2.6.1 Tensor Basis Vector Position

Understanding where tensor basis vectors are positioned within their algebraic framework and how this positioning is fundamental to tensor algebra operations.

Tensor Basis Vector Position is the ordered slot occupied by a particular basis vector or dual basis covector factor within a basis tensor product, distinct from the index value labeling which specific basis element fills that slot. It governs which argument of the underlying multilinear map each factor corresponds to, and it is a structural feature entirely separate from the numerical value of the index attached to the factor occupying that position.

Position Versus Index Value

Two Independent Pieces of Data

Let $V$ be a vector space with basis $e_{1}, \dots, e_{n}$ and dual basis $e^{1}, \dots, e^{n}$ . A basis tensor product

e_{i_{1}} \otimes e_{i_{2}} \otimes \dots \otimes e_{i_{r}}

involves, for each factor, two independent pieces of information: its position, meaning whether it is the first, second, or $k$ -th factor in the product, and its index value, meaning which specific basis vector, $e_{1}$ through $e_{n}$ , occupies that position.

Notation Reflecting This Separation

The subscript position of $i_{k}$ in the multi-index records the position within the tensor product, while the value that $i_{k}$ takes, from $1$ to $n$ , records the index value at that position.

Position and the Multilinear Map's Argument Slots

Correspondence to the Domain

A tensor of type $(r, s)$ is a multilinear map on an ordered domain of $r$ covector arguments followed by $s$ vector arguments. The position of each factor in a basis tensor product corresponds directly to which argument of this domain that factor is designed to pair with under evaluation:

(e_{i_{1}} \otimes \dots \otimes e_{i_{r}} \otimes e^{j_{1}} \otimes \dots) (ω^{1}, \dots, v_{1}, \dots) = e_{i_{1}} (ω^{1}) \dots e_{i_{r}} (ω^{r}) e^{j_{1}} (v_{1}) \dots

The factor in the first position pairs only with the first covector argument, the factor in the second position pairs only with the second, and so on, so position determines pairing, independent of which index value labels the factor.

Consequences of Changing Position

Reordering Produces a Different Element

Exchanging the positions of two factors of the same variance type generally produces a different tensor, even when the index values involved are unchanged, because the resulting multilinear map pairs with the domain arguments differently:

e_{i} \otimes e_{j} \neq e_{j} \otimes e_{i}

whenever $i \neq j$ , since the two sides act differently on an input tuple where the first and second arguments are distinct basis covectors.

Position and Symmetry

Whether a tensor is unaffected by a change of position, that is, whether it is invariant under permuting the factors occupying its like-type positions, is exactly the defining condition for that tensor to be symmetric; a tensor that changes sign under such a permutation is antisymmetric. Position is therefore the structural feature against which symmetry properties are stated, rather than a feature that symmetric or antisymmetric tensors lack.

Position Within a Fixed Variance Type

Contravariant Positions Are Mutually Interchangeable in Range

All $r$ contravariant positions independently range over the same basis vector set, and all $s$ covariant positions independently range over the same dual basis covector set. Position never mixes a contravariant slot with a covariant one; a basis vector never occupies a position reserved for a dual basis covector, and conversely.

Position Count Equals Tensor Order

The number of contravariant positions equals $r$ and the number of covariant positions equals $s$ , so the total count of positions in a basis tensor product is $r + s$ , matching the order of the tensor type exactly.

Position Under Change of Basis

Position Is Preserved, Values Transform

When the underlying basis of $V$ is changed, the transformation law relates the value occupying a given position in the new basis to a linear combination of values occupying that same position in the old basis. Position itself, as a slot in the tensor product, is unaffected by the change of basis; only the index values attached to each position undergo transformation.

Position-Wise Application of the Transformation Law

This is why the tensor transformation law is applied one position at a time, contracting the transition matrix, or its inverse, against exactly the index value occupying that position, without any position affecting or being affected by the transformation applied to a different position.

Position in Coordinate Notation

Multi-Index Ordering Reflects Position

In the coordinate expression $T_{j_{1} \dots j_{s}}^{i_{1} \dots i_{r}}$ , the left-to-right order in which the index labels are written mirrors the left-to-right order of positions in the corresponding basis tensor product, so that reading the notation left to right recovers the correct pairing between each component index and the argument slot it governs.