10.3.1 Tensor Basis Transformation Source Basis

Understanding how tensor basis transformations work within the source basis framework in algebra.

Tensor Basis Transformation Source Basis is the specific role played by the original, starting basis within the basis transformation process, referring to the basis in which a tensor's components are already known and from which every subsequent step of the transformation departs.

Defining Role of the Source Basis

The Starting Point of Every Transformation

The source basis ({e_i}) is, by definition, the basis relative to which the tensor's components are given before any transformation is applied; every quantity appearing without a prime in the standard transformation notation refers to this source basis.

v = v^{i} e_{i}

Requirements the Source Basis Must Satisfy

Like any basis, the source basis must be linearly independent and span the full vector space, with exactly (n) vectors matching the space's dimension; a set failing to meet these requirements cannot serve as a valid source basis, and the transformation process cannot legitimately begin from it.

\det ([e_{1} \dots e_{n}]) \neq 0

The Source Basis in Matrix Construction

Supplying the Reference Frame for A

The change-of-basis matrix (A) is defined by expressing each target basis vector as a linear combination of the source basis vectors, so the source basis directly supplies the frame of reference against which every entry of (A) is measured.

e_{i}' = A_{i}^{j} e_{j}

Changing which basis is designated as the source, even while keeping the same pair of bases, would produce the reciprocal matrix (A^{-1}) in place of (A), showing that the labeling of source versus target is a convention, not an intrinsic property of the two bases themselves.

The Source Basis and Component Labeling

Unprimed Symbols Belong to the Source Basis

By the standard notational convention, unprimed component symbols, such as (v^i) or (T^i_{\ j}), are always understood to be measured relative to the source basis, while primed symbols refer to the target basis reached after transformation.

v^{i} (source basis components)

Consistency of Labeling Across a Calculation

Maintaining the same designation of which basis is the source throughout an entire calculation is essential; switching which basis is treated as the source partway through a derivation, without correspondingly adjusting every matrix and component symbol, is a frequent source of sign or transformation-direction errors.

Source Basis Versus Target Basis

A Relative, Not Absolute, Distinction

Whether a given basis counts as the source or the target depends entirely on the direction of the transformation currently under consideration; a basis serving as the source in one transformation may equally serve as the target in the reverse transformation, with no change to the basis itself.

{e_{i}} source in one direction, target in the reverse

Visual Illustration

Why Identifying the Source Basis Carefully Matters

Correctly identifying the source basis, and consistently treating its components as unprimed throughout a calculation, is what anchors every subsequent step of the basis transformation process. Since the roles of source and target are purely relative to the direction of transformation being performed, a practitioner must fix this labeling explicitly at the outset of a calculation to avoid inadvertently reversing the direction of the transformation partway through.