6.19.4 Tensor Vector Transformation Pattern

Tensor Vector Transformation Pattern is the specific rule, v'^i = B^i_k v^k, that dictates how the components of a vector change when the underlying basis is replaced by another, using the inverse B of the transition matrix A that relates the new basis vectors to the old ones. This pattern is the archetype of contravariant transformation, the simplest case in which the general transformation law for type (p, q) tensors is applied, and understanding it thoroughly is the essential prerequisite for understanding every more elaborate transformation pattern encountered later in tensor algebra.

Deriving the Pattern from Basis Invariance

Setting Up the Two Bases

Let {e_i} be an original basis and {e'_i} a new basis related by a transition matrix A, so that e'_i = A^k_i e_k, with A invertible since both collections are bases of the same space. Denote the inverse matrix by B = A^{-1}.

Requiring the Vector to Remain Unchanged

A fixed vector v can be expanded in either basis, v = v^i e_i = v'^j e'_j. Substituting the relation between the bases into the second expansion gives:

Comparing this to the original expansion v = v^k e_k and using the linear independence of the basis {e_k} forces the coefficients of each e_k to match exactly:

Solving this relation for the new components using the inverse matrix B produces the vector transformation pattern:

Why the Pattern Uses the Inverse Matrix

Compensating for the Change in Basis Scale

If the new basis vectors e'_i are, for instance, twice as long as the old ones in some direction, then representing the same fixed vector v in the new basis requires correspondingly smaller coefficients in that direction, exactly half as large, to compensate; this compensating, inverse relationship between the scale of the basis and the scale of the components is the intuitive content of the pattern, and it is captured precisely by the matrix B = A^{-1} appearing in the formula.

The Term Contravariant

This inverse relationship, in which the components vary opposite to the basis vectors, is the origin of the term "contravariant" applied to vector components, distinguishing this pattern sharply from the covariant transformation pattern of covectors, which uses A directly rather than its inverse.

Verifying Consistency of the Pattern

Composing Two Changes of Basis

If a second change of basis, with transition matrix A', is applied after the first, the vector transformation pattern composes correctly: transforming first by B = A^{-1} and then by B' = (A')^{-1} gives the same result as transforming directly by (AA')^{-1}, confirming that the pattern respects the group structure of composing changes of basis, as it must for the notion of "the same vector in a new basis" to be well defined regardless of how many successive changes are performed.

Returning to the Original Basis

Applying the transformation pattern with A and then applying it again with A^{-1} returns the original components exactly, since B A = A^{-1} A = I, confirming that the pattern is invertible and that no information about the vector is lost by changing basis and then changing back.

Diagram of the Vector Transformation Pattern

The Vector Pattern as the Foundation for General Contravariant Behavior

Extending to Multiple Contravariant Slots

The vector transformation pattern, using one factor of B, is repeated once for each additional upper index when transforming a tensor of higher contravariant count, so a type (2,0) tensor uses two factors of B, one for each of its two slots, following exactly the same derivation as above applied independently to each index.

The Vector Pattern as the Simplest Nontrivial Case

Because the vector transformation pattern involves only a single index and a single factor of B, it serves as the clearest and simplest setting in which to understand why contravariant tensors transform the way they do, before the added notational complexity of multiple indices, multiple factors, and possible symmetry considerations enters the picture for higher-order tensors.