10.6.1 Tensor Inverse Basis Source Frame

Tensor Inverse Basis Source Frame is the designation given to the new basis, that is, the basis reached by the forward change of basis, when that basis is treated as the starting point of the inverse basis change rule. It names the frame from which the inverse transformation departs, in contrast to the frame at which the inverse transformation arrives, and its purpose is to remove any ambiguity about which of the two bases involved in a change of basis is serving as the origin of a given transformation step.

Role Within the Inverse Rule

Identifying the Starting Frame of the Reverse Transformation

Every basis change rule, forward or inverse, requires a clearly identified source and a clearly identified target. For the inverse basis change rule, the source frame is the basis that resulted from the original forward change, commonly marked with a prime, and the target frame is the original unprimed basis being recovered.

Here the primed basis vector on the right side is the inverse basis source frame, since it is the quantity from which the unprimed basis vector on the left is computed.

Distinguishing Source From Target Across Directions

Because the same two bases participate in both the forward and inverse rules, the labels of source and target reverse depending on which rule is being applied. Recognizing which basis currently plays the role of source frame prevents the coefficient matrix from being applied in the wrong direction, which would otherwise produce components or basis vectors that do not correspond to any consistent tensor.

Properties of the Source Frame

Determined by the Prior Forward Change

The inverse basis source frame is never chosen independently; it is always the basis that was already constructed as the output of a prior forward change of basis. Its properties, including its relationship to the original basis through the coefficient matrix, are entirely inherited from that forward construction.

Anchoring the Inverse Coefficient Matrix

The inverse matrix used in the inverse basis change rule acts specifically on components and vectors expressed in the source frame. This anchoring means the inverse coefficient matrix cannot be applied to quantities expressed in any other basis without first re-expressing them in the source frame.

The primed contravariant component on the right is again the quantity read from the source frame, feeding into the computation of the unprimed component in the target frame.

Role in Chained Basis Changes

When a sequence of basis changes is performed, each intermediate basis in turn plays the role of source frame for the inverse step that returns to the basis immediately preceding it. Keeping track of which basis is the current source frame at each stage of the chain is what allows a long sequence of changes to be reversed correctly, one step at a time.

Schematic Representation

The diagram places the source frame on the right, representing the basis produced by the earlier forward change, with the arrow of the inverse basis change rule pointing from that source frame back toward the target frame on the left.

Distinction From the Forward Source Frame

Reversal of Roles

In the forward change of basis, the original unprimed basis serves as the source and the new primed basis serves as the target. The inverse basis source frame is precisely the reversal of this assignment: the primed basis, previously the target, becomes the source, and the unprimed basis, previously the source, becomes the target.

Necessity for Correct Matrix Selection

Because the forward matrix and its inverse are different objects acting in opposite directions, correctly identifying the inverse basis source frame is a prerequisite for selecting the inverse matrix rather than the forward matrix when computing a reverse transformation, ensuring the resulting basis vectors and components are consistent with the original basis being recovered.