9.3.3 Tensor Basis Span Requirement

Tensor Basis Span Requirement is the condition, alongside linear independence, that a collection of vectors must satisfy in order to serve as a basis: that every vector in the space be expressible as some linear combination of the chosen vectors, with no vector in the space left unreachable by any combination of them. It is the half of the basis definition responsible for completeness — ensuring the expansion formula underlying tensor components has something to say about every vector and every tensor of the relevant type, rather than only about the vectors that happen to lie in a smaller subspace.

Stating the Requirement Precisely

Every Vector Must Be Reachable

A collection {e₁, ..., eₖ} of vectors in V satisfies the span requirement if, for every vector v ∈ V, there exist scalars c¹, ..., cᵏ such that v = c¹e₁ + ⋯ + cᵏeₖ; equivalently, the span of the collection — the set of all such linear combinations — equals the entire space V, not merely some proper subset of it.

Distinct From, but Paired With, Linear Independence

The span requirement says nothing by itself about whether the chosen vectors are linearly independent; a collection can span the space while containing redundant vectors expressible in terms of the others. A basis requires both properties simultaneously — the span requirement guaranteeing completeness, and linear independence guaranteeing no redundancy — and a collection satisfying only the span requirement, without independence, is called a spanning set rather than a basis.

Consequences of the Span Requirement for Tensor Expansion

Guaranteeing the Expansion Formula Applies Universally

The entire justification for writing an arbitrary tensor as T = T^{i}_{\ j} eᵢ ⊗ eʲ rests on the span requirement holding for the underlying vector space basis: without it, some vectors (and hence some tensors built from them) would have no expression in terms of the chosen basis vectors at all, and the component-based expansion would be incomplete, covering only part of the space of tensors rather than all of it.

Failure of the Span Requirement Narrows Expansion Scope

If a chosen collection of vectors fails the span requirement — spanning only a proper subspace W ⊂ V — then the corresponding tensor basis expansion scope is correspondingly reduced to tensors built entirely from W, exactly as described when an incomplete expansion set is used deliberately or by oversight; the span requirement is therefore the precise condition whose failure is responsible for that narrowed scope.

Verifying the Span Requirement in Practice

Counting Argument in Finite Dimensions

In a finite-dimensional space of dimension n, a standard shortcut for verifying the span requirement is available: a collection of exactly n linearly independent vectors automatically spans the space, so once independence and the correct count are confirmed, the span requirement follows without a separate direct check; this shortcut fails, however, for a collection with fewer than n vectors, no matter how independent they are, since fewer than n independent vectors can never span an n-dimensional space.

Direct Verification When the Count Is Insufficient or Uncertain

When the number of proposed basis vectors is not already known to equal the space's dimension, the span requirement must be checked directly, typically by expressing a general vector of the space as an undetermined linear combination of the proposed vectors and confirming that the resulting system of equations for the coefficients always has a solution, for every choice of the vector being expanded.

Diagram of a Spanning Versus a Non-Spanning Collection

The Span Requirement Within the Basis System Structure

A Prerequisite for the Entire Structure to Be Well-Defined

Because the dual basis, the induced tensor product bases, and the whole basis system structure are built on top of the primal basis, the span requirement for the primal basis is a prerequisite for every derived piece of that structure to be correctly defined; a primal collection failing to span V produces a dual basis, if one is even defined relative to it, that likewise fails to pair correctly with all of V*, propagating the deficiency through the entire structure rather than confining it to the primal basis alone.

Restoring the Requirement When It Fails

When a proposed collection of vectors is found to fail the span requirement, the standard remedy is to add additional, carefully chosen vectors — linearly independent of the ones already present — until the enlarged collection does span the space, at which point it becomes a genuine basis and the full apparatus of tensor component expansion becomes available for the whole space rather than only for the subspace the original, deficient collection happened to reach.