2.8 Tensor Basis Span Structure

Tensor Basis Span Structure is the specific instance of the general span relation obtained by taking the generating set to be the induced basis tensor products of a tensor space, characterized by the fact that this particular span not only equals the entire tensor space but does so with the minimal possible number of generators, none of which can be removed without shrinking the span. It is the structural fact that unifies the spanning property of a tensor basis with the general theory of spans, showing a tensor basis to be the most efficient possible spanning set for its tensor space.

The Span Equals the Whole Space

Setting

Let $$V$$ be a vector space of dimension $$n$$ over a field $$F$$, with basis $$e_1,\cdots ,e_n$$ and dual basis $$e^1,\cdots ,e^n$$, and let $$B$$ denote the set of induced basis tensor products of a fixed type $$\left(r,s\right)$$. The tensor basis span structure is the statement

established by the tensor basis spanning property, guaranteeing that every tensor arises as some finite linear combination of the elements of $$B$$.

Minimality of the Spanning Set

No Generator Is Redundant

Because the elements of $$B$$ are also linearly independent, none is expressible as a linear combination of the others, so by the span relation governing removal of a generator, discarding any single element of $$B$$ strictly shrinks the span, and the reduced set no longer spans $$T_s^r\left(V\right)$$.

Achieving the Dimension Bound

Since any finite spanning set of size $$m$$ satisfies $$\text{dim}{T}_s^r\left(V\right)\le m$$, and $$B$$ has exactly $$n^{r+s}$$ elements while achieving equality in this bound, $$B$$ attains the smallest size any spanning set of $$T_s^r\left(V\right)$$ could possibly have.

Comparison to Larger and Smaller Spanning Sets

Proper Supersets Remain Spanning but Are Not Minimal

Any set $$B\cup \left\{T\right\}$$, formed by adjoining an additional tensor $$T\in T_s^r\left(V\right)$$ to $$B$$, still spans $$T_s^r\left(V\right)$$ by monotonicity of the span relation, but is necessarily linearly dependent, since $$T$$ already lies in $$\text{span}\left(B\right)$$ and is therefore expressible in terms of the elements of $$B$$.

Proper Subsets Fail to Span

Any proper subset of $$B$$ has fewer than $$n^{r+s}$$ elements and therefore cannot span a space of that dimension, by the same dimension bound applied in reverse; some tensor of $$T_s^r\left(V\right)$$, specifically an omitted basis tensor product, necessarily falls outside the span of any such subset.

Uniqueness of Coefficients as a Consequence

Beyond Mere Spanning

The span relation alone, applied to a spanning set that is not independent, guarantees only that every tensor is reachable, without guaranteeing a unique choice of coefficients. The tensor basis span structure improves on this: because $$B$$ is simultaneously spanning and independent, the coefficients expressing any given tensor as a linear combination of $$B$$ are unique, and these unique coefficients are precisely the components of that tensor.

Span Structure Across Different Choices of Underlying Basis

Each Basis of V Yields a Distinct Minimal Spanning Set

A different basis of $$V$$ induces a different set $$B^{\prime }$$ of basis tensor products, related to $$B$$ by the standard tensor transformation law. Both $$B$$ and $$B^{\prime }$$ independently satisfy the tensor basis span structure, each spanning $$T_s^r\left(V\right)$$ minimally, since the property of being a minimal spanning set is a property of the tensor space itself, achieved by infinitely many different specific sets, one for each choice of underlying basis.

Invariant Dimension Across These Choices

Regardless of which basis of $$V$$ is used to construct the induced spanning set, its size is always exactly $$n^{r+s}$$, confirming that the minimal spanning cardinality, the dimension of $$T_s^r\left(V\right)$$, is an invariant of the tensor space, unaffected by which particular minimal spanning set is examined.

Practical Significance of the Structure

Basis Tensor Products as the Canonical Reference Spanning Set

Among the many spanning sets available for a tensor space, the induced basis tensor products are singled out as the canonical choice precisely because of this span structure: minimal, independent, and yielding unique coordinates, properties not guaranteed by an arbitrary spanning set drawn from elsewhere in the tensor space.

Foundation for Coordinate-Based Computation

Every technique that computes with tensors via components, rather than via their defining multilinear maps, relies on the tensor basis span structure to guarantee that the resulting coordinate description is both complete, since the span covers the whole space, and unambiguous, since the coefficients achieving that coverage are unique.