7.9 Tensor Covector Component Case

Tensor Covector Component Case is the description of how the numerical entries of a covector (a rank-one covariant tensor, also called a one-form or dual vector) are represented relative to a chosen basis of the dual space, and how those entries transform when the underlying basis of the vector space is changed. A covector is a linear functional acting on vectors, and its components arise as the values it assigns to each basis vector of the space on which it acts.

Structural Setting

The Dual Space

Given a vector space (V) with basis ({e_1, e_2, \ldots, e_n}), the dual space (V^{*}) consists of all linear functionals (\omega : V \to \mathbb{R}). The dual basis ({e^1, e^2, \ldots, e^n}) is defined by the property that each dual basis element returns one on its matching basis vector and zero on every other.

Definition of a Covector Component

A covector (\omega \in V^{*}) is written in terms of the dual basis with scalar coefficients (\omega_i), called its covariant components.

Each component is obtained by evaluating the covector directly on the corresponding basis vector of (V).

Index Placement and Notation

Lower Index Convention

Covector components always carry a lower (subscript) index, in contrast to vector components, which carry an upper (superscript) index. This index placement is not a stylistic choice; it encodes the transformation behavior of the quantity under a change of basis.

Component Table Representation

For a fixed basis, a covector's component case can be exhibited as an ordered list, matching the dimension of the space:

This ordered list is basis-dependent: the same covector produces a different component case in a different basis, even though the underlying linear functional is unchanged.

Transformation Behavior

Change of Basis for Vectors

Suppose the basis of (V) changes according to a transformation matrix (A), so that the new basis vectors are expressed in terms of the old ones:

Covariant Transformation Law

Covector components transform using the same matrix that relates the old basis vectors to the new ones, which is why they are called covariant: they transform "with" the basis rather than "against" it.

Contrast with Contravariant Components

Vector components, expressed with the inverse transformation matrix, change oppositely to the basis vectors:

This distinction is the structural reason a covector component case and a vector component case cannot be freely interchanged without an explicit metric or duality pairing.

Illustrative Case in Two Dimensions

Original Basis

Consider a covector acting on a two-dimensional space with basis ({e_1, e_2}), producing the component case:

Rescaled Basis

If the basis is rescaled so that (e_1' = 2e_1) and (e_2' = e_2), the covariant transformation law dictates that the first component is halved, since the covector must still return the same value when applied to the original vectors expressed in the new basis:

This inverse relationship between the scaling of a basis vector and the scaling of the corresponding covector component is the defining signature of the covector component case.

Invariance of the Underlying Functional

Basis-Independent Value

Although the numerical entries of the component case change under a basis transformation, the value the covector assigns to any fixed vector remains identical, since the contraction of covariant components with contravariant components is invariant.

Role in Tensor Algebra

The covector component case is the prototype for the covariant slot of every higher-rank tensor: any tensor index that transforms with the same rule as (\omega_i) is classified as a covariant index, and the tensor's full component case is built by combining such indices with the contravariant components that transform according to the inverse rule.