{\begin{pmatrix}3&1\\-1&3\end{pmatrix}}{\begin{pmatrix}1&0\\0&-1\end{pmatrix}}{\begin{pmatrix}3&-1\\1&3\end{pmatrix}}={\frac {1}{5}}{\begin{pmatrix}4&-3\\-3&-4\end{pmatrix}}.$ As a quick check of our work, we should verify that indeed $A{\begin{pmatrix}3\\-1\end{pmatrix}}={\begin{pmatrix}3\\-1\end{pmatrix}}$ and $A{\begin{pmatrix}1\\3\end{pmatrix}}=-{\begin{pmatrix}1\\3\end{pmatrix}}.$

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