Answered: In Exercises 13–16, use a rectangular…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

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$### Linear Transformation Example **Question 15:** Consider the linear transformation $ T(\mathbf{x}) $ defined as follows: \[ T(\mathbf{x}) = \begin{bmatrix} 0 & 0 \\ 0 & 1 \end{bmatrix} \begin{bmatrix} x_1 \\ x_2 \end{bmatrix} \] **Explanation:** This equation represents a linear transformation of a vector $\mathbf{x}$ in $\mathbb{R}^2$. The transformation is applied by multiplying the vector $\mathbf{x}$ by the given matrix. Here, $\mathbf{x} = \begin{bmatrix} x_1 \\ x_2 \end{bmatrix}$ is any vector in $\mathbb{R}^2$, and $ T(\mathbf{x}) $ is the resulting vector after the transformation. ### Detailed Matrix Multiplication: To understand how the transformation works, let's perform the matrix multiplication step-by-step. \[ T(\mathbf{x}) = \begin{bmatrix} 0 & 0 \\ 0 & 1 \end{bmatrix} \begin{bmatrix} x_1 \\ x_2 \end{bmatrix} = \begin{bmatrix} 0 \cdot x_1 + 0 \cdot x_2 \\ 0 \cdot x_1 + 1 \cdot x_2 \end{bmatrix} = \begin{bmatrix} 0 \\ x_2 \end{bmatrix} \] As a result: \[ T(\mathbf{x}) = \begin{bmatrix} 0 \\ x_2 \end{bmatrix} \] ### Analysis: - The transformation $ T(\mathbf{x}) $ maps the x-component of the input vector (represented by $ x_1 $) to zero. - The y-component (represented by $ x_2 $) remains unchanged. ### Visual Interpretation: If $\mathbf{x}$ was plotted in a 2D Cartesian coordinate system, this transformation would effectively collapse any point along the x-axis to the origin (0, 0), while maintaining the y-component's value. Thus, every input vector is projected onto the y-axis. Understanding these transformations is crucial in fields like computer graphics, engineering, and physics, where manipulating and$

Transcribed Image Text:### Linear Transformation Example **Question 15:** Consider the linear transformation $ T(\mathbf{x}) $ defined as follows: \[ T(\mathbf{x}) = \begin{bmatrix} 0 & 0 \\ 0 & 1 \end{bmatrix} \begin{bmatrix} x_1 \\ x_2 \end{bmatrix} \] **Explanation:** This equation represents a linear transformation of a vector $\mathbf{x}$ in $\mathbb{R}^2$. The transformation is applied by multiplying the vector $\mathbf{x}$ by the given matrix. Here, $\mathbf{x} = \begin{bmatrix} x_1 \\ x_2 \end{bmatrix}$ is any vector in $\mathbb{R}^2$, and $ T(\mathbf{x}) $ is the resulting vector after the transformation. ### Detailed Matrix Multiplication: To understand how the transformation works, let's perform the matrix multiplication step-by-step. \[ T(\mathbf{x}) = \begin{bmatrix} 0 & 0 \\ 0 & 1 \end{bmatrix} \begin{bmatrix} x_1 \\ x_2 \end{bmatrix} = \begin{bmatrix} 0 \cdot x_1 + 0 \cdot x_2 \\ 0 \cdot x_1 + 1 \cdot x_2 \end{bmatrix} = \begin{bmatrix} 0 \\ x_2 \end{bmatrix} \] As a result: \[ T(\mathbf{x}) = \begin{bmatrix} 0 \\ x_2 \end{bmatrix} \] ### Analysis: - The transformation $ T(\mathbf{x}) $ maps the x-component of the input vector (represented by $ x_1 $) to zero. - The y-component (represented by $ x_2 $) remains unchanged. ### Visual Interpretation: If $\mathbf{x}$ was plotted in a 2D Cartesian coordinate system, this transformation would effectively collapse any point along the x-axis to the origin (0, 0), while maintaining the y-component's value. Thus, every input vector is projected onto the y-axis. Understanding these transformations is crucial in fields like computer graphics, engineering, and physics, where manipulating and

**Vector Transformations in Rectangular Coordinates**

In Exercises 13–16, use a rectangular coordinate system to plot the vectors **u** and **v** and their images under the given transformation **T**.

Given vectors:
\[ \mathbf{u} = \begin{bmatrix} 5 \\ 2 \end{bmatrix}, \quad \mathbf{v} = \begin{bmatrix} -2 \\ 4 \end{bmatrix} \]

1. **Plotting the Vectors**:
- Plot vector **u** starting from the origin (0,0) to the point (5, 2).
- Plot vector **v** starting from the origin (0,0) to the point (-2, 4).

2. **Transformation**:
- For each vector, apply the given transformation **T** to find the new images of **u** and **v**.
- Make sure to create a separate and reasonably large sketch for each exercise.

3. **Geometrical Description of Transformation**:
- Describe how the transformation **T** geometrically affects any vector **x** in $\mathbb{R}^2$.

**Note**: Include detailed steps and sketches to illustrate the transformation clearly.

By following these instructions, you will visualize how transformation **T** modifies vectors in a two-dimensional space.

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