Let {𝐱1, 𝐱2, …, 𝐱n} be any set of vectors in a vector space V. Show that Span(𝐱1, 𝐱2, …, 𝐱n) is

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Let {𝐱1, 𝐱2, …, 𝐱n} be any set of vectors in a vector space...

Let $\left\{\mathbf{x}_{1}, \mathbf{x}_{2}, \ldots, \mathbf{x}_{n}\right\}$ be any set of vectors in a vector space $V$. Show that $\operatorname{Span}\left(\mathbf{x}_{1}, \mathbf{x}_{2}, \ldots, \mathbf{x}_{n}\right)$ is a subspace of $V$.

Key Concepts

Closure Properties

Closure properties in the context of vector spaces refer to the requirement that the set is closed under vector addition and scalar multiplication. Verifying that a set satisfies these properties is essential when determining if it qualifies as a subspace, as is commonly done when proving that the span of a set of vectors forms a subspace.

Span

The span of a set of vectors is the set comprising all possible linear combinations of those vectors. It represents the smallest subspace containing the given set and is fundamental in understanding how vectors can generate entire subspaces within a vector space.

Linear Combination

A linear combination is an expression constructed from a set of vectors by multiplying each vector by a corresponding scalar and summing the results. This concept is critical as it underpins the definition of span and plays a central role in the study of linear independence, bases, and dimension.

Vector Space

A vector space is a mathematical structure composed of a set of vectors along with two operations, vector addition and scalar multiplication, which satisfy specific axioms such as associativity, commutativity, the existence of a zero vector, and distributivity. These properties allow for the systematic study of vector operations and form the foundational framework for linear algebra.

Subspace

A subspace is a subset of a vector space that is itself a vector space under the operations of addition and scalar multiplication inherited from the larger vector space. To be classified as a subspace, the subset must include the zero vector and be closed under both vector addition and scalar multiplication.

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