2.7 Tensor Linear Combination Structure

Tensor Linear Combination Structure is the general framework governing finite sums of tensors of a fixed type, each weighted by a scalar coefficient, built entirely from tensor addition and scalar action and forming the mechanism by which every element of a tensor space, simple or otherwise, is ultimately expressed. It generalizes the familiar notion of a linear combination from ordinary vectors to multilinear objects, while remaining subject to exactly the same algebraic rules that govern linear combinations in any vector space.

Definition

The General Form

Let $$V$$ be a vector space over a field $$F$$, and let $$T_1,\cdots ,T_N\in T_s^r\left(V\right)$$ be tensors of a common type $$\left(r,s\right)$$. A linear combination of these tensors is any expression

with coefficients $${\alpha }_k\in F$$, built by repeated application of tensor addition and scalar action to the given tensors.

Requirement of a Common Type

The structure only admits tensors sharing the same type $$\left(r,s\right)$$ as terms of a single linear combination, since addition itself is defined only between tensors of matching type; there is no linear combination structure mixing tensors of different types without first mapping them into a common space by some other operation.

Closure of the Combination

Every Combination Remains in the Same Space

Because $$T_s^r\left(V\right)$$ is closed under addition and scalar action, any linear combination of tensors of type $$\left(r,s\right)$$ is again a tensor of that same type:

Well-Definedness

Since addition is commutative and associative, the value of a linear combination does not depend on the order in which its terms are summed or grouped, so the sum notation above denotes a single, unambiguous tensor regardless of how the summation is carried out step by step.

Trivial and Nontrivial Combinations

The Trivial Combination

Setting every coefficient to $$0$$ produces the zero tensor regardless of which tensors $$T_1,\cdots ,T_N$$ are involved; this is called the trivial linear combination.

Nontrivial Vanishing Combinations

A linear combination is nontrivial if at least one coefficient is nonzero. Whether a nontrivial combination of a given set of tensors can still equal the zero tensor is precisely the question of linear independence: a set is linearly independent if and only if no nontrivial combination of its elements vanishes.

Span as the Set of All Combinations

Definition of Span

Given a set of tensors $$S\subset T_s^r\left(V\right)$$, the span of $$S$$ is the set of all linear combinations of finitely many elements of $$S$$:

Span as a Subspace

Because the linear combination structure is closed under further linear combinations, in the sense that a linear combination of linear combinations is again a linear combination of the original elements, $$\text{span}\left(S\right)$$ is itself a linear subspace of $$T_s^r\left(V\right)$$, the smallest subspace containing every element of $$S$$.

Linear Combination Structure of Basis Tensors

The Distinguished Combination

Every tensor $$T$$ admits a linear combination expression using the basis tensor products as the underlying set, with the components of $$T$$ serving as coefficients. Among all sets whose span is the full tensor space, the basis tensor products form the unique such set, up to reordering, that achieves this while remaining linearly independent, making the coordinate description of a tensor exactly this distinguished linear combination.

Simple Tensors Versus General Linear Combinations

A simple tensor is a single tensor product, requiring no linear combination at all beyond a trivial one-term sum, while a tensor of higher rank genuinely requires a linear combination of several simple tensors, none of which alone reproduces the given tensor. The linear combination structure is precisely what accommodates such higher-rank tensors within the same algebraic framework as simple ones.

Preservation Under Linear Operations

Behavior Under a Change of Basis

A linear combination expressed relative to one basis remains a linear combination relative to any other basis, since the change-of-basis transformation acts linearly on components, and a linear combination of tensors transforms by applying that same linear transformation to each term, then recombining, term by term, the transformed coefficients.

Behavior Under Contraction and the Tensor Product

Operations that are themselves linear in each tensor argument, such as contraction or one factor of a tensor product, distribute over a linear combination exactly as scalar multiplication distributes over addition of field elements, since multilinearity of these operations guarantees that applying them to a linear combination equals the corresponding linear combination of the operation applied to each term individually.