7.8.4 Tensor Vector Component Transformation Behavior

Tensor Vector Component Transformation Behavior describes how vector components change under coordinate transformations in tensor algebra.

Tensor Vector Component Transformation Behavior is the precise rule describing how a vector's components change when the basis of the underlying vector space is changed, characterized by the use of the inverse of the change-of-basis matrix, a behavior that qualifies the components as contravariant.

Definition and Scope

The Transformation Formula

If a new basis (e_i') relates to an old basis by (e_i' = A_i^{\ k} e_k), a vector's components transform according to

v^{'} = {(A^{- 1})}_{k}^{i} v^{k}

using the inverse of the matrix (A) that relates the new basis vectors to the old ones, rather than (A) itself.

Why the Inverse Appears

The inverse appears because the vector (v = v^i e_i) must remain the same object regardless of basis; if the basis vectors are scaled up by a factor, the coefficients multiplying them must scale down by the same factor to keep their product, the vector itself, unchanged. This compensating relationship between the change in the basis and the change in the components is what defines contravariant behavior.

Structural Properties

Composition of Successive Changes of Basis

Applying two changes of basis in sequence, first by (A) and then by (B), transforms a vector's components by the inverse of the combined matrix (BA), consistent with applying the individual inverse transformations in the reversed order:

v^{''} = {(B A)}^{- 1} v = A^{- 1} B^{- 1} v

a consistency requirement that any valid transformation law for tensor components must satisfy.

Invariance of the Vector Itself

Although the components change under this transformation, the underlying vector (v) does not; transformation behavior describes how the representation of (v) changes, not any change to (v) as an abstract object, a distinction central to the entire notion of basis dependence in tensor algebra.

Contrast With Covector Transformation

The defining feature separating vector transformation behavior from covector transformation behavior is exactly this use of the inverse: covector components transform using the matrix (A) directly, so that the dual pairing between a vector and a covector, which multiplies a component of each together and sums, remains invariant, since the factors of (A) and (A^{-1}) introduced by the two transformation laws cancel.

ω_{i}' v^{i}' = ω_{i} v^{i}

Role Within Tensor Algebra

Defining Criterion for the Contravariant Label

Vector component transformation behavior is precisely what the label contravariant refers to throughout tensor algebra; any object whose components transform with the inverse of the change-of-basis matrix, regardless of what it represents physically, is classified as carrying an upper, contravariant index by virtue of this behavior alone.

Generalization to Higher-Rank Tensors

The transformation behavior of a single upper index in a vector is the building block from which the transformation law of any tensor with multiple upper indices is constructed, with one factor of the inverse matrix applied per upper index, so that understanding this single-index case in full is sufficient to derive the general rule for tensors of arbitrary rank.