4.10.4 Tensor Vector Valued Basis Expansion

Tensor Vector Valued Basis Expansion is a method to represent multilinear transformations using vector-valued bases in tensor algebra.

Tensor Vector Valued Basis Expansion is the representation of a vector-valued multilinear map as a finite sum of basis tensor products, each weighted by a coefficient that is itself a vector in the target space W rather than a scalar in the base field. It extends the ordinary basis expansion of a ordinary tensor, in which a scalar-valued tensor is written as a linear combination of basis tensor products with scalar coefficients, to the setting where the codomain is a vector space, so that the coefficients attached to each basis tensor product become vectors of W instead of numbers.

Setting Up the Expansion

The Argument Spaces and Their Bases

Let V_1, ..., V_k be finite-dimensional vector spaces, each equipped with a basis, and let M : V_1 × ... × V_k → W be a multilinear map into a finite-dimensional vector space W. Fixing a basis e^{(1)}_1, ..., e^{(1)}_{n_1} for V_1, a basis e^{(2)}_1, ..., e^{(2)}_{n_2} for V_2, and so on through V_k, every argument v_r ∈ V_r decomposes uniquely as

v_{r} = \sum_{i_{r} = 1}^{n_{r}} c_{r}^{i_{r}} e_{i_{r}}^{(r)}

where the c_r^{i_r} are ordinary scalar coordinates of v_r relative to the chosen basis of V_r.

Applying Multilinearity

Because M is linear in each of its k arguments separately, substituting the basis expansion of every argument and distributing M across each sum produces a single expansion of M(v_1, ..., v_k) as a sum over all combinations of basis indices, with the scalar coordinates factored out in front of M evaluated on the basis vectors themselves.

The Vector-Valued Basis Expansion Formula

Isolating the Vector Coefficients

Carrying out this substitution gives the tensor vector-valued basis expansion:

M (v_{1}, \dots, v_{k}) = \sum_{i_{1}, \dots, i_{k}}^{⋮} c_{1}^{i_{1}} \dots c_{k}^{i_{k}} M (e_{i_{1}}^{(1)}, \dots, e_{i_{k}}^{(k)})

where each term M(e^{(1)}_{i_1}, ..., e^{(k)}_{i_k}) is itself a fixed vector in W, since it is the map evaluated on basis vectors rather than on the general arguments v_r. These fixed vectors are the vector-valued coefficients of the expansion, and there are exactly n_1 × n_2 × ⋯ × n_k of them, one for each combination of basis indices.

Naming the Coefficient Vectors

It is convenient to name each of these coefficient vectors directly:

w_{i_{1} \dots i_{k}} = M (e_{i_{1}}^{(1)}, \dots, e_{i_{k}}^{(k)}) \in W

so that the expansion reads compactly as a sum of scalar products c_1^{i_1} ⋯ c_k^{i_k} multiplying vectors w_{i_1 ... i_k} of W, rather than multiplying scalars as in the ordinary tensor case.

Relation to the Tensor Role

Coefficients as Components of a Single Tensor

The collection of vectors w_{i_1 ... i_k} is precisely the component data of the tensor obtained by applying the tensor role construction to M, since expanding W in its own basis w_1, ..., w_m writes each w_{i_1 ... i_k} as a further linear combination ∑_a M~^a_{i_1 ... i_k} w_a, recovering the full component array of M~ with its free output index a alongside the argument indices i_1, ..., i_k.

Why the Expansion Is Unique

Because the chosen bases of V_1, ..., V_k are linearly independent, the basis tensor products e^{(1)}_{i_1} ⊗ ... ⊗ e^{(k)}_{i_k} are also linearly independent in the corresponding tensor product space, so the assignment of a vector coefficient w_{i_1 ... i_k} to each combination of indices is uniquely determined by M, and no two distinct choices of coefficients can represent the same map.

Basis Change and the Expansion

Effect of Changing the Argument Bases

If the basis of any V_r is changed, the scalar coordinates c_r^{i_r} transform by the corresponding change-of-basis matrix, and the vector coefficients w_{i_1 ... i_k} must be recomputed by evaluating M on the new basis vectors; the two effects combine so that the overall value M(v_1, ..., v_k) is unchanged, since it depends only on the arguments themselves and not on the basis used to expand them.

Effect of Changing the Output Basis

Changing the basis of W leaves the vector coefficients w_{i_1 ... i_k} themselves as vectors, unaffected in their identity, but re-expresses each one in terms of new basis vectors of W, which changes the numerical component array M~^a_{i_1 ... i_k} according to the ordinary transformation rule for a single free upper index tied to W.

Diagrammatic Summary

The diagram shows the expansion as a sum, indexed by the basis indices of each argument space, of scalar coordinate products multiplying vector coefficients that live in W, matching the vector-valued basis expansion formula.