10.9.2 Tensor Covector Component Covariant Behavior

Tensor covector components transform covariantly under coordinate changes, preserving geometric relationships in multilinear algebra.

Tensor Covector Component Covariant Behavior is the qualitative pattern, following directly from the covector component change rule, in which a covector's components scale in the same sense as the scaling of the basis vectors under a change of basis, so that stretching the basis vectors causes the covector's components to grow and shrinking the basis vectors causes them to shrink as well, in exact parallel with the basis vectors themselves. It is the conceptual counterpart to the formal transformation formula, describing in qualitative terms why the components of a covector are said to vary together with, rather than contrary to, the basis.

The Behavior Described

Same-Sense Scaling Relationship

When a basis vector is rescaled by some factor, the covector component associated with that direction rescales by the same factor rather than its reciprocal, matching the direction of change experienced by the basis vector itself.

ω_{i}' = λ ω_{i}

This simple scalar case, where the basis vector along a given direction is stretched by a single factor, illustrates the general principle: the covector component adjusts by that same factor rather than by its reciprocal.

General Case Through the Forward Matrix

For a general change of basis, not merely a uniform rescaling, the same same-sense relationship is captured precisely by the forward coefficient matrix appearing in the full covector component change rule, generalizing the simple direct scaling to a full linear transformation.

ω_{i}' = A_{i}^{j} ω_{j}

Origin of the Term Covariant

Shared Variation

The word covariant describes components that vary together with the basis vectors they accompany, in contrast to contravariant components, which vary contrary to the basis vectors using the inverse matrix rather than the forward matrix directly.

Necessity for Invariant Pairing With Vectors

This same-sense variation is required so that the scalar result of pairing a covector with a vector, which involves the inverse-transforming components of the vector, remains unchanged under a change of basis. If the covector components instead transformed contrary to the basis, matching the vector's pattern, the pairing would fail to remain invariant.

Illustrating the Behavior

Uniform Stretching Example

If every basis vector is stretched by the same factor, every component of every covector expressed relative to that basis grows by exactly that same factor, mirroring the stretching of the basis directly rather than compensating for it.

Non-Uniform Change Example

If the basis vectors are stretched by different amounts or rotated relative to one another, the covector components no longer scale by a single common factor, but each component still adjusts precisely according to the forward matrix, tracking the basis change directly rather than inversely.

Schematic Representation

The diagram compares a short basis vector paired with a proportionally small covector component to a long basis vector paired with a proportionally large covector component, illustrating the same-sense relationship at the heart of covariant behavior.

Role Within the Broader Framework

Tensor Covector Component Covariant Behavior is the qualitative description underlying the covector component change rule, and it generalizes directly to every lower index of a tensor of arbitrary rank, each of which exhibits the same shared variation relative to the basis vectors of the corresponding vector space.