2. let where a) why why be ([^]) = [24] 2x L: IR→ L did we use just ь) із sketch Sketch the the linear word transformation images " √ = [2], * [³], [²] Label operator ketel on this "Operator" here? ? 3 of 3, 3, W on plane [* same Planes Lat
2. let where a) why why be ([^]) = [24] 2x L: IR→ L did we use just ь) із sketch Sketch the the linear word transformation images " √ = [2], * [³], [²] Label operator ketel on this "Operator" here? ? 3 of 3, 3, W on plane [* same Planes Lat
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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Question
![Sure! Here’s a transcription suitable for an educational website:
---
**Problem 2: Linear Operator and Vector Sketching**
Let \( L: \mathbb{R}^2 \rightarrow \mathbb{R}^2 \) be a linear operator where
\[
L \begin{pmatrix} x \\ y \end{pmatrix} = \begin{pmatrix} -2y \\ 2x \end{pmatrix}
\]
**a) Terminology Question:**
Why did we use the word "operator" here? Why not just "transformation"?
**b) Vector Sketching:**
1. Sketch the vectors \(\vec{u} = \begin{pmatrix} 1 \\ 2 \end{pmatrix}\), \(\vec{v} = \begin{pmatrix} -1 \\ 3 \end{pmatrix}\), and \(\vec{w} = \begin{pmatrix} 3 \\ 2 \end{pmatrix}\). Label them.
2. Sketch the images of \(\vec{u}\), \(\vec{v}\), and \(\vec{w}\) on the same plane. Label them.
(*Note: The sketches should be done on the provided Cartesian plane.*)
**c) Visual Analysis:**
Describe what happened visually to the size and position of all the vectors.
---
**Explanation for the Diagram:**
The provided coordinate plane is a simple Cartesian grid where vectors will be plotted. Begin by plotting the original vectors \(\vec{u}\), \(\vec{v}\), and \(\vec{w}\). Then, apply the linear operator \(L\) to these vectors to obtain their images, and plot these new vectors on the same grid.
For example, to find the image of \(\vec{u}\) under \(L\), compute:
\[
L\begin{pmatrix} 1 \\ 2 \end{pmatrix} = \begin{pmatrix} -2(2) \\ 2(1) \end{pmatrix} = \begin{pmatrix} -4 \\ 2 \end{pmatrix}
\]
Repeat similar calculations for \(\vec{v}\) and \(\vec{w}\). Analyze how the vectors' positions change and note any differences in size or direction.
---](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F0f5c8177-d167-495b-a75f-4d9ab8b89388%2F781f80d6-00f5-48d6-ace1-122919322b57%2Fidhahb8_processed.jpeg&w=3840&q=75)
Transcribed Image Text:Sure! Here’s a transcription suitable for an educational website:
---
**Problem 2: Linear Operator and Vector Sketching**
Let \( L: \mathbb{R}^2 \rightarrow \mathbb{R}^2 \) be a linear operator where
\[
L \begin{pmatrix} x \\ y \end{pmatrix} = \begin{pmatrix} -2y \\ 2x \end{pmatrix}
\]
**a) Terminology Question:**
Why did we use the word "operator" here? Why not just "transformation"?
**b) Vector Sketching:**
1. Sketch the vectors \(\vec{u} = \begin{pmatrix} 1 \\ 2 \end{pmatrix}\), \(\vec{v} = \begin{pmatrix} -1 \\ 3 \end{pmatrix}\), and \(\vec{w} = \begin{pmatrix} 3 \\ 2 \end{pmatrix}\). Label them.
2. Sketch the images of \(\vec{u}\), \(\vec{v}\), and \(\vec{w}\) on the same plane. Label them.
(*Note: The sketches should be done on the provided Cartesian plane.*)
**c) Visual Analysis:**
Describe what happened visually to the size and position of all the vectors.
---
**Explanation for the Diagram:**
The provided coordinate plane is a simple Cartesian grid where vectors will be plotted. Begin by plotting the original vectors \(\vec{u}\), \(\vec{v}\), and \(\vec{w}\). Then, apply the linear operator \(L\) to these vectors to obtain their images, and plot these new vectors on the same grid.
For example, to find the image of \(\vec{u}\) under \(L\), compute:
\[
L\begin{pmatrix} 1 \\ 2 \end{pmatrix} = \begin{pmatrix} -2(2) \\ 2(1) \end{pmatrix} = \begin{pmatrix} -4 \\ 2 \end{pmatrix}
\]
Repeat similar calculations for \(\vec{v}\) and \(\vec{w}\). Analyze how the vectors' positions change and note any differences in size or direction.
---
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