2.6.2 Tensor Basis Vector Label

The Tensor Basis Vector Label identifies and organizes basis vectors in tensor spaces, essential for tensor algebra representation and manipulation.

Tensor Basis Vector Label is the individual index value, such as the integer $1$ through $n$ , attached to a basis vector or dual basis covector for the sole purpose of naming which specific element of the basis is being referred to. It is distinct from the basis vector itself, which is a genuine element of the vector space, and distinct from the position that vector occupies within a tensor product, being instead simply the identifying tag that lets one basis vector be told apart from another.

Labels as Identifiers

Setting

Let $V$ be a vector space of dimension $n$ with a fixed ordered basis $e_{1}, \dots, e_{n}$ . The label attached to the vector $e_{i}$ is the integer $i$ ; it is a bookkeeping symbol, not part of the algebraic structure of $V$ itself.

Distinguishing the Label from the Vector

The vector $e_{3}$ , for example, is an element of $V$ , participating in sums and scalar multiples like any vector. The label $3$ attached to it carries no such algebraic content; it is a name used in writing and referring to that vector, and it plays no role in any vector space operation performed on $e_{3}$ .

Labels and Dual Labels

Matching Labels Across Dual Bases

The dual basis covector $e^{i}$ carries the same label $i$ as the basis vector $e_{i}$ it is dual to, by convention, so that the duality relation can be written compactly:

e^{i} (e_{j}) = δ_{j}^{i}

This shared labeling convention is what allows an upper label and a lower label to be recognized as referring to a dual pair without additional bookkeeping.

Labels Are Assigned, Not Intrinsic

Nothing about the vector space forces the label $i$ onto $e_{i}$ other than the ordering chosen for the basis; relabeling the same basis with a different bijection to $1, \dots, n$ produces the identical set of vectors under a permuted set of labels.

Labels in Tensor Basis Products

Compound Labels

A basis tensor product carries one label per factor, assembled into a multi-index:

e_{i_{1}} \otimes \dots \otimes e_{i_{r}} \otimes e^{j_{1}} \otimes \dots \otimes e^{j_{s}}

The compound label $(i_{1}, \dots, i_{r}, j_{1}, \dots, j_{s})$ identifies exactly one basis tensor product among the $n^{r + s}$ total, with each of its constituent labels naming the individual factor occupying that particular position.

Free Labels Versus Dummy Labels

When such a label appears once in an expression, it is a free label, naming a specific component of a specific tensor. When a label appears twice, once upper and once lower, the Einstein summation convention interprets it as a dummy label, meaning it does not identify a single basis element at all but instead indicates that a sum over every possible label value is to be taken.

Renaming Dummy Labels

Invariance Under Relabeling

Because a dummy label represents a summation over all its possible values, the specific symbol chosen for it carries no meaning: replacing every occurrence of a dummy label with a different symbol not otherwise in use leaves the value of the expression unchanged:

T_{i} e^{i} = T_{k} e^{k}

This freedom to rename dummy labels is used routinely to avoid label collisions when combining two separate expressions that happen to use the same dummy label for unrelated sums.

Restriction on Renaming Free Labels

A free label, by contrast, cannot be renamed independently on only one side of an equation without also renaming it on the other side, since doing so would refer to a different component or a different argument of the tensor being described.

Labels Under a Change of Basis

Same Label Set, Different Meaning

When the underlying basis is changed, the same set of labels $1, \dots, n$ is typically reused for the new basis ${\tilde{e}}_{1}, \dots, {\tilde{e}}_{n}$ , but the label $i$ now identifies a different vector than it did under the original basis. The tilde or other decoration distinguishing ${\tilde{e}}_{i}$ from $e_{i}$ is what disambiguates which basis a given label is being read against.

Labels Do Not Transform

It is the components, not the labels, that transform under a change of basis; the labels $1$ through $n$ remain the same finite set of names throughout, serving as a fixed indexing scheme applied consistently to whichever basis is currently under discussion.

Practical Role of Labels

Enabling Unambiguous Reference

The primary function of a tensor basis vector label is to make it possible to refer unambiguously to a single component, a single basis vector, or a single dual basis covector, without restating the full vector or covector each time. All subsequent index notation, the summation convention, the transformation law, and the placement of upper and lower indices, depends on labels functioning reliably as such unambiguous names.