10.8 Tensor Vector Component Change Rule

Tensor Vector Component Change Rule is the specific instance of the tensor component transformation law that applies to a rank-one contravariant tensor, stating that the components of a vector transform under a change of basis by contraction with the inverse of the change-of-basis matrix, in contrast to the basis vectors themselves, which transform directly with the forward matrix. It is the simplest nontrivial case of the general transformation law and serves as the foundational example from which the transformation behavior of covectors and higher-rank tensors is understood by comparison and contrast.

Statement of the Rule

The Transformation Formula

Given a change of basis described by a forward coefficient matrix, the new components of a vector are obtained from its old components by contracting with the inverse of that matrix.

Reverse Direction of the Formula

Conversely, the old components can be recovered from the new components by contracting with the forward matrix itself, following the inverse basis change rule applied specifically to a vector.

Why the Rule Takes This Form

Contrast With Basis Vector Transformation

Because the vector itself, as a geometric or algebraic object, must remain unchanged under a change of basis, and because the basis vectors transform with the forward matrix, the components must transform with the inverse matrix so that the two transformations cancel when the components are recombined with the new basis vectors.

Origin of the Term Contravariant

This opposite, or contrary, transformation behavior relative to the basis vectors is precisely the reason a vector's components are described as contravariant: they vary in a manner contrary to the variation of the basis itself, increasing in magnitude when the basis vectors shrink and decreasing when the basis vectors grow.

Properties of the Rule

Linearity in the Original Components

The vector component change rule is linear in the old components, meaning the new components of a sum of two vectors equal the sum of the new components of each vector individually, and the new components of a scaled vector equal the same scale applied to the new components of the original vector.

Dependence Only on the Matrix, Not the Vector

The specific numerical values that appear in the transformation, namely the entries of the inverse coefficient matrix, depend only on the chosen change of basis and not on which particular vector is being transformed, so the same inverse matrix applies uniformly to every vector defined on the same space.

Consistency Under Composition of Basis Changes

Applying the vector component change rule for a change from one basis to a second, followed by the rule for a change from the second basis to a third, yields the same result as applying the rule once for a direct change from the first basis to the third, with the two inverse matrices composing by ordinary matrix multiplication.

Schematic Representation

The blue arrow represents a single fixed vector whose components differ when read against the black basis compared to the red basis, with the two sets of components related precisely by the vector component change rule.

Position Within the Broader Framework

The vector component change rule is the rank-one contravariant case of the general tensor component transformation law, and it directly informs the more general rule by fixing the pattern that an upper index is always paired with the inverse coefficient matrix, a pattern that extends unchanged to every contravariant index of a tensor of any rank.