10.3.2 Tensor Basis Transformation Target Basis

Tensor Basis Transformation Target Basis is the specific role played by the destination basis within the basis transformation process, referring to the new basis into which a tensor's components are being converted and relative to which every primed quantity in the transformation is measured.

Defining Role of the Target Basis

The Destination of the Transformation

The target basis ({e'_i}) is the basis toward which the transformation is directed; every primed component symbol produced by the transformation law refers specifically to this target basis.

Requirements the Target Basis Must Satisfy

The target basis must independently satisfy the same requirements as any basis, being linearly independent and spanning the space with exactly (n) vectors; a proposed target set that fails either requirement cannot serve as a valid destination for the transformation, regardless of how it is related to the source basis.

The Target Basis in Matrix Construction

Supplied as the Output of Expansion

Each vector of the target basis is what the change-of-basis matrix expands in terms of the source basis, meaning the target basis vectors are the objects being described, while the source basis vectors are the objects doing the describing.

The Target Basis Determines the Column or Row Structure of A

Because each target basis vector corresponds to one full row or column of coefficients in (A), the number and identity of the target basis vectors directly determines the structure of the matrix used throughout the remainder of the transformation.

Reaching the Target Basis From Components

Applying the Transformation to Arrive at Target Components

Once (A) and its inverse are known, the target basis components of any tensor are obtained by applying the general component law, with the specific matrix or inverse factor chosen according to whether each index is contravariant or covariant.

The Target Basis as the Next Source Basis

Once the transformation is complete, the former target basis frequently becomes the new source basis for any subsequent transformation, such as a second change of basis chained after the first; this relabeling reflects the earlier observation that source and target roles are relative to a specific transformation step, not fixed properties of a basis.

Distinguishing the Target Basis From Related Concepts

Not to Be Confused With the Dual Basis

The target basis is a second basis of the same vector space (V), related to the source basis by the matrix (A); it should not be confused with the dual basis of either the source or target basis, which is a distinct basis of the dual space (V^*) associated with each of them individually.

Target Basis Versus Target Coordinate System

In the setting of curvilinear coordinates, the counterpart of a target basis is a target coordinate system, with the corresponding basis vectors defined as tangent directions to that coordinate system's coordinate lines rather than as a fixed, globally constant set of vectors.

Visual Illustration

Why Correctly Identifying the Target Basis Matters

Explicitly fixing which basis is the target, and consistently using primed notation for every quantity measured relative to it, prevents the transformation process from becoming ambiguous partway through a calculation. Precisely because the target basis of one transformation can become the source basis of the next, careful and explicit labeling at each individual step is what keeps a chain of successive basis transformations consistent and correctly ordered.