9.14 Tensor Basis Change Operation
The Tensor Basis Change Operation explains how tensor components transform under basis changes, maintaining their intrinsic properties across coordinate systems.
Tensor Basis Change Operation is the procedure by which the basis vectors, dual basis covectors, and component array of a tensor are transformed from one basis to another while the tensor itself, as an abstract object, remains unchanged. It is the general operation that relates any two valid representations of the same tensor arising from different choices of basis.
Purpose of the Operation
Preserving the Tensor While Changing Its Representation
A tensor exists independently of any basis, but its components can only be written down once a basis is chosen. The basis change operation allows the same tensor to be re-expressed relative to a new basis, producing a new set of components that describes exactly the same object as before.
Necessity in Practice
Because different problems or coordinate systems naturally suggest different bases, the ability to move a tensor's representation from one basis to another is essential whenever results obtained in one frame must be compared with, or combined with, results expressed in a different frame.
Mechanics of the Operation
The Transformation Matrix
The operation is carried out using a transformation matrix that expresses each new basis vector as a linear combination of the old basis vectors. This matrix, together with its inverse, provides the coefficients needed to convert components from the old basis to the new one and back again.
Transformation of Contravariant Components
Contravariant components, carrying upper indices, transform using the inverse of the matrix that transforms the basis vectors, since these components must compensate for the transformation of the basis vectors to leave the tensor itself unchanged.
Transformation of Covariant Components
Covariant components, carrying lower indices, transform using the same matrix that transforms the basis vectors, since dual basis covectors transform in the way needed to preserve the duality relation with the new basis vectors.
General Mixed Tensors
For a tensor with both upper and lower indices, the operation applies the inverse transformation matrix to each contravariant index and the transformation matrix to each covariant index, combining these individually for every index carried by the tensor.
Properties of the Operation
Invertibility
The basis change operation is always invertible, since the transformation matrix relating any two valid bases is itself invertible, and applying the inverse transformation returns every basis vector, dual basis covector, and component to its original value.
Composability
Successive basis changes compose according to matrix multiplication: transforming from one basis to a second and then from the second to a third produces the same result as transforming directly from the first basis to the third using the product of the two transformation matrices.
Invariance of Tensorial Content
Throughout the operation, quantities built from full contraction of all indices, such as scalars obtained from a tensor paired completely with dual tensors, remain numerically unchanged, since these invariants do not depend on the basis in which the tensor happens to be expressed.
Scope of Application
Applies Uniformly to All Tensor Types
The basis change operation applies to scalars, vectors, covectors, and general higher-order tensors alike, with the transformation rule for each type determined entirely by the number and placement of its upper and lower indices.
Foundation for Coordinate Transformations
When bases arise from coordinate systems, the basis change operation underlies the transformation of tensor components between coordinate charts, making it a foundational tool wherever tensors are used to describe quantities independent of a particular choice of coordinates.