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Find a single matrix for the transformation that is equivalent to doing the following four transformations of the plane in succession: 1. Shear by a factor of 2 in the x-direction. 2. Reflection over the line y = x. 3. Expand by a factor of 7 in the y-direction. 4. Rotate counter-clockwise by 90 degrees. X 0 0 -7 14

Find a single matrix for the transformation that is equivalent to doing the following four transformations of the plane in succession: 1. Shear by a factor of 2 in the x-direction. 2. Reflection over the line y = x. 3. Expand by a factor of 7 in the y-direction. 4. Rotate counter-clockwise by 90 degrees. X 0 0 -7 14

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: Matrix Transformations on the Plane** **Problem Statement:** Find a single matrix for the transformation that is equivalent to doing the following four transformations of the plane in succession: 1. **Shear by a factor of 2 in the \( x \)-direction.** 2. **Reflection over the line \( y = x \).** 3. **Expand by a factor of 7 in the \( y \)-direction.** 4. **Rotate counter-clockwise by 90 degrees.** **Resultant Matrix:** \[ \begin{bmatrix} 0 & 7 \\ 0 & 14 \end{bmatrix} \] **Explanation:** This problem requires finding a composite matrix that represents the sequential operations of shearing, reflecting, expanding, and rotating. Each of these transformations can be represented by a matrix, and the final transformation is obtained by multiplying these matrices in the order of the operations. The final 2x2 matrix is: \[ \begin{bmatrix} 0 & 7 \\ 0 & 14 \end{bmatrix} \] This matrix represents the cumulative effect of the transformations listed.
Transcribed Image Text:**Title: Matrix Transformations on the Plane** **Problem Statement:** Find a single matrix for the transformation that is equivalent to doing the following four transformations of the plane in succession: 1. **Shear by a factor of 2 in the \( x \)-direction.** 2. **Reflection over the line \( y = x \).** 3. **Expand by a factor of 7 in the \( y \)-direction.** 4. **Rotate counter-clockwise by 90 degrees.** **Resultant Matrix:** \[ \begin{bmatrix} 0 & 7 \\ 0 & 14 \end{bmatrix} \] **Explanation:** This problem requires finding a composite matrix that represents the sequential operations of shearing, reflecting, expanding, and rotating. Each of these transformations can be represented by a matrix, and the final transformation is obtained by multiplying these matrices in the order of the operations. The final 2x2 matrix is: \[ \begin{bmatrix} 0 & 7 \\ 0 & 14 \end{bmatrix} \] This matrix represents the cumulative effect of the transformations listed.
### Matrix Transformation Problem **Objective:** Find a single matrix for the transformation that is equivalent to performing the following four transformations of the plane in succession: 1. **Shear by a factor of 2 in the x-direction.** 2. **Reflection over the line \( y = x \).** 3. **Expand by a factor of 7 in the y-direction.** 4. **Rotate counter-clockwise by 90 degrees.** **Solution Structure:** - Construct individual matrices for each transformation. - Multiply these matrices in the correct order to find the overall transformation matrix. **Diagram Explanation:** - There are no explicit graphs or diagrams provided. However, the solution may involve a step-by-step process of matrix multiplication which will ultimately lead to a single 2x2 matrix representing the overall transformation.
Transcribed Image Text:### Matrix Transformation Problem **Objective:** Find a single matrix for the transformation that is equivalent to performing the following four transformations of the plane in succession: 1. **Shear by a factor of 2 in the x-direction.** 2. **Reflection over the line \( y = x \).** 3. **Expand by a factor of 7 in the y-direction.** 4. **Rotate counter-clockwise by 90 degrees.** **Solution Structure:** - Construct individual matrices for each transformation. - Multiply these matrices in the correct order to find the overall transformation matrix. **Diagram Explanation:** - There are no explicit graphs or diagrams provided. However, the solution may involve a step-by-step process of matrix multiplication which will ultimately lead to a single 2x2 matrix representing the overall transformation.
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