2.10.5 Tensor Coordinate Vector Change Context

Understanding how tensor coordinates transform under vector changes, essential for maintaining tensor invariance across different bases.

Tensor Coordinate Vector Change Context is the framework describing how the coordinate vector of a fixed abstract vector transforms when the ordered basis used to describe the vector space is replaced by a different ordered basis, using an explicit change of basis matrix to translate between the two coordinate descriptions. This context is essential whenever a computation begins in one basis but must be interpreted, compared, or continued in another.

Formal Statement

Change of Basis Matrix

Given two ordered bases of the same vector space, there exists an invertible matrix, called the change of basis matrix, whose columns are the coordinate vectors of the new basis vectors expressed in terms of the old basis.

P = [\begin{matrix} [_{b_{1}}]_{B_{old}} & \dots & [_{b_{n}}]_{B_{old}} \end{matrix}]

Coordinate Transformation Formula

Applying this matrix to a coordinate vector expressed relative to the new basis recovers the coordinate vector of the same abstract vector relative to the old basis.

[_{v}]_{B_{old}} = P [_{v}]_{B_{new}}

Invertibility and Reversibility

Guaranteed Invertibility of the Change Matrix

Because both sets of vectors involved are bases of the same vector space, the change of basis matrix is always invertible, guaranteeing that the transformation between coordinate systems can be reversed without loss of information.

Reverse Transformation

Applying the inverse of the change of basis matrix converts coordinates from the old basis back to the new basis, completing a consistent round trip between the two coordinate descriptions.

[_{v}]_{B_{new}} = P^{- 1} [_{v}]_{B_{old}}

What Is Preserved and What Changes

The Underlying Vector Is Preserved

Throughout any change of basis, the actual vector being represented never changes; only its numerical coordinate description shifts to reflect the new choice of basis.

Linear Maps Transform Alongside Coordinates

Matrices representing linear maps must be transformed by conjugation with the change of basis matrix when the basis changes, ensuring that the action of the map on vectors remains consistent regardless of which coordinate system is used to describe the input and output.

Role in Tensor Construction

Coordinated Change Across Tensor Factors

When a tensor is built from several vector spaces, a change of basis in any one factor space requires a corresponding coordinate transformation in the tensor's components, applied along the index associated with that particular factor.

Necessity for Comparing Tensors Across Bases

Coordinate vector change context is what makes it possible to compare or combine tensor data computed in different bases, by first transforming all relevant coordinate and component data into a shared basis.

Summary of Key Properties

Systematic Translation Between Coordinate Systems

Tensor Coordinate Vector Change Context provides a systematic, invertible procedure for translating coordinate descriptions of a vector between any two chosen bases.

Foundation for Basis-Independent Reasoning

By making explicit how coordinates change under a change of basis, this context allows conclusions to be drawn about properties of vectors and tensors that do not depend on any particular basis, even while all computations are performed in coordinates.