X E R? : x= t 3 t is a real number. %3D
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
Describe geometrically W c R^2 by sketiching it's graph
if you could explain plz
![The image displays a mathematical expression defining a set \( W \).
\[
W = \left\{ \mathbf{x} = \begin{bmatrix} x_1 \\ x_2 \\ x_3 \end{bmatrix}, \, x_1 = -x_2 - x_3 \right\}
\]
This notation represents a set \( W \), where vectors \( \mathbf{x} \) are described in a 3-dimensional space. The vector \( \mathbf{x} \) consists of three components \( x_1 \), \( x_2 \), and \( x_3 \). The condition \( x_1 = -x_2 - x_3 \) constrains the values of \( x_1 \) based on the values of \( x_2 \) and \( x_3 \). This expression describes a plane in three-dimensional space.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fbb7f1ce8-2552-49b5-ba1d-534a58de11ae%2F681231bc-1a71-4dde-a046-5af24b462a06%2Fljspvi_processed.jpeg&w=3840&q=75)
Transcribed Image Text:The image displays a mathematical expression defining a set \( W \).
\[
W = \left\{ \mathbf{x} = \begin{bmatrix} x_1 \\ x_2 \\ x_3 \end{bmatrix}, \, x_1 = -x_2 - x_3 \right\}
\]
This notation represents a set \( W \), where vectors \( \mathbf{x} \) are described in a 3-dimensional space. The vector \( \mathbf{x} \) consists of three components \( x_1 \), \( x_2 \), and \( x_3 \). The condition \( x_1 = -x_2 - x_3 \) constrains the values of \( x_1 \) based on the values of \( x_2 \) and \( x_3 \). This expression describes a plane in three-dimensional space.
![The image contains a mathematical expression defining a set \( W \) in \(\mathbb{R}^2\):
\[
W = \left\{ \mathbf{x} \in \mathbb{R}^2 : \mathbf{x} = t \begin{bmatrix} 1 \\ 3 \end{bmatrix}, \, t \text{ is a real number.} \right\}
\]
**Explanation:**
This expression describes a line in the 2-dimensional real coordinate space, \(\mathbb{R}^2\). The set \( W \) consists of all vectors \(\mathbf{x}\) that can be represented as a scalar multiple \( t \) of the vector \(\begin{bmatrix} 1 \\ 3 \end{bmatrix}\), where \( t \) is any real number. This means that the line passes through the origin and has a direction vector of \(\begin{bmatrix} 1 \\ 3 \end{bmatrix}\), which indicates that for every 1 unit moved horizontally, it moves 3 units vertically.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fbb7f1ce8-2552-49b5-ba1d-534a58de11ae%2F681231bc-1a71-4dde-a046-5af24b462a06%2Fdnqx13_processed.jpeg&w=3840&q=75)
Transcribed Image Text:The image contains a mathematical expression defining a set \( W \) in \(\mathbb{R}^2\):
\[
W = \left\{ \mathbf{x} \in \mathbb{R}^2 : \mathbf{x} = t \begin{bmatrix} 1 \\ 3 \end{bmatrix}, \, t \text{ is a real number.} \right\}
\]
**Explanation:**
This expression describes a line in the 2-dimensional real coordinate space, \(\mathbb{R}^2\). The set \( W \) consists of all vectors \(\mathbf{x}\) that can be represented as a scalar multiple \( t \) of the vector \(\begin{bmatrix} 1 \\ 3 \end{bmatrix}\), where \( t \) is any real number. This means that the line passes through the origin and has a direction vector of \(\begin{bmatrix} 1 \\ 3 \end{bmatrix}\), which indicates that for every 1 unit moved horizontally, it moves 3 units vertically.
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