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Are the two linear transformations the same? Yes ● S: R² → R², shear such that H maps to H then vertical scale by a factor of 1 T: R² → R², vertical scale by a factor of then shear such that 7 maps to 7°

Are the two linear transformations the same? Yes ● S: R² → R², shear such that H maps to H then vertical scale by a factor of 1 T: R² → R², vertical scale by a factor of then shear such that 7 maps to 7°

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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**Title: Exploring Linear Transformations: Shearing and Scaling** **Question:** Are the two linear transformations the same? **Answer:** Yes ✅ **Explanation:** We examine two transformations, \( S \) and \( T \), both mapping from \(\mathbb{R}^2\) to \(\mathbb{R}^2\). 1. **Transformation \( S \):** - **Shear:** The vector \(\begin{bmatrix} 1 \\ 0 \end{bmatrix}\) is transformed to \(\begin{bmatrix} 1 \\ 7 \end{bmatrix}\). - **Vertical Scaling:** After shearing, a vertical scaling by a factor of \(\frac{1}{7}\) is applied. 2. **Transformation \( T \):** - **Vertical Scaling:** Initially, a vertical scaling by a factor of \(\frac{1}{7}\) is applied. - **Shear:** Following the scaling, a shear transforms the vector \(\begin{bmatrix} 1 \\ 0 \end{bmatrix}\) to \(\begin{bmatrix} 1 \\ 1 \end{bmatrix}\). **Conclusion:** Despite the different order of operations, both transformations ultimately result in the same effect on the plane, confirming their equivalence as indicated by the selection "Yes" with a checkmark icon.
Transcribed Image Text:**Title: Exploring Linear Transformations: Shearing and Scaling** **Question:** Are the two linear transformations the same? **Answer:** Yes ✅ **Explanation:** We examine two transformations, \( S \) and \( T \), both mapping from \(\mathbb{R}^2\) to \(\mathbb{R}^2\). 1. **Transformation \( S \):** - **Shear:** The vector \(\begin{bmatrix} 1 \\ 0 \end{bmatrix}\) is transformed to \(\begin{bmatrix} 1 \\ 7 \end{bmatrix}\). - **Vertical Scaling:** After shearing, a vertical scaling by a factor of \(\frac{1}{7}\) is applied. 2. **Transformation \( T \):** - **Vertical Scaling:** Initially, a vertical scaling by a factor of \(\frac{1}{7}\) is applied. - **Shear:** Following the scaling, a shear transforms the vector \(\begin{bmatrix} 1 \\ 0 \end{bmatrix}\) to \(\begin{bmatrix} 1 \\ 1 \end{bmatrix}\). **Conclusion:** Despite the different order of operations, both transformations ultimately result in the same effect on the plane, confirming their equivalence as indicated by the selection "Yes" with a checkmark icon.
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