8.14.2 Tensor Index Range Basis Dependence

Tensor Index Range Basis Dependence is the principle that the number of values an index ranges over is fixed by the number of basis vectors chosen to span the underlying vector space, so that the size of every index range is, at bottom, a count of basis elements rather than an independent property of the space itself. Because every finite-dimensional vector space admits many different bases, all sharing the same cardinality, the index range determined by any one basis is invariant under a change of basis, even though the specific components an index labels are not.

The Basis as the Origin of the Range

Range Equals Number of Basis Vectors

For a vector space spanned by a basis ${e_1, e_2, \dots, e_n}$, an index attached to a vector component ranges exactly over ${1, 2, \dots, n}$ because there are exactly $n$ basis vectors to select among. A vector $v$ expressed in this basis is written

and the index $i$ ranges over exactly as many values as there are basis vectors $e_i$ to pair its components with; the range is not an independently chosen number but a direct readout of the basis's cardinality.

Basis Cardinality Equals Dimension

Because every basis of a finite-dimensional vector space has the same number of elements — the dimension $n$ of the space, by the basis exchange theorem — the index range determined by basis dependence always agrees numerically with the dimension-dependent range discussed elsewhere in tensor index notation. Basis dependence and dimension dependence yield the identical numerical bound, but basis dependence explains that bound in terms of a concrete spanning set rather than an abstract invariant of the space.

Invariance of the Range Under Change of Basis

Range Size Is Basis-Independent

Although the specific components a vector or tensor has depend heavily on which basis is chosen, the size of the index range itself does not change when the basis changes, since any two bases of the same space contain the same number of vectors. Switching from a basis ${e_i}$ to a new basis ${e_{i'}}$ related by

leaves the index $i'$ ranging over exactly the same set ${1, \dots, n}$ as the original index $i$ did, since the transformation matrix $\Lambda$ is square and invertible, preserving the count of basis vectors.

What Does Change: Components, Not Range

What genuinely differs between bases is the numerical value each component takes, not the range the index runs over. A vector's components $v^{i}$ in one basis and $v^{i'}$ in another basis will typically be different numbers, related by the transformation law $v^{i'} = \Lambda^{i'}_{j} v^{j}$, but both indices $i$ and $i'$ range over the identical set of $n$ values.

Basis Dependence for Dual and Mixed Objects

Dual Basis Shares the Same Range

The dual basis ${e^{1}, \dots, e^{n}}$, used to express covector components, is constructed to have exactly as many elements as the original basis ${e_1, \dots, e_n}$, one dual basis covector for each original basis vector. As a result, an index on a covector, such as the $i$ in $\omega_i$, ranges over the same set of values as an index on a vector in the same space, since both ranges trace back to bases of equal cardinality.

Consistency Across Tensor Products

When forming tensor products of a vector space with itself or with its dual, the basis for the resulting product space is built from all combinations of the original basis vectors, and each index attached to a slot of the resulting tensor still ranges over the same $n$ values inherited from the single underlying basis, regardless of how many tensor factors are combined. Basis dependence guarantees that every index in a tensor equation, no matter how many factors are involved, ultimately traces back to the same fixed basis cardinality.

Basis Dependence Versus Coordinate Domain

Basis as the Algebraic Layer, Coordinates as the Geometric Layer

Basis dependence explains the range of an index in purely algebraic terms — as a count of spanning vectors in a vector space — while the coordinate domain of an index attaches a geometric or physical label, such as a specific coordinate direction, to each value in that range. The two perspectives describe the same numerical range from different angles: basis dependence explains why the range has the size it does, while the coordinate domain explains what each value in that range represents geometrically once a coordinate system is fixed.

Role Within Index Range Notation

Basis dependence supplies the most fundamental justification for why tensor index ranges take the values they do: an index range is, at its root, nothing more than an enumeration of the vectors in whatever basis has been chosen for the space. Recognizing this basis-theoretic origin clarifies why the range size is invariant under change of basis even while individual components are not, and it grounds both dimension dependence and coordinate-domain interpretations of index range in a single, more primitive algebraic fact about vector spaces and their bases.