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Show that a subset W of a vector space V is a subspace of V if and only if span[W] = W.

Show that a subset W of a vector space V is a subspace of V if and only if span[W] = W.

Calculus: Early Transcendentals
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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**Title: Understanding Subspaces in Vector Spaces** **Concept:** A subset \( W \) of a vector space \( V \) is a subspace of \( V \) if and only if the span of \( W \), denoted by \( \text{span}[W] \), is equal to \( W \). **Explanation:** In the context of vector spaces, a subspace is a subset of a vector space that is itself a vector space under the inherited operations. For \( W \) to be considered a subspace: 1. **Non-emptiness and Zero Vector**: \( W \) must contain the zero vector. 2. **Closure under Addition**: If \( \mathbf{u}, \mathbf{v} \in W \), then \( \mathbf{u} + \mathbf{v} \in W \). 3. **Closure under Scalar Multiplication**: If \( \mathbf{u} \in W \) and \( c \) is a scalar, then \( c \mathbf{u} \in W \). The span of \( W \), \( \text{span}[W] \), is defined as the set of all linear combinations of vectors in \( W \). If \(\text{span}[W] = W\), this ensures all vectors in \( W \) can be represented as linear combinations of themselves, which satisfies the necessary conditions for \( W \) to be a subspace. This relationship highlights the fundamental property of subspaces, ensuring that it is not only closed under vector additions and scalar multiplications but also fully generated by its own vectors.
Transcribed Image Text:**Title: Understanding Subspaces in Vector Spaces** **Concept:** A subset \( W \) of a vector space \( V \) is a subspace of \( V \) if and only if the span of \( W \), denoted by \( \text{span}[W] \), is equal to \( W \). **Explanation:** In the context of vector spaces, a subspace is a subset of a vector space that is itself a vector space under the inherited operations. For \( W \) to be considered a subspace: 1. **Non-emptiness and Zero Vector**: \( W \) must contain the zero vector. 2. **Closure under Addition**: If \( \mathbf{u}, \mathbf{v} \in W \), then \( \mathbf{u} + \mathbf{v} \in W \). 3. **Closure under Scalar Multiplication**: If \( \mathbf{u} \in W \) and \( c \) is a scalar, then \( c \mathbf{u} \in W \). The span of \( W \), \( \text{span}[W] \), is defined as the set of all linear combinations of vectors in \( W \). If \(\text{span}[W] = W\), this ensures all vectors in \( W \) can be represented as linear combinations of themselves, which satisfies the necessary conditions for \( W \) to be a subspace. This relationship highlights the fundamental property of subspaces, ensuring that it is not only closed under vector additions and scalar multiplications but also fully generated by its own vectors.
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