Describe geometrically W C R² bu sketching its graph: (a) X1 W = = 3x2 :X= x1 = X2
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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Please explain
![The image shows a set definition:
\[ W = \left\{ \mathbf{x} \in \mathbb{R}^2 : \mathbf{x} = t \begin{bmatrix} 1 \\ 3 \end{bmatrix}, \, t \text{ is a real number.} \right\} \]
**Explanation:**
This notation defines a subset \( W \) within the two-dimensional real number space \( \mathbb{R}^2 \). The elements \( \mathbf{x} \) of this subset are vectors that can be expressed as scalar multiples of the vector
\[ \begin{bmatrix} 1 \\ 3 \end{bmatrix} \]
where \( t \) is any real number. Essentially, this describes a line through the origin in the direction of the vector \([1, 3]^T\).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fbb7f1ce8-2552-49b5-ba1d-534a58de11ae%2F5207d7d9-eeb8-4e77-b0d6-4be1264dff2b%2Ffntveh_processed.jpeg&w=3840&q=75)
Transcribed Image Text:The image shows a set definition:
\[ W = \left\{ \mathbf{x} \in \mathbb{R}^2 : \mathbf{x} = t \begin{bmatrix} 1 \\ 3 \end{bmatrix}, \, t \text{ is a real number.} \right\} \]
**Explanation:**
This notation defines a subset \( W \) within the two-dimensional real number space \( \mathbb{R}^2 \). The elements \( \mathbf{x} \) of this subset are vectors that can be expressed as scalar multiples of the vector
\[ \begin{bmatrix} 1 \\ 3 \end{bmatrix} \]
where \( t \) is any real number. Essentially, this describes a line through the origin in the direction of the vector \([1, 3]^T\).
![---
**Problem Statement:**
Describe geometrically \( W \subseteq \mathbb{R}^2 \) by sketching its graph:
(a)
\[
W = \left\{ \mathbf{x} \in \mathbb{R}^2 : \mathbf{x} = \begin{bmatrix} x_1 \\ x_2 \end{bmatrix} , \, x_1 = 3x_2 \right\}
\]
**Explanation:**
The problem asks to geometrically describe the subset \( W \) of the two-dimensional real plane \( \mathbb{R}^2 \). The subset \( W \) is defined as the set of all vectors \(\mathbf{x} = \begin{bmatrix} x_1 \\ x_2 \end{bmatrix}\) such that the first component \( x_1 \) is three times the second component \( x_2 \).
To sketch this graphically, one should draw the line represented by the equation \( x_1 = 3x_2 \) in the Cartesian coordinate system. This line passes through the origin and has a slope of 3, indicating that for every unit increase in \( x_2 \), \( x_1 \) increases by 3 units.
---](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fbb7f1ce8-2552-49b5-ba1d-534a58de11ae%2F5207d7d9-eeb8-4e77-b0d6-4be1264dff2b%2Fbor5l2v_processed.jpeg&w=3840&q=75)
Transcribed Image Text:---
**Problem Statement:**
Describe geometrically \( W \subseteq \mathbb{R}^2 \) by sketching its graph:
(a)
\[
W = \left\{ \mathbf{x} \in \mathbb{R}^2 : \mathbf{x} = \begin{bmatrix} x_1 \\ x_2 \end{bmatrix} , \, x_1 = 3x_2 \right\}
\]
**Explanation:**
The problem asks to geometrically describe the subset \( W \) of the two-dimensional real plane \( \mathbb{R}^2 \). The subset \( W \) is defined as the set of all vectors \(\mathbf{x} = \begin{bmatrix} x_1 \\ x_2 \end{bmatrix}\) such that the first component \( x_1 \) is three times the second component \( x_2 \).
To sketch this graphically, one should draw the line represented by the equation \( x_1 = 3x_2 \) in the Cartesian coordinate system. This line passes through the origin and has a slope of 3, indicating that for every unit increase in \( x_2 \), \( x_1 \) increases by 3 units.
---
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