2.7.2 Tensor Linear Combination Vector Terms

Explore how tensor linear combinations form vector terms, foundational in algebraic structures and tensor algebra applications.

Tensor Linear Combination Vector Terms is the description of the individual tensors $T_{1}, \dots, T_{N}$ that appear as the objects being combined in a linear combination, each treated purely in its capacity as an element of a vector space, regardless of whether it is a simple tensor, a general tensor, or itself already a combination of others. Where the scalar coefficients supply the weighting, the vector terms supply the substance of the combination, and understanding their required properties, and the freedom permitted in choosing them, is what determines what a given linear combination can and cannot produce.

Requirements on the Terms

Common Type Requirement

Let $V$ be a vector space over a field $F$ . Every vector term appearing in a single linear combination must belong to the same tensor space $T_{s}^{r} (V)$ , of fixed type $(r, s)$ :

T_{1}, \dots, T_{N} \in T_{s}^{r} (V)

This requirement follows from the definition of tensor addition, which is undefined between tensors of differing type, so terms of mismatched type simply cannot coexist within a single sum.

No Restriction on Internal Structure

Beyond sharing a common type, a vector term is subject to no further restriction: it may be a simple tensor, a general tensor already expressed as a sum of several simple tensors, the zero tensor, or even another linear combination of other tensors. The linear combination structure treats every admissible term identically, purely as a point of the vector space $T_{s}^{r} (V)$ , without regard to how that point happens to be constructed.

Terms Versus the Underlying Multilinear Content

Opacity of Internal Structure

Within the linear combination, a vector term's internal description as a multilinear map, or its factorization if it is simple, plays no role; only its identity as an element of $T_{s}^{r} (V)$ matters for how it participates in the sum. Two different-looking expressions denoting the same tensor may be used interchangeably as a term, since the combination depends only on which element of the vector space each term denotes.

Consequence for Simplification

Because of this opacity, a vector term may always be replaced by any equal tensor without altering the value of the linear combination, licensing substitutions and simplifications of individual terms independently of the surrounding sum.

Repetition and Redundancy Among Terms

Repeated Terms

The same tensor may appear more than once among the vector terms of a combination; if $T_{j} = T_{k}$ for $j \neq k$ , their contributions may be merged into a single term with coefficient $α_{j} + α_{k}$ , without changing the resulting tensor.

Redundant Terms Under Dependence

If the vector terms are linearly dependent, some term is itself expressible as a combination of the others, meaning it contributes nothing to the reachable set of combinations beyond what the remaining terms already provide. Recognizing such redundancy is what allows a spanning set of vector terms to be reduced to a linearly independent, and therefore minimal, generating set.

The Number of Terms

Finiteness

A linear combination, by definition, involves only finitely many vector terms, even when the tensor space itself, or a spanning set under consideration, is described by an infinite family of candidate tensors. This finiteness restriction is what keeps the linear combination structure well-defined as an ordinary finite sum, rather than requiring any notion of convergence or limit.

Selecting Terms from a Larger Set

When vector terms are drawn from a larger set $S \subset T_{s}^{r} (V)$ , possibly infinite, a specific linear combination always selects only a finite subset of $S$ to serve as its terms; the span of $S$ is the union, over every such finite selection, of all linear combinations formed from that selection.

Special Choices of Vector Terms

Basis Tensor Products as Terms

Choosing the vector terms to be exactly the basis tensor products of an induced tensor basis produces the coordinate expansion of a tensor, with the components serving as coefficients; this is the most structured choice of terms, since basis tensor products are simultaneously spanning and independent.

Arbitrary Spanning Terms

Choosing the vector terms to be any spanning set, not necessarily independent, still allows every tensor of the space to be reached as some linear combination, though the coefficients achieving a given tensor may no longer be unique, since dependence among the terms permits multiple coefficient assignments to yield the same result.

A Single Term

A linear combination may consist of a single vector term, in which case it reduces to a scalar multiple of that one tensor, the simplest nontrivial instance of the general structure.

Terms and the Resulting Subspace

The Span Generated by a Set of Terms

The set of all tensors obtainable as a linear combination of a fixed collection of vector terms is the span of that collection, a subspace of $T_{s}^{r} (V)$ whose dimension is bounded above by the number of distinct, linearly independent terms among those supplied.

Terms Determine Reachability

Any tensor lying outside the span of a given collection of vector terms cannot be produced by any linear combination of those terms, regardless of how the scalar coefficients are chosen; reachability is governed entirely by which vector terms are made available, while the coefficients only determine which point within the resulting span is selected.