7.6.3 Tensor Component Basis Assigned Value

Tensor Component Basis Assigned Value is the specific number produced at a given index address once a particular basis has been chosen, treated as a value handed to that position by the act of choosing that basis rather than as an intrinsic, basis-free property of the tensor.

Definition and Scope

Assignment as the Result of a Choice

A tensor by itself, as a multilinear map, does not carry numbers; numbers appear only once a basis is selected and the tensor is expanded against it. The basis assigned value at a given index tuple is precisely the coefficient produced by that expansion:

for a ((0,2)) tensor evaluated on basis vectors (e_i) and (e_j), making explicit that the value at position ((i,j)) is the result of applying the tensor to that specific pair of basis vectors, and would differ for a different basis.

Distinguishing Assignment From Inherent Content

Calling a value basis assigned emphasizes that the number is contingent on the basis, not fixed. Reporting a component's numerical value without stating the basis it was assigned under leaves the number without a clear referent, since another basis would assign a different number to nominally the same index position.

Structural Properties

Recomputing Assigned Values Under a New Basis

Changing the basis reassigns every value according to the tensor's transformation law, replacing the old basis assigned values with new ones computed directly from the tensor's action on the new basis vectors and covectors, or equivalently through the standard change-of-basis formula applied to the old values:

Special Bases Producing Simple Assigned Values

Certain choices of basis assign particularly simple values to a tensor's components, such as a basis aligned with a symmetric tensor's principal axes assigning zero to every off-diagonal position, leaving only diagonal entries nonzero. The existence of such a favorable basis is itself a structural property of the tensor, even though the specific favorable values are, like any others, basis assigned.

Assigned Values Versus Invariant Quantities

Not every number derived from a tensor is a basis assigned value in this sense; quantities built to be independent of basis, such as a trace or a determinant, are not reassigned under a change of basis and stand apart from the individual component values, which do change with every choice of basis.

Role Within Tensor Algebra

Clarifying the Interpretation of Numerical Data

Framing a component as a basis assigned value is what justifies treating an array of numbers as representing a tensor only relative to a stated basis, preventing the common error of comparing or combining component arrays that were assigned under different, unstated bases as though they described the same object directly.

Foundation for the Reconstruction Process

Basis assigned values, taken together across every index address, are exactly the data required to reconstruct the tensor from the chosen basis, so the notion of an assigned value and the notion of tensor reconstruction are two views of the same underlying relationship between a tensor, a basis, and the numbers that basis produces.