3.6.3 Tensor Covector Evaluation Linearity

Tensor covector evaluation linearity explores how linear transformations act on covectors, preserving structure in tensor algebra.

Tensor Covector Evaluation Linearity is the property that the evaluation pairing between a covector and a vector respects addition and scalar multiplication separately in each of its two arguments, making the pairing a bilinear operation rather than merely a general function of two variables. This linearity is not an incidental feature; it is the defining requirement that a covector must satisfy to belong to the dual space in the first place, and it is what allows the evaluation operation to be manipulated algebraically using the ordinary rules of linear algebra.

Linearity in the Vector Argument

Additivity

For a fixed covector f in V* and any two vectors v_1, v_2 in V, evaluation distributes over vector addition:

f (v_{1} + v_{2}) = f (v_{1}) + f (v_{2})

This property is inherited directly from the definition of a linear functional: f is, by construction, required to be additive on V in order to qualify as an element of V*.

Homogeneity

For a fixed covector f and any scalar c in the field F, evaluation respects scalar multiplication:

f (c v) = c f (v)

Additivity together with homogeneity is exactly what it means for f to be linear on V, so linearity in the vector argument is guaranteed automatically by the definition of the dual space, not by any extra assumption placed on the evaluation operation itself.

Linearity in the Covector Argument

Additivity of Covectors

The set V* of linear functionals on V is itself a vector space, with addition defined pointwise: for f_1, f_2 in V*, the sum f_1 + f_2 is the functional whose value on any v is f_1(v) + f_2(v). Evaluation therefore also distributes over addition in its first argument:

(f_{1} + f_{2}) (v) = f_{1} (v) + f_{2} (v)

Homogeneity of Covectors

Scalar multiplication of a covector f by c is likewise defined pointwise, (cf)(v) = c \cdot f(v), so evaluation is homogeneous in the covector argument:

(c f) (v) = c (f (v))

Because the vector space structure of V* is itself built from these pointwise operations, linearity of evaluation in the covector slot is, again, a direct and automatic consequence of how V* is constructed, rather than a separate assumption.

Bilinearity as a Combined Statement

The Full Bilinear Property

Combining both slots, for scalars a, b, c, d in F, vectors v_1, v_2 in V, and covectors f_1, f_2 in V*, the evaluation operation satisfies

(a f_{1} + b f_{2}) (c v_{1} + d v_{2})

= a c f_{1} (v_{1}) + a d f_{1} (v_{2}) + b c f_{2} (v_{1}) + b d f_{2} (v_{2})

This expansion is exactly analogous to distributing multiplication over a sum, and it justifies why the covector-vector pairing behaves, in every algebraic sense, like a generalized product rather than an arbitrary two-variable function.

Why Bilinearity, Not Just Linearity in One Slot, Is Required

A function that is linear only in the vector argument would fail to capture the dual space structure, and a function that is linear only in the covector argument would fail to be a well-defined evaluation of a functional. It is the joint bilinearity, linear separately in each of the two arguments while the other is held fixed, that classifies this pairing as a (1, 1) tensor under the general definition of a tensor as a multilinear map.

Consequences of Linearity in Component Form

Linearity Justifies the Summation Formula

Given a basis e_1, ..., e_n of V with dual basis e^1, ..., e^n, any vector decomposes as v = v^i e_i and any covector as f = f_i e^i. Applying linearity in the vector argument repeatedly to expand v into its basis components, and linearity in the covector argument to expand f, yields exactly the componentwise sum

f (v) = f_{i} v^{i}

This shows that the entire coordinate formula for evaluation, used throughout tensor calculus, is a direct consequence of linearity rather than an independent postulate.

Diagrammatic Summary

The diagram shows both arguments of the evaluation operation being decomposed by linearity into scalar-weighted sums, which distribute through the pairing to produce a single expanded scalar result.