Understanding the Span of a Subset and Its Subspace Property

In the realm of linear algebra, the concept of a span plays a critical role in the exploration of vector spaces and subspaces. Specifically, if X is any subset of a vector space V, does the span of X have to be a subspace of V? This article will delve into this question and provide a comprehensive explanation, supported by mathematical definitions and proofs.

Definition of Span

The span of a set X, denoted as span(X), is defined as the set of all linear combinations of the vectors in X. Formally, it can be represented as:

span(X)  { Σi1n c_i x_i | x_i ∈ X, c_i ∈ F, n ∈ N }

Here, F is the field over which the vector space V is defined, such as the real or complex numbers.

Subspace Criteria and Span(X)

To prove that span(X) is a subspace of V, we need to verify that it satisfies the following three properties:

Contains the Zero Vector

The zero vector in V can be expressed as a linear combination of vectors in X by taking all coefficients c_i 0. Hence, the zero vector is in span(X).

Closed under Addition

If u and v are in span(X), then there exist linear combinations of vectors in X such that u Σi1n a_i x_i and v Σj1m b_j y_j. Their sum u v can also be expressed as a linear combination of vectors in X since the addition of linear combinations remains within span(X).

Closed under Scalar Multiplication

If u is in span(X), then u Σi1n c_i x_i for some c_i ∈ F. For any scalar k ∈ F, k〈u〉 k(Σi1n c_i x_i) Σi1n (k c_i) x_i is also a linear combination of vectors in X, hence k〈u〉 ∈ span(X).

Given that all three conditions are satisfied, it follows that span(X) is indeed a subspace of V.

Proof and Notes on Span(X)

Lets consider an example where X {x_1, x_2, ..., x_m}. If we take two linear combinations, u_1 Σi1m λ_i x_i and u_2 Σi1m μ_i x_i, and consider any two scalars α, β ∈ F, then:

α u_1   β u_2  Σi1m (α λ_i   β μ_i) x_i

From this, it is clear that α u_1 β u_2 is a linear combination of the vectors in X, thus α u_1 β u_2 ∈ span(X).

From this, we observe that for any subset W of V, if W is closed under linear combinations of vectors, then it satisfies the criteria to be a subspace of V. If X is a finite set as mentioned, and V is finitely generated with dimension n, then:

If m n, then W span(X) is a proper subspace of V. If m n, then W V, and X forms a basis for V.

In the second alternative, the notation for X can be changed to A [a_1, a_2, ..., a_n], and A is a basis for V.

Conclusion

Yes, the span of any subset X of a vector space V is indeed a subspace of V. This property is fundamental in linear algebra and has wide-ranging applications in mathematics and various scientific fields. Understanding this concept is crucial for deepening one's knowledge in the areas of linear algebra and vector spaces.