9.12 Tensor Basis Expansion Operation

Tensor Basis Expansion Operation is the concrete, step-by-step procedure that takes a tensor and a chosen basis as its input and produces, as its output, the finite sum of basis elements weighted by coefficients that reconstructs the tensor exactly, comprising the determination of each coefficient followed by the assembly of those coefficients with their matching basis elements into a single sum; it is the operational counterpart to the more general expansion role, specifying exactly how expansion is to be carried out rather than only why it is valid.

The Two Steps of the Operation

Determining Each Coefficient

The first step of the expansion operation applies the component assignment mechanism, pairing the tensor with each combination of basis and dual-basis elements in turn, to determine the coefficient that will multiply the corresponding basis element in the final sum.

Assembling the Weighted Sum

The second step of the operation multiplies each basis element by its determined coefficient and sums the results over every admissible index combination, producing a single tensor equal to the one that was expanded.

Requirements for the Operation to Succeed

A Complete Basis Must Be Available

The expansion operation requires a genuine basis of the tensor space in question, spanning it completely; if the elements supplied do not span the space, some tensors will have no combination of coefficients capable of reproducing them, and the operation will fail to reconstruct such tensors.

Every Coefficient Must Be Correctly Determined

Because the operation's second step depends entirely on the coefficients determined in its first step, any mistake made while pairing the tensor against the basis and dual-basis elements propagates directly into an incorrect final sum, one that fails to equal the original tensor.

Restricting the Operation Using Component Support

Skipping Vanishing Terms

Once a tensor's component support has been identified, the expansion operation can be restricted to sum only over those index combinations known to have a nonzero coefficient, producing the same result as summing over every combination while avoiding unnecessary terms.

Efficiency Gains From a Well Chosen Basis

Because a well selected basis tends to concentrate a tensor's component support into a small number of index combinations, carrying out the expansion operation using such a basis requires assembling fewer terms than would be required using a poorly chosen basis, directly linking the practical cost of the operation to the earlier basis selection stage.

Diagram of the Expansion Operation

Consequences of Treating Expansion as an Operation

It Makes the Procedure Repeatable and Checkable

Defining expansion as a concrete two-step operation allows it to be applied repeatedly to different tensors within the same basis, and allows each application to be checked step by step, first verifying the determined coefficients and then verifying the assembled sum, rather than relying on the expansion role's justification alone.

It Connects Directly to Array Construction

Because the coefficient determination step of the expansion operation coincides with component assignment, and the resulting coefficients are exactly what array construction gathers into a completed structure, the expansion operation and the coordinate representation process share their initial steps, differing only in how their outputs are ultimately used.