2.6.5 Tensor Basis Vector Expansion Role

Tensor Basis Vector Expansion Role enables representation of tensors through basis vectors, forming the foundation for tensor algebra operations and transformations.

Tensor Basis Vector Expansion Role is the function a basis vector performs as a term being summed, weighted by a coefficient, in the reconstruction of an arbitrary vector or tensor from its coordinates, complementing the coordinate role of extracting those coefficients in the first place. Where the coordinate role reads a number out of a given vector, the expansion role builds a vector back up out of numbers, and the two roles together form a matched pair of inverse operations centered on the same basis vector.

The Expansion Formula

Setting

Let $V$ be a vector space of dimension $n$ over a field $F$ , with basis $e_{1}, \dots, e_{n}$ . Given any coordinates $v^{1}, \dots, v^{n}$ in $F$ , the basis vectors serve as the terms of the expansion

v = v^{i} e_{i}

with summation implied over $i$ . Here each $e_{i}$ plays the role of a fixed direction contributed to the sum in an amount governed by its accompanying coefficient $v^{i}$ .

Extension to Tensors

The same expansion role extends to tensor spaces: a tensor $T \in T_{s}^{r} (V)$ is reconstructed from its components by

T = T_{j_{1} \dots}^{i_{1} \dots} e_{i_{1}} \otimes \dots \otimes e^{j_{s}}

where the basis tensor products play the role of the terms, and the components of $T$ play the role of the coefficients weighting those terms.

Expansion as the Inverse of Coordinate Extraction

Round-Trip Consistency

The expansion role and the coordinate role are mutually inverse in the following sense: expanding a vector from its coordinates and then extracting coordinates back out returns the original coordinates,

e^{k} (v^{i}, e_{i}) = v^{i} e^{k} (e_{i}) = v^{i} δ_{i}^{k} = v^{k}

confirming that no information is lost when passing from a vector to its coordinates and back through the expansion.

The Expansion Map

The expansion role can be packaged as a map

ψ : F^{n} \to V

sending a coordinate tuple to the corresponding linear combination of basis vectors; this map is exactly the inverse of the coordinate assignment map built from the dual basis, since the two operations undo one another.

Linearity of the Expansion Role

Compatibility with Coefficient Addition

Because the basis vectors are held fixed while only the coefficients vary, expanding a sum of coordinate tuples equals the sum of the individual expansions:

(a^{i} + b^{i}) e_{i} = a^{i} e_{i} + b^{i} e_{i}

Compatibility with Coefficient Scaling

Similarly, scaling every coefficient before expansion produces the same result as expanding first and then scaling:

(α, v^{i}) e_{i} = α (v^{i}, e_{i})

These two properties confirm that the expansion role is fulfilled through a linear operation, consistent with the vector space operations already defined on $V$ .

Uniqueness of the Expansion

No Ambiguity in the Terms Used

The independence property of the basis guarantees that the coefficients appearing in an expansion are the only ones that reproduce a given vector: no alternative choice of coefficients, using the same basis vectors as terms, yields the same sum unless every coefficient matches exactly. The expansion role of each basis vector is therefore tied to a single, non-interchangeable coefficient for any fixed target vector.

Contribution Independent of Other Terms

Each basis vector's contribution to the expansion is determined solely by its own coefficient, unaffected by the coefficients attached to the other basis vectors in the same sum, a direct consequence of linear independence: altering one coefficient changes only the corresponding term, never compensating for or interacting with any other term in the expansion.

Expansion Role in Constructing Simple and General Tensors

Building Simple Tensors from Components

A simple tensor's expansion is a special case in which only the basis tensor products corresponding to a single choice of factors contribute, while a general tensor's expansion draws on multiple basis tensor products simultaneously, each fulfilling its expansion role independently and summing to produce the full tensor.

Term-by-Term Construction

In this way, the expansion role reduces the task of specifying an arbitrary tensor of type $(r, s)$ to the much simpler task of specifying $n^{r + s}$ individual scalar weights, one per basis tensor product, with the basis tensor products themselves supplying all of the structural, multilinear content of the resulting object.