Is W a subspace of the vector space? If not, state why. (Select all that apply.) W is the set of all vectors in R2 whose components are rational numbers. O w is a subspace of R2. O w is not a subspace of R2 because it is not closed under addition. O wis not a subspace of R2 because it is not closed under scalar multiplication.
Is W a subspace of the vector space? If not, state why. (Select all that apply.) W is the set of all vectors in R2 whose components are rational numbers. O w is a subspace of R2. O w is not a subspace of R2 because it is not closed under addition. O wis not a subspace of R2 because it is not closed under scalar multiplication.
Elementary Linear Algebra (MindTap Course List)
8th Edition
ISBN:9781305658004
Author:Ron Larson
Publisher:Ron Larson
Chapter4: Vector Spaces
Section4.5: Basis And Dimension
Problem 65E: Find a basis for the vector space of all 33 diagonal matrices. What is the dimension of this vector...
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![**Is \( W \) a Subspace of the Vector Space?**
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*Question:*
Is \( W \) a subspace of the vector space? If not, state why. (Select all that apply.)
*Information Given:*
\( W \) is the set of all vectors in \( \mathbb{R}^2 \) whose components are rational numbers.
*Options:*
- [ ] \( W \) is a subspace of \( \mathbb{R}^2 \).
- [ ] \( W \) is not a subspace of \( \mathbb{R}^2 \) because it is not closed under addition.
- [ ] \( W \) is not a subspace of \( \mathbb{R}^2 \) because it is not closed under scalar multiplication.
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*Explanation:*
To determine if \( W \) is a subspace, we need to verify if it satisfies the conditions for a subspace. Specifically, we need to check if \( W \) is closed under addition and scalar multiplication. If \( W \) fails any of these conditions, it is not a subspace.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F42f67b29-db3a-470e-8f57-9956af53c593%2F0a6be123-54b5-482e-a0bd-0d079cc57a1d%2Fdk60xwn_processed.png&w=3840&q=75)
Transcribed Image Text:**Is \( W \) a Subspace of the Vector Space?**
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*Question:*
Is \( W \) a subspace of the vector space? If not, state why. (Select all that apply.)
*Information Given:*
\( W \) is the set of all vectors in \( \mathbb{R}^2 \) whose components are rational numbers.
*Options:*
- [ ] \( W \) is a subspace of \( \mathbb{R}^2 \).
- [ ] \( W \) is not a subspace of \( \mathbb{R}^2 \) because it is not closed under addition.
- [ ] \( W \) is not a subspace of \( \mathbb{R}^2 \) because it is not closed under scalar multiplication.
---
*Explanation:*
To determine if \( W \) is a subspace, we need to verify if it satisfies the conditions for a subspace. Specifically, we need to check if \( W \) is closed under addition and scalar multiplication. If \( W \) fails any of these conditions, it is not a subspace.
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