9.14.1 Tensor Basis Change Source System

The Tensor Basis Change Source System explains how to transform tensor components under basis changes, essential for understanding coordinate-independent tensor algebra.

Tensor Basis Change Source System is the original basis of a vector space, together with its associated dual basis, from which a transformation to a new basis is initiated. It represents the starting reference frame whose vectors, dual covectors, and tensor components serve as the known quantities that a basis change operation takes as input.

Role Within a Basis Change

Starting Point of the Transformation

Every basis change operation requires two systems: the system being transformed from and the system being transformed to. The source system is the former, providing the basis vectors, dual basis covectors, and component values that already describe the tensor before the transformation is applied.

{e_{i}} \to {{\bar{e}}_{i}}

Here the unbarred basis vectors denote the source system, and the barred basis vectors denote the target system reached after the change.

Reference for the Transformation Matrix

The transformation matrix that carries out a basis change is defined by expressing each new basis vector as a linear combination of the source system's basis vectors. The source system therefore supplies the fixed reference against which the coefficients of this matrix are determined.

{\bar{e}}_{i} = A_{i}^{j} e_{j}

Components of the Source System

Source Basis Vectors

The source system includes an ordered set of basis vectors spanning the vector space, each assigned a fixed contravariant index that labels the corresponding component slot of any tensor expanded in this system.

Source Dual Basis

Alongside the basis vectors, the source system includes the dual basis covectors satisfying the duality relation with the source basis vectors, which are used to label the covariant component slots of tensors expanded in this system.

Source Components

Any tensor already expressed in the source system carries a definite array of components relative to the source basis vectors and dual basis covectors. These are the values that must be transformed once a new basis is introduced.

Requirements on the Source System

Well-Defined Basis

The source system must genuinely constitute a basis, meaning its vectors are linearly independent and span the entire vector space, since a basis change operation presumes a valid starting basis to transform away from.

Fixed Throughout the Operation

The source system must remain fixed and unchanged for the duration of the transformation. Allowing the source system to shift midway through the calculation would make the transformation matrix and the resulting new components inconsistent with either system.

Distinguishing Source from Target

Directionality of the Change

The designation of a system as the source rather than the target is a matter of directionality chosen for the specific transformation being described; the same basis can serve as the source in one transformation and as the target in another, depending on which direction the change is being carried out.

Symmetric Relationship Between the Two Systems

Although the source system is treated as fixed and known while the target system is being reached, the relationship between the two is invertible: reversing the roles and applying the inverse transformation matrix recovers the source system starting from the target system.