2.8.4 Tensor Span Subspace Relation

Tensor Span Subspace Relation explores how tensor spans generate subspaces, linking algebraic structures with geometric interpretations in multilinear algebra.

Tensor Span Subspace Relation is the correspondence between the span of a set of vectors and the subspace structure it generates, expressed by the fact that the span of any subset of a vector space is itself a subspace, and moreover the smallest subspace that contains every vector of that subset. This relation ties the operation of spanning to the broader lattice of subspaces within the vector space used for tensor construction.

Formal Statement

Span Is Always a Subspace

For any subset of a vector space, the span of that subset satisfies the subspace criteria: it contains the zero vector, it is closed under addition, and it is closed under scalar multiplication.

S \subseteq V \Rightarrow span (S) is a subspace of V

Minimality Among Containing Subspaces

Among every subspace of the vector space that contains the original set, the span is the smallest one, meaning it is contained in any other subspace that also contains the set.

S \subseteq W, W a subspace of V \Rightarrow span (S) \subseteq W

Span as Intersection of Containing Subspaces

Equivalent Characterization

The span of a set can equivalently be described as the intersection of all subspaces that contain the set, since any such intersection is itself a subspace and, by the minimality property, must coincide with the span.

span (S) = ⋂_{W subspace, S \subseteq W} W

Consistency of the Two Descriptions

Describing the span either as the set of linear combinations or as the intersection of containing subspaces produces the same subspace, giving two complementary ways to reason about which vectors belong to the span.

Monotonicity of the Span Operation

Larger Generating Sets Yield Larger or Equal Spans

If one set of vectors is contained in another, the span of the first is contained in the span of the second, since every linear combination available using the smaller set remains available when the larger set is used.

Redundant Vectors Do Not Change the Span

Adding a vector that already lies within the span of an existing set to that set does not enlarge the span, because the added vector contributes no linear combination not already reachable.

Role in Tensor Basis Span Structure

Anchoring Basis Vectors to a Concrete Subspace

Within the tensor basis span structure, the span subspace relation identifies precisely which subspace a chosen collection of basis vectors is responsible for representing, giving a concrete target that basis coverage must match.

Nesting of Subspaces in Tensor Construction

When multiple related spanning sets are used across stages of tensor construction, the span subspace relation explains how the resulting subspaces nest inside one another according to how the generating sets are related by inclusion.

Summary of Key Properties

Span as the Canonical Minimal Subspace

The span subspace relation establishes the span of a set as the canonical, minimal subspace built from that set, unifying the generative view of linear combinations with the structural view of subspace containment.

Structural Foundation for Later Constructions

This relation underlies later constructions in tensor algebra, including coordinate reach and dimension structure, since those constructions depend on knowing exactly which subspace a given spanning set determines.