i.e., Show that the set of vectors in R³ the components of which sum up to zero, S = {v € R³: v₁ + 0₂ +03=0} (v; are components of v) is vector subspace of R³ of dimension 2. Examples of vectors in S are U = 1 2 7 -Q. V=
i.e., Show that the set of vectors in R³ the components of which sum up to zero, S = {v € R³: v₁ + 0₂ +03=0} (v; are components of v) is vector subspace of R³ of dimension 2. Examples of vectors in S are U = 1 2 7 -Q. V=
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Question
![**Vector Subspaces in \(\mathbb{R}^3\)**
---
### Problem Statement:
Show that the set of vectors in \(\mathbb{R}^3\) whose components sum up to zero is a vector subspace of \(\mathbb{R}^3\).
---
### Mathematically:
Consider the set
\[ S = \{ v \in \mathbb{R}^3 : v_1 + v_2 + v_3 = 0 \} \]
where \( v_i \) are the components of \( v \).
This set \( S \) must be shown to be a vector subspace of \(\mathbb{R}^3\) with a dimension of 2.
### Examples of Vectors in \( S \):
Two examples of vectors in \( S \) are:
\[ u = \begin{bmatrix}
-3 \\
1 \\
2
\end{bmatrix}, \quad v = \begin{bmatrix}
1 \\
1 \\
-2
\end{bmatrix}. \]
---
### Explanation:
Vectors \( u \) and \( v \) are presented as examples that belong to the set \( S \) because the sum of their components equals zero:
For \( u \):
\[ -3 + 1 + 2 = 0 \]
For \( v \):
\[ 1 + 1 + (-2) = 0 \]
Thus, both vectors satisfy the condition \( v_1 + v_2 + v_3 = 0 \) necessary for membership in the subspace \( S \).
The set \( S \) has been determined to be a vector subspace of \(\mathbb{R}^3\) with dimension 2. This implies that \( S \) can be spanned by two linearly independent vectors.
---
By understanding this subspace, one can further explore linear algebra concepts such as basis, dimension, and vector spaces in \( \mathbb{R}^n \).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fdf4abcde-0790-4b6d-b8ff-52c0ea5b10aa%2Fba7039d2-c473-4e83-a35d-2e0da831a4d9%2Fi5bjbp_processed.png&w=3840&q=75)
Transcribed Image Text:**Vector Subspaces in \(\mathbb{R}^3\)**
---
### Problem Statement:
Show that the set of vectors in \(\mathbb{R}^3\) whose components sum up to zero is a vector subspace of \(\mathbb{R}^3\).
---
### Mathematically:
Consider the set
\[ S = \{ v \in \mathbb{R}^3 : v_1 + v_2 + v_3 = 0 \} \]
where \( v_i \) are the components of \( v \).
This set \( S \) must be shown to be a vector subspace of \(\mathbb{R}^3\) with a dimension of 2.
### Examples of Vectors in \( S \):
Two examples of vectors in \( S \) are:
\[ u = \begin{bmatrix}
-3 \\
1 \\
2
\end{bmatrix}, \quad v = \begin{bmatrix}
1 \\
1 \\
-2
\end{bmatrix}. \]
---
### Explanation:
Vectors \( u \) and \( v \) are presented as examples that belong to the set \( S \) because the sum of their components equals zero:
For \( u \):
\[ -3 + 1 + 2 = 0 \]
For \( v \):
\[ 1 + 1 + (-2) = 0 \]
Thus, both vectors satisfy the condition \( v_1 + v_2 + v_3 = 0 \) necessary for membership in the subspace \( S \).
The set \( S \) has been determined to be a vector subspace of \(\mathbb{R}^3\) with dimension 2. This implies that \( S \) can be spanned by two linearly independent vectors.
---
By understanding this subspace, one can further explore linear algebra concepts such as basis, dimension, and vector spaces in \( \mathbb{R}^n \).
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