1.3.1 Multilinear Tensor Structure

Multilinear Tensor Structure combines linear spaces into a framework for multilinear algebra, essential in physics and engineering.

Multilinear Tensor Structure is the property, foundational to the entire theory of tensors, that a tensor acts as a map taking several vector or covector arguments and producing a scalar in such a way that the map is linear in each individual argument separately while every other argument is held fixed. This property, called multilinearity, is what distinguishes tensors from more general functions of several vector variables, and it is the structural feature from which the transformation laws, the tensor product construction, and the component notation of tensor algebra all follow.

Definition of Multilinearity

The Multilinear Map

A tensor of type (p, q) can be defined directly as a multilinear map

T : \times_{i = 1}^{p} V^{*} \times \times_{j = 1}^{q} V \to F

meaning it takes p covectors and q vectors as input and returns a single scalar in the field F.

Linearity in Each Argument Separately

Multilinearity requires that fixing every argument except one, the map behaves as an ordinary linear function of the remaining argument. For a single vector argument v_k held among otherwise fixed arguments,

T (\dots, a v_{k} + b w_{k}, \dots) = a T (\dots, v_{k}, \dots) + b T (\dots, w_{k}, \dots)

for any scalars a, b in F and any vectors v_k, w_k in V, with the identical property holding independently for each of the other p + q - 1 argument slots.

Contrast with Linearity in All Arguments Jointly

Not the Same as a Single Linear Map

Multilinearity is a weaker and different condition than linearity of the whole map viewed as a function of all arguments bundled together into one large vector; a multilinear map is generally not linear as a function on the direct sum of its argument spaces, since scaling every argument at once scales the output by the product of the scaling factors, not by a single common factor.

T (c v_{1}, \dots, c v_{q}) = c^{q} T (v_{1}, \dots, v_{q})

illustrating that scaling every one of q vector arguments by c scales the output by c raised to the power q, not simply by c.

Multilinearity and the Component Representation

Why Components Suffice to Determine the Map

A key consequence of multilinearity is that the entire behavior of a tensor on arbitrary vector and covector arguments is completely determined once its values on a basis are known. Because each argument can be expanded in a basis and the map distributed linearly over that expansion, the tensor's action on any collection of arguments reduces to a sum of scalar multiples of its basis values, which are exactly the tensor's components.

T (u, v) = T (u^{i} e_{i}, v^{j} e_{j}) = u^{i} v^{j} T (e_{i}, e_{j}) = u^{i} v^{j} T_{i j}

This expansion shows exactly how the finite set of components T_{ij} encodes the entire multilinear map, since applying T to any pair of vectors reduces to this single sum weighted by their respective components.

Multilinearity as the Source of the Transformation Law

Deriving the Transformation Rule

The rule governing how tensor components change under a change of basis is not an independent postulate but a direct consequence of multilinearity applied to the change-of-basis expansion itself. Writing the new basis vectors as linear combinations of the old ones and substituting into the multilinear expansion of the tensor's action produces exactly the transformation law relating old and new components.

Multilinearity Guarantees Basis Independence

Because the multilinear map itself is defined without reference to any basis, its value on any specific set of vector and covector arguments is a single well-defined scalar regardless of which basis is used to compute it, which is precisely what guarantees that the component transformation law preserves the tensor as a single coherent object across all bases.

Diagrammatic Summary

The diagram represents a multilinear tensor structure as a map T that accepts several vector and covector arguments and produces a single scalar output, being linear in each individual argument while all the others remain fixed.