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

MATH221 April 2009

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Question 07 (a)
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 a basis for $\mathbb {R} ^{2}$ consisting of eigenvectors of A.

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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. We first note that as T is a reflection mapping, it leaves the length of a vector unchanged. This is important as it allows us to determine the eigenvalues and eigenvectors without ever having to explicitly determine what T is. Since T leaves the length of a vector unchanged, the only eigenvalues are 1 and -1. The eigenvalue 1 corresponds to a vector which is invariant under the reflection; this vector must therefore lie on the line $x_{1}+3x_{2}=0$. Now the vector with coordinates $x_{1}=3,x_{2}=-1$ lies on the line $x_{1}+3x_{2}=0$, so we conclude ${\begin{pmatrix}3\\-1\end{pmatrix}}$ is an eigenvector corresponding to the eigenvalue 1. The eigenvalue -1 corresponds to vectors which, under the reflection, has a 180 degree reversal in orientation. Such vectors necessarily must lie on the line perpendicular to the line $x_{1}+3x_{2}=0$. The equation for this perpendicular line is given by $3x_{1}-x_{2}=0$, and the vector ${\begin{pmatrix}1\\3\end{pmatrix}}$ lies on this perpendicular line. Therefore ${\begin{pmatrix}1\\3\end{pmatrix}}$ is an eigenvector corresponding to the eigenvalue -1. Hence, we conclude $\left\{{\begin{pmatrix}3\\-1\end{pmatrix}},{\begin{pmatrix}1\\3\end{pmatrix}}\right\}$ is a basis for $\mathbb {R} ^{2}$ consisting of eigenvectors of T.

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