Science:Math Exam Resources/Courses/MATH221/December 2007/Question 08 (c)

MATH221 December 2007

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Question 08 (c)
Let $T:\mathbb {R} ^{2}\to \mathbb {R} ^{2}$ be the linear map given by $T(v)=5v$, for all $v\in \mathbb {R} ^{2}$ (T is a dilation). Find a basis of $\mathbb {R} ^{2}$ consisting of eigenvalues of T, or explain why this is not possible.

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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. Note that every non-zero vector in $\mathbb {R} ^{2}$ is an eigenvector for T (corresponding to the eigenvalue 5). In particular, the standard basis $\{{\begin{pmatrix}1\\0\end{pmatrix}},{\begin{pmatrix}0\\1\end{pmatrix}}\}$ is a basis of $\mathbb {R} ^{2}$ consisting of eigenvectors of T.

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