2.5.5 Tensor Basis Change Context

Understanding how tensor bases transform under change of basis in multilinear algebra.

Tensor Basis Change Context is the setting in which two distinct bases of the same underlying vector space are simultaneously in play, together with the transition data relating them, that must be established before any statement about how tensor components transform can be made precise. It supplies the frame in which questions such as "how do components change" or "is this expression basis-independent" become well-posed, by fixing exactly which two bases are being compared and how they relate.

The Two-Basis Setting

Original and New Basis

Let $V$ be a vector space of dimension $n$ over a field $F$ . A basis change context consists of two bases,

e_{1}, \dots, e_{n}

referred to as the original basis, and

{\tilde{e}}_{1}, \dots, {\tilde{e}}_{n}

referred to as the new basis, together with the dual bases $e^{1}, \dots, e^{n}$ and ${\tilde{e}}^{1}, \dots, {\tilde{e}}^{n}$ determined by each.

Transition Matrix

The context is completed by specifying the transition matrix $A_{k}^{i}$ , defined by expressing each new basis vector in terms of the original basis:

{\tilde{e}}_{k} = A_{k}^{i} e_{i}

with summation over $i$ , and its inverse ${(A)}_{i}^{- 1} k_{k}$ , satisfying $A_{k}^{i} {(A^{- 1})}_{i}^{l} = δ_{k}^{l}$ , which expresses the original basis vectors in terms of the new ones.

Why Invertibility Is Required

Both Sets Must Remain Bases

The transition matrix $A_{k}^{i}$ is required to be invertible, since both the original and new sets must independently satisfy the spanning and independence properties of a basis. A singular matrix would produce a new set that fails to span $V$ or fails to be independent, placing it outside the basis change context entirely.

The General Linear Group

The set of all admissible transition matrices for a fixed dimension $n$ forms the general linear group $GL (n, F)$ , so a basis change context may equivalently be described as a choice of element of this group, acting on the original basis to produce the new one.

Consequences for the Dual Basis

Induced Transformation

The transition matrix relating the two bases of $V$ determines a corresponding relation between the two dual bases of $V^{*}$ , obtained by requiring that the duality relation ${\tilde{e}}^{i} ({\tilde{e}}_{j}) = δ_{j}^{i}$ continue to hold in the new basis, forcing the dual basis covectors to transform using the inverse transition matrix rather than the transition matrix itself.

Opposite Transformation Behavior

This is the origin of the distinction between covariant and contravariant transformation behavior within the basis change context: basis vectors transform using $A_{k}^{i}$ , while dual basis covectors transform using its inverse, and this opposition is a structural consequence of maintaining the duality relation across the change of basis, not an independent assumption.

Extending the Context to Tensor Spaces

Two Induced Tensor Bases

Once a basis change context is fixed for $V$ , it induces two bases for any tensor space $T_{s}^{r} (V)$ , one built from tensor products of $e_{i}$ and $e^{j}$ , and one built from tensor products of ${\tilde{e}}_{i}$ and ${\tilde{e}}^{j}$ , so the same basis change context that relates two bases of $V$ simultaneously relates two bases of every tensor space over $V$ .

Fixed Tensor, Two Coordinate Descriptions

Within this context, a single tensor $T$ acquires two coordinate assignments, one relative to each induced tensor basis, and the basis change context is precisely what makes it meaningful to compare these two descriptions and to ask how one is obtained from the other.

What the Context Does Not Yet Specify

The Transformation Law Itself

The basis change context establishes the participants, the two bases of $V$ , their dual bases, and the transition matrix, but does not by itself state the explicit formula for how the components of a given tensor $T$ in one induced basis relate to its components in the other. That explicit formula is a further result derived within this context, built from repeated application of $A_{k}^{i}$ and its inverse, one factor per index of the tensor.

Invariance as the Guiding Question

The basis change context exists to make precise the guiding question of tensor algebra: which quantities remain the same object regardless of which basis is used to describe them, and which quantities are artifacts of a particular coordinate choice. Every statement about basis-independence or basis-dependence of a tensor construction presupposes such a context, since without two bases to compare, there is nothing for invariance or variance to be asserted about.

Practical Instances of the Context

Rotations and Coordinate Frames

A common instance of this context arises when $V$ is a physical space and the transition matrix $A_{k}^{i}$ represents a rotation or other change of reference frame, in which case the basis change context describes precisely how measurements of a physical tensorial quantity recorded in one frame relate to measurements recorded in another.

Change of Coordinate Chart

In settings where $V$ arises as a tangent space parametrized by local coordinates, the basis change context corresponds to a change of coordinate chart, with the transition matrix given by the Jacobian of the coordinate transformation, extending the same algebraic pattern to a differential-geometric setting.