3.14.1 Tensor Covector Change of Basis Behavior

Understanding how tensor covectors transform under basis changes is key to mastering their behavior in multilinear algebra.

Tensor Covector Change of Basis Behavior is the step-by-step mechanics of recomputing a covector's components when a specific new basis replaces an old one, covering how the change-of-basis matrix is built, how it is applied to covector components, and how the process differs in practice from the parallel computation carried out for ordinary vectors. Where covector transformation behavior states the general rule that covector components transform with the matrix A, the change of basis behavior focuses on the concrete procedure of carrying out that recomputation from start to finish.

The Change-of-Basis Matrix

Building the Matrix from New Basis Vectors

Given an old basis e_1, ..., e_n of V and a new basis f_1, ..., f_n, the change-of-basis matrix A is built by expressing each new basis vector as a linear combination of the old ones and reading off the coefficients as the columns, or rows depending on convention, of A:

f_{i} = \sum_{j = 1}^{n} A_{i}^{j} e_{j}

This matrix A is invertible, since it relates two bases of the same space, and its inverse A^{-1} recovers the old basis vectors from the new ones.

Computing the Inverse Once

For covector components, the relevant matrix in the transformation formula is A itself, matching the matrix used to build the new basis vectors. This is a practical simplification compared to the vector case, where the inverse A^{-1} must first be computed before the transformation of vector components can be carried out. When only covector transformations are needed, the extra step of inverting A can be skipped entirely.

Procedure for Transforming Covector Components

Step 1: Identify the Old Components

Begin with a covector φ expressed in the old dual basis, φ = φ_1 e^1 + ... + φ_n e^n, so the starting data is the list of numbers φ_1, ..., φ_n.

Step 2: Apply the Matrix A

Compute the new components φ'_i by applying the matrix A to the list of old components:

φ_{i}^{'} = \sum_{j = 1}^{n} A_{i}^{j} φ_{j}

This step is a direct matrix-vector multiplication, treating the old components φ_j as a column of numbers and multiplying by A to produce the new components φ'_i.

Step 3: Interpret the Result

The resulting numbers φ'_1, ..., φ'_n are the components of the same covector φ, but now expressed relative to the new dual basis f^1, ..., f^n. The covector itself has not changed; only its numerical description has been updated to match the new coordinate system.

Worked Example in Two Dimensions

Setting Up the Bases

Let V be two-dimensional with old basis e_1, e_2 and new basis f_1 = e_1 + e_2, f_2 = e_1 - e_2. The change-of-basis matrix is:

A = (\begin{matrix} 1 & 1 \\ 1 & -1 \end{matrix})

since f_1 = 1·e_1 + 1·e_2 and f_2 = 1·e_1 + (-1)·e_2 give the rows of A.

Applying the Matrix to a Sample Covector

Suppose φ has old components φ_1 = 3, φ_2 = 1. Applying A:

φ_{1}^{'} = 1 \cdot 3 + 1 \cdot 1 = 4

φ_{2}^{'} = 1 \cdot 3 + (-1) \cdot 1 = 2

so φ has new components φ'_1 = 4, φ'_2 = 2 relative to the dual basis of f_1, f_2.

Checking Consistency with a Fixed Vector

Take a vector v = e_1, which has old components v^1 = 1, v^2 = 0, so φ(v) = 3·1 + 1·0 = 3. Expressed in the new basis, e_1 = ½f_1 + ½f_2, so v has new components v'^1 = ½, v'^2 = ½. Recomputing the pairing with the new components on both sides gives φ'(v') = 4·½ + 2·½ = 3, matching the original value and confirming the change of basis was carried out consistently.

Behavior Under Successive Changes of Basis

Composing Two Changes

If a first change of basis is described by matrix A, taking e to f, and a second change is described by matrix B, taking f to g, the covector components transform successively by A and then by B. Applying the two transformations in sequence is equivalent to applying a single transformation by the product matrix B·A, since covector components transform in the same order as the composition of the basis changes themselves.

Returning to the Original Basis

If a basis is changed and then changed back to the original basis, the covector components must return to their original values. This is guaranteed because the second change-of-basis matrix is A^{-1}, the inverse of the first, so applying A followed by A^{-1} to the components reproduces the identity transformation, leaving the original components φ_i unchanged.

Diagrammatic Summary

The diagram lays out the three-step procedure as a left-to-right pipeline, starting from the old components, passing through multiplication by the matrix A, and producing the new components φ'_i relative to the replaced basis.