10.7.1 Tensor Component Law Basis Input

Understanding how tensor components transform under basis changes is foundational to mastering tensor algebra in physics and mathematics.

Tensor Component Law Basis Input is the pair of bases, together with the change-of-basis matrix and its inverse relating them, that the tensor component transformation law requires as given data before it can produce any transformed component array. It is the collection of prerequisite information, distinct from the tensor's own components, that must be supplied to the law in order for the transformation to be carried out at all.

Composition of the Input

The Old Basis and the New Basis

The most basic elements of the basis input are the two bases themselves, the one in which the tensor's components are already known and the one in which its components are to be found. Without both of these specified, the component transformation law has no defined starting point or destination.

The Change-of-Basis Matrix

Alongside the two bases, the basis input includes the matrix of coefficients expressing the new basis vectors as linear combinations of the old basis vectors. This matrix is what actually enters the component transformation law as the forward factor applied to contravariant indices.

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

The Inverse Matrix

Because the law requires a separate factor for covariant indices, the basis input must also supply, or make derivable, the inverse of the change-of-basis matrix, which is contracted against every lower index of the tensor being transformed.

{(A^{- 1})}_{j}^{i}

Requirements on the Input

Invertibility of the Matrix

The change-of-basis matrix supplied as part of the basis input must be invertible, since the component transformation law relies on both the forward matrix and its inverse being well defined. A singular matrix would fail to supply a valid basis input, because the corresponding candidate new basis would not actually be a basis.

Matching Dimension to the Tensor's Vector Space

The basis input must be defined on the same vector space, of the same dimension, as the one underlying the tensor being transformed. A change-of-basis matrix defined on a different or mismatched vector space cannot serve as valid input to the component transformation law for that tensor.

Sufficiency Without the Tensor's Own Components

The basis input, by itself, contains no information about the particular tensor being transformed; it is entirely determined by the two bases involved. This means the same basis input can be reused as the input to the component transformation law for any number of different tensors defined on the same vector space.

Role Within the Transformation Process

Supplying the Matrix Factors Applied to Each Index

Once the basis input is fixed, the component transformation law simply reads off, for each index of the tensor, whether to apply the forward matrix or the inverse matrix from that same fixed input, without needing any additional basis information.

T_{j}^{i}' = {(A^{- 1})}_{k}^{i} A_{j}^{l} T_{l}^{k}

Reusability Across Multiple Applications of the Law

Because the basis input does not depend on which tensor is being transformed, a single basis input, once established for a particular change of basis, can be applied repeatedly to transform scalars, vectors, covectors, and higher-rank tensors defined on the same underlying space, without recomputing the matrix or its inverse each time.

Schematic Representation

The diagram shows the two components of the basis input, the pair of bases and their associated matrix and inverse, feeding together into the component transformation law as its required prerequisite data.