2.7.4 Tensor Linear Combination Coordinate Form

Tensor Linear Combination Coordinate Form expresses tensors as linear combinations in coordinate systems, foundational in algebraic structures and tensor analysis.

Tensor Linear Combination Coordinate Form is the explicit formula, relative to a fixed basis, for computing the components of a linear combination of tensors directly from the components of its individual terms and their scalar coefficients, without needing to reconstruct any of the tensors as multilinear maps along the way. It is the computational realization of the linear combination structure, translating an abstract sum of weighted tensors into ordinary arithmetic performed on arrays of numbers.

The Coordinate Formula

Setting

Let $V$ be a vector space of dimension $n$ over a field $F$ , with basis $e_{1}, \dots, e_{n}$ and dual basis $e^{1}, \dots, e^{n}$ . Let $T_{1}, \dots, T_{N} \in T_{s}^{r} (V)$ have components ${(T_{k})}_{j_{1} \dots}^{i_{1} \dots}$ , and let $α_{1}, \dots, α_{N} \in F$ . The components of the linear combination

T = \sum_{k = 1}^{N} α_{k} T_{k}

are given by

T_{j_{1} \dots}^{i_{1} \dots} = \sum_{k = 1}^{N} α_{k} {(T_{k})}_{j_{1} \dots}^{i_{1} \dots}

for every admissible index combination.

Derivation of the Formula

Reduction to the Two Elementary Operations

The formula follows directly from the componentwise descriptions of tensor addition and scalar action established individually for those two operations: scalar action multiplies every component of $T_{k}$ by $α_{k}$ , and tensor addition then sums the resulting component arrays entrywise across all $k$ . No additional principle beyond these two componentwise rules is needed to justify the combined formula.

Verification by Evaluation

The formula may equivalently be verified by evaluating $T$ directly on basis inputs:

T_{j_{1} \dots}^{i_{1} \dots} = T (e^{i_{1}}, \dots, e_{j_{1}}, \dots) = \sum_{k = 1}^{N} α_{k} T_{k} (e^{i_{1}}, \dots, e_{j_{1}}, \dots)

using the definitions of addition and scalar action as pointwise operations on the multilinear maps involved.

Matrix and Array Interpretation

Combining Component Arrays

If each tensor $T_{k}$ is represented by its full component array, the coordinate form of the linear combination is obtained by scaling each array by its coefficient $α_{k}$ and summing the resulting arrays entrywise, exactly as a linear combination of ordinary coordinate vectors in $F^{n^{r + s}}$ would be computed.

Compatibility with the Coordinate Assignment Isomorphism

This agreement is not a coincidence: since the coordinate assignment map $φ$ sending a tensor to its component array is a linear isomorphism, it must send a linear combination of tensors to the corresponding linear combination of their component arrays,

φ (\sum_{k = 1}^{N} α_{k} T_{k}) = \sum_{k = 1}^{N} α_{k} φ (T_{k})

which is precisely the coordinate form of the linear combination stated above.

Coordinate Form Relative to Basis Tensor Products

The Distinguished Case

When each $T_{k}$ is itself one of the basis tensor products, its own component array has a single entry equal to $1$ and all others equal to $0$ , so the general coordinate form of the linear combination reduces exactly to the familiar coordinate expansion of a single tensor $T$ in terms of its own components, confirming that the coordinate expansion of a tensor is a special case of the general linear combination coordinate formula.

Basis Independence of the Formula's Validity

Consistency Across Bases

The coordinate form of a linear combination holds in every basis, not merely the one initially chosen, since the componentwise rules for addition and scalar action from which it is derived hold in every basis; applying the standard tensor transformation law to both sides of the formula, term by term, shows that the same relationship between combined and individual components persists after any change of basis.

Practical Use

Computing a Combination Without Reconstructing Multilinear Maps

The coordinate form allows a linear combination of tensors to be computed entirely from their component arrays, without ever evaluating any of the tensors as multilinear maps on arbitrary vector or covector inputs, making it the standard method by which linear combinations of tensors are carried out in explicit calculation.

Solving for Coefficients

Given a target tensor's components and the components of a proposed set of terms, the coordinate form also supplies the system of linear equations, one equation per admissible index combination, that any valid coefficients $α_{1}, \dots, α_{N}$ expressing the target as a combination of those terms must satisfy.