9.14.2 Tensor Basis Change Target System

The Tensor Basis Change Target System explains how tensor components transform under basis changes, enabling coordinate-independent calculations in multilinear algebra.

Tensor Basis Change Target System is the new basis of a vector space, together with its induced dual basis, toward which a tensor's representation is converted during a basis change operation. It is the destination reference frame whose basis vectors, dual covectors, and resulting component values are produced as the output of the transformation.

Role Within a Basis Change

Endpoint of the Transformation

While the source system supplies the starting basis and known components, the target system is the basis into which everything is converted. Every quantity produced by the basis change operation, the new basis vectors, the new dual basis covectors, and the new components, belongs to the target system.

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

Defined Through the Transformation Matrix

Each basis vector of the target system is defined as a specific linear combination of the source system's basis vectors, with the coefficients of that combination given by the transformation matrix. The target system is therefore fully determined once the transformation matrix is specified.

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

Components of the Target System

Target Basis Vectors

The target system includes its own ordered set of basis vectors, distinct from those of the source system, spanning the same vector space and assigned their own set of contravariant indices for labeling component slots.

Target Dual Basis

Corresponding to the target basis vectors, the target system includes an induced dual basis of covectors satisfying the duality relation with the target basis vectors, used to label covariant component slots for any tensor expressed in the target system.

Target Components

Once the transformation is applied, the tensor acquires a new array of components relative to the target system, distinct in value from the source system's components but representing the same underlying tensor.

Requirements on the Target System

Must Also Be a Valid Basis

The target system must itself be a genuine basis, with linearly independent vectors spanning the full vector space; equivalently, the transformation matrix relating it to the source system must be invertible.

Determined, Not Arbitrary, Given the Matrix

Once the transformation matrix is fixed, the target system is completely determined and cannot be chosen independently. Selecting a different target system corresponds to selecting a different transformation matrix from the outset.

Relationship to the Source System

Reversibility

Because the transformation matrix defining the target system is invertible, the roles of source and target can always be reversed: treating the target system as the new source and applying the inverse transformation matrix recovers the original source system.

Chained Target Systems

A target system reached from one source system can itself serve as the source system for a further basis change, producing a new target system in turn. Such chains of basis changes compose through multiplication of the successive transformation matrices, allowing a direct transformation to be computed between any two systems in the chain.