Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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8.
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![**Determine whether the given set S is a subspace of \( \mathbb{R}^3 \).**
\[ S = \{ (x, y, z) \in \mathbb{R}^3 : z = 0 \} \]
**Explanation:**
This statement is asking us to determine if the set \( S \), which consists of all points \((x, y, z)\) in three-dimensional space where the \( z \)-coordinate is zero, forms a subspace of \(\mathbb{R}^3\).
**Key Points:**
- The condition \( z = 0 \) indicates that the set \( S \) represents the xy-plane in \(\mathbb{R}^3\).
- A subspace must satisfy three main conditions:
1. It must contain the zero vector.
2. It must be closed under vector addition.
3. It must be closed under scalar multiplication.
If these three conditions are verified, then \( S \) can be concluded as a subspace of \(\mathbb{R}^3\).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F86847a45-621f-4eb2-b143-0cde4fac410f%2F38ed1f6a-1342-40ce-a9cd-8f648c1a0582%2F1pmqg2i_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Determine whether the given set S is a subspace of \( \mathbb{R}^3 \).**
\[ S = \{ (x, y, z) \in \mathbb{R}^3 : z = 0 \} \]
**Explanation:**
This statement is asking us to determine if the set \( S \), which consists of all points \((x, y, z)\) in three-dimensional space where the \( z \)-coordinate is zero, forms a subspace of \(\mathbb{R}^3\).
**Key Points:**
- The condition \( z = 0 \) indicates that the set \( S \) represents the xy-plane in \(\mathbb{R}^3\).
- A subspace must satisfy three main conditions:
1. It must contain the zero vector.
2. It must be closed under vector addition.
3. It must be closed under scalar multiplication.
If these three conditions are verified, then \( S \) can be concluded as a subspace of \(\mathbb{R}^3\).
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