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T: R² R² rotates points (about the origin) through 31/2 radians (in the counterclockwise direction).

T: R² R² rotates points (about the origin) through 31/2 radians (in the counterclockwise direction).

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Assume that T is a linear transformation. Find the standard matrix of T. 

### Rotation Transformation in the Plane Let \( T : \mathbb{R}^2 \rightarrow \mathbb{R}^2 \) be a transformation that rotates points around the origin. The transformation \( T \) rotates points by an angle of \( \frac{3\pi}{2} \) radians in the counterclockwise direction. In mathematical terms, given a point \((x, y)\) in the plane, the transformation \( T \) maps this point to a new point \((x', y')\) after rotating it \( \frac{3\pi}{2} \) radians counterclockwise around the origin. This type of transformation is essential in various fields including geometry, physics, and computer graphics, where understanding how points move under rotation is crucial. For a visual explanation, imagine the \( x \)-\( y \) coordinate plane with the point \((1, 0)\). After applying the rotation transformation \( T \), the point will move along the circle centered at the origin by \( \frac{3\pi}{2} \) radians counterclockwise, resulting in the new position of \((0, -1)\).
Transcribed Image Text:### Rotation Transformation in the Plane Let \( T : \mathbb{R}^2 \rightarrow \mathbb{R}^2 \) be a transformation that rotates points around the origin. The transformation \( T \) rotates points by an angle of \( \frac{3\pi}{2} \) radians in the counterclockwise direction. In mathematical terms, given a point \((x, y)\) in the plane, the transformation \( T \) maps this point to a new point \((x', y')\) after rotating it \( \frac{3\pi}{2} \) radians counterclockwise around the origin. This type of transformation is essential in various fields including geometry, physics, and computer graphics, where understanding how points move under rotation is crucial. For a visual explanation, imagine the \( x \)-\( y \) coordinate plane with the point \((1, 0)\). After applying the rotation transformation \( T \), the point will move along the circle centered at the origin by \( \frac{3\pi}{2} \) radians counterclockwise, resulting in the new position of \((0, -1)\).
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