2.5.4 Tensor Basis Coordinate Assignment

Tensor Basis Coordinate Assignment defines how vectors and tensors are expressed in a basis, establishing coordinates through linear combinations in algebraic structures.

Tensor Basis Coordinate Assignment is the explicit procedure, and the resulting linear isomorphism, by which every tensor in a fixed-type tensor space is assigned a unique array of scalar coordinates relative to a chosen basis of the underlying vector space and its dual. It is the operational counterpart of the tensor basis spanning and independence properties: those properties guarantee that such an assignment exists and is well-defined, while the assignment itself is the map that carries that guarantee into practice.

The Assignment Map

Setting

Let $V$ be a vector space of dimension $n$ over a field $F$ , with basis $e_{1}, \dots, e_{n}$ and dual basis $e^{1}, \dots, e^{n}$ . The coordinate assignment map

φ : T_{s}^{r} (V) \to F^{n^{r + s}}

sends a tensor $T$ to the array of its components,

φ (T) = (T_{j_{1} \dots j_{s}}^{i_{1} \dots i_{r}})

where each component is computed by evaluation on basis inputs.

Assignment Procedure

The assignment is carried out by a fixed, mechanical procedure: for each of the $n^{r + s}$ admissible index tuples $(i_{1}, \dots, i_{r}, j_{1}, \dots, j_{s})$ , substitute the corresponding basis vectors and dual basis covectors into $T$ and record the resulting scalar as the coordinate at that index position.

Ordering Multi-Indices into a Single Array

From Multi-Index to Linear Position

Although each coordinate is naturally labeled by a multi-index $(i_{1}, \dots, j_{s})$ , the target space $F^{n^{r + s}}$ is a single flat coordinate space. A fixed ordering convention, such as treating the multi-index as digits of a base- $n$ numeral, converts each multi-index into a single linear position, so that the assignment is unambiguous as a map into an ordinary coordinate vector space.

Convention Dependence

This linear ordering is a matter of convention rather than mathematical necessity; any bijection between the set of multi-indices and the integers from $1$ to $n^{r + s}$ yields an equally valid assignment map, differing only by a permutation of coordinate slots.

Linearity of the Assignment

Compatibility with Addition

The assignment respects tensor addition:

φ (S + T) = φ (S) + φ (T)

since evaluating a sum of tensors on a fixed input equals the sum of the individual evaluations.

Compatibility with Scalar Action

The assignment likewise respects the scalar action:

φ (α, T) = α φ (T)

These two properties establish that the assignment map $φ$ is linear.

The Assignment as an Isomorphism

Injectivity

The tensor basis independence property shows that the assignment is injective: if $φ (T) = 0$ , then $T$ , being the corresponding linear combination of basis tensor products with all vanishing coefficients, is itself the zero tensor.

Surjectivity

The tensor basis spanning property shows that the assignment is surjective: every array of scalars corresponds to some tensor, namely the linear combination of basis tensor products with those scalars as coefficients.

Resulting Isomorphism

Injectivity and surjectivity together establish that $φ$ is a linear isomorphism:

T_{s}^{r} (V) ≅ F^{n^{r + s}}

identifying the abstract tensor space with an ordinary coordinate space of dimension $n^{r + s}$ .

The Inverse Assignment

Reconstructing a Tensor from Coordinates

The inverse map $φ^{- 1}$ sends a coordinate array back to the tensor it represents:

φ^{- 1} (c_{j_{1} \dots}^{i_{1} \dots}) = c_{j_{1} \dots}^{i_{1} \dots} e_{i_{1}} \otimes \dots \otimes e^{j_{s}}

so that assignment and reconstruction are mutually inverse operations, neither losing nor adding information about the tensor.

Dependence on the Chosen Basis

The Assignment Is Not Canonical

The coordinate assignment map depends on the choice of basis of $V$ ; a different basis yields a different assignment map $φ^{'}$ , related to the original by the standard tensor transformation law.

Composing Two Assignments

Given two bases with change-of-basis matrix $A_{i}^{k}$ , the composite map $φ^{'} \circ φ^{- 1}$ , carrying coordinates in one assignment to coordinates in the other, is precisely the linear transformation built from $A_{i}^{k}$ and its inverse applied to each contravariant and covariant index respectively, showing that different coordinate assignments of the same tensor are always related by an explicit, invertible linear substitution.

Practical Significance

Reducing Tensor Algebra to Array Algebra

Because the assignment is a linear isomorphism, any computation involving tensor addition and scalar action may be carried out entirely on coordinate arrays using ordinary array arithmetic, with the assurance that the result, mapped back through the inverse assignment, equals what would have been obtained by working directly with the tensors as multilinear maps.

Basis-Dependent Bookkeeping

The coordinate assignment supplies the bookkeeping device, upper and lower indices attached to a labeled array, through which more advanced tensor operations, such as the tensor product and contraction, are subsequently expressed and computed in explicit form.