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Describe geometrically the effect of the transformation T. Let A = [1] Define a transformation T by T(x) = Ax. O Contraction Transformation O Shear Transformation O Reflection through x axis O DilationTransformation

Describe geometrically the effect of the transformation T. Let A = [1] Define a transformation T by T(x) = Ax. O Contraction Transformation O Shear Transformation O Reflection through x axis O DilationTransformation

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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### Matrix Equation Problem **Problem Statement:** Suppose \( A = \begin{bmatrix} 4 & 5 & -1 \\ 0 & 12 & -4 \\ -3 & 6 & 4 \end{bmatrix} \) that gives a non-trivial solution. **Options:** - \(\circ\) (8, 2, 3) - \(\circ\) (2, -1, 3) - \(\circ\) (-1, 1, 1) - \(\circ\) (-2, 1, -3) **Instructions for students:** Analyze the given matrix \(A\) and identify the solution vector that satisfies \(A \mathbf{x} = \mathbf{0}\) for a non-trivial solution, where \(\mathbf{x} \neq \mathbf{0}\). Choose the correct option from the list provided. For this exercise, practice your skills in: 1. Solving systems of linear equations. 2. Performing matrix operations. 3. Identifying non-trivial solutions in homogeneous systems. Understanding these concepts is essential for mastery in linear algebra and its applications in various fields such as physics, engineering, and computer science.
Transcribed Image Text:### Matrix Equation Problem **Problem Statement:** Suppose \( A = \begin{bmatrix} 4 & 5 & -1 \\ 0 & 12 & -4 \\ -3 & 6 & 4 \end{bmatrix} \) that gives a non-trivial solution. **Options:** - \(\circ\) (8, 2, 3) - \(\circ\) (2, -1, 3) - \(\circ\) (-1, 1, 1) - \(\circ\) (-2, 1, -3) **Instructions for students:** Analyze the given matrix \(A\) and identify the solution vector that satisfies \(A \mathbf{x} = \mathbf{0}\) for a non-trivial solution, where \(\mathbf{x} \neq \mathbf{0}\). Choose the correct option from the list provided. For this exercise, practice your skills in: 1. Solving systems of linear equations. 2. Performing matrix operations. 3. Identifying non-trivial solutions in homogeneous systems. Understanding these concepts is essential for mastery in linear algebra and its applications in various fields such as physics, engineering, and computer science.
### Geometric Transformation Overview #### Topic: Describing the Geometric Effect of a Transformation In this lesson, we explore the geometric effect of a given transformation matrix on vectors in a plane. **Matrix Definition:** Let \( A \) be the transformation matrix: \[ A = \begin{pmatrix} 1 & -5 \\ 0 & 1 \end{pmatrix} \] **Transformation Function:** Define a transformation \( T \) by \( T(x) = Ax \). #### Question: Which of the following best describes the geometric effect of the transformation \( T \)? - ○ Contraction Transformation - ○ Shear Transformation - ○ Reflection through x-axis - ○ Dilation Transformation - ○ Reflection through y-axis ### Explanation: The matrix \( A \) described causes a specific geometric transformation which we need to identify from the provided options. For detailed analysis: - **Matrix Analysis**: The matrix \( A \) keeps the x-coordinate unchanged but adds a multiple of the y-coordinate \(-5y\) to the x-coordinate. - This form is typical of a **shear transformation**, where the lines parallel to the x-axis remain unchanged but are all shifted horizontally by varying amounts dependent on their y-values. Hence, the correct option is: - **Shear Transformation**
Transcribed Image Text:### Geometric Transformation Overview #### Topic: Describing the Geometric Effect of a Transformation In this lesson, we explore the geometric effect of a given transformation matrix on vectors in a plane. **Matrix Definition:** Let \( A \) be the transformation matrix: \[ A = \begin{pmatrix} 1 & -5 \\ 0 & 1 \end{pmatrix} \] **Transformation Function:** Define a transformation \( T \) by \( T(x) = Ax \). #### Question: Which of the following best describes the geometric effect of the transformation \( T \)? - ○ Contraction Transformation - ○ Shear Transformation - ○ Reflection through x-axis - ○ Dilation Transformation - ○ Reflection through y-axis ### Explanation: The matrix \( A \) described causes a specific geometric transformation which we need to identify from the provided options. For detailed analysis: - **Matrix Analysis**: The matrix \( A \) keeps the x-coordinate unchanged but adds a multiple of the y-coordinate \(-5y\) to the x-coordinate. - This form is typical of a **shear transformation**, where the lines parallel to the x-axis remain unchanged but are all shifted horizontally by varying amounts dependent on their y-values. Hence, the correct option is: - **Shear Transformation**
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