Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN: 9780470458365
Author: Erwin Kreyszig
Publisher: Wiley, John & Sons, Incorporated
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Question
Let \( T: \mathbb{R}^2 \rightarrow \mathbb{R}^2 \) be such that \[ T \begin{bmatrix} x_1 \\ x_2 \end{bmatrix} = \begin{bmatrix} \cos \theta & -\sin \theta \\ \sin \theta & \cos \theta \end{bmatrix} \begin{bmatrix} x_1 \\ x_2 \end{bmatrix} \] Show that \( T \) is an orthogonal linear transformation.
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Transcribed Image Text:Let \( T: \mathbb{R}^2 \rightarrow \mathbb{R}^2 \) be such that \[ T \begin{bmatrix} x_1 \\ x_2 \end{bmatrix} = \begin{bmatrix} \cos \theta & -\sin \theta \\ \sin \theta & \cos \theta \end{bmatrix} \begin{bmatrix} x_1 \\ x_2 \end{bmatrix} \] Show that \( T \) is an orthogonal linear transformation.
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