2.6.4 Tensor Basis Vector Coordinate Role

Understanding how basis vectors and coordinates interact in tensor algebra is key to grasping tensor structure and transformation properties.

Tensor Basis Vector Coordinate Role is the specific function a basis vector performs in generating and defining the coordinate system relative to which tensors are described, distinct from its role as a mere factor in tensor products: each basis vector marks out one coordinate axis, has trivially known coordinates of its own, and, through its dual, supplies the mechanism by which any other vector or tensor's coordinates are extracted.

Marking a Coordinate Axis

Setting

Let $V$ be a vector space of dimension $n$ over a field $F$ , with ordered basis $e_{1}, \dots, e_{n}$ . Relative to this basis, every vector $v \in V$ is assigned coordinates $v^{1}, \dots, v^{n}$ via $v = v^{i} e_{i}$ , and the vector $e_{k}$ is, by definition, the direction along which only the $k$ -th coordinate varies: the set of vectors $t e_{k}$ , for $t \in F$ , constitutes the $k$ -th coordinate axis.

Basis Vectors as Axis Generators

Because there are exactly $n$ basis vectors and exactly $n$ coordinate axes, each basis vector generates precisely one axis, and the full set of basis vectors together generates the entire coordinate grid relative to which every point of $V$ is located.

The Self-Coordinates of a Basis Vector

Coordinates of a Basis Vector Relative to Itself

The coordinates of the basis vector $e_{k}$ , relative to the same basis, are given directly by the Kronecker delta:

e_{k} = δ_{k}^{i} e_{i}

meaning $e_{k}$ has coordinate $1$ in its own slot and coordinate $0$ in every other slot. This trivial self-coordinate is what makes basis vectors the simplest possible reference points from which the coordinates of an arbitrary vector are measured by comparison.

Elementary Tensors and Their Coordinates

The same triviality extends to tensor spaces: the basis tensor product $e_{i_{1}} \otimes \dots \otimes e^{j_{s}}$ has, relative to the induced tensor basis, a single component equal to $1$ and all other components equal to $0$ , generalizing the coordinate role of a basis vector to every tensor type.

Extracting Coordinates via the Dual Pairing

Basis Vectors as Targets of Evaluation

While a basis vector marks an axis, it is the dual basis covector $e^{i}$ that performs the extraction of the $i$ -th coordinate from an arbitrary vector, via $e^{i} (v) = v^{i}$ . The coordinate role of the basis vector $e_{i}$ and the coordinate-extracting role of its dual $e^{i}$ are complementary: one marks where the axis points, the other reads off how far along that axis a given vector lies.

Generalization to Tensor Coordinates

For a tensor $T \in T_{s}^{r} (V)$ , the same pairing mechanism, applied to a full input tuple of basis vectors and dual basis covectors, extracts each component of $T$ :

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

so that basis vectors, filling the covariant argument slots, are exactly the probes needed to extract the vector-type coordinates of a tensor.

Basis Vectors as Coordinate Reference Points

Comparison Structure

Every coordinate assigned to a vector or tensor is fundamentally a statement of proportion or combination relative to the basis vectors: a coordinate value expresses how much of a given basis vector's direction is present, and the coordinate role of the basis vector is to serve as that fixed unit of comparison against which such proportions are measured.

Unchanging Reference Under Fixed Basis

As long as the basis is held fixed, the coordinate role of each basis vector remains fixed as well; every coordinate statement made about any vector or tensor is made relative to this same unchanging set of reference directions, which is what allows different tensors' coordinates to be meaningfully compared, added, or combined term by term.

Coordinate Role Under Change of Basis

Reassignment of Axes

When the basis is changed to ${\tilde{e}}_{1}, \dots, {\tilde{e}}_{n}$ , the coordinate role previously played by $e_{i}$ is taken over by ${\tilde{e}}_{i}$ , which now marks the $i$ -th axis of the new coordinate system, generally pointing in a different direction within $V$ than the original $e_{i}$ did.

Invariance of the Role Itself

Although the specific vector performing the coordinate role changes with the basis, the role itself, generating one axis, having trivial self-coordinates, and pairing with the dual basis to extract coordinates from other vectors, is preserved under any change of basis, since these are structural features of what it means to be a basis vector, not features tied to any particular choice of $V$ 's basis.