3.14.3 Tensor Covector Component Transformation

Tensor Covector Component Transformation describes how covector components change under coordinate transformations in tensor algebra.

Tensor Covector Component Transformation is the index-level formula expressing how the numerical components of a covector change under a change of basis, derived directly from requiring that the covector's expansion in the old dual basis and its expansion in the new dual basis describe the same fixed covector. This transformation is the computational heart of covector theory: while the abstract covector itself does not change, its component representation does, and the transformation formula is the precise recipe connecting the two representations.

Deriving the Formula from Expansion Equality

The Covector Has Two Valid Expansions

A covector φ in V* can be expanded in either the old dual basis e^1, ..., e^n or the new dual basis f^1, ..., f^n, and both expansions must describe the same object:

φ = \sum_{j = 1}^{n} φ_{j} e^{j} = \sum_{i = 1}^{n} φ_{i}^{'} f^{i}

Substituting the Relation Between Dual Bases

Using the fact that f^i = (A^{-1})^i_k e^k, substitute into the right-hand expansion:

\sum_{i = 1}^{n} φ_{i}^{'} \sum_{k = 1}^{n} {(A^{- 1})}_{k}^{i} e^{k} = \sum_{k = 1}^{n} (\sum_{i = 1}^{n} {(A^{- 1})}_{k}^{i} φ_{i}^{'}) e^{k}

Comparing Coefficients

Since e^1, ..., e^n is a basis, the coefficients of each e^k on both sides of the original expansion equality must match, giving φ_k equal to the bracketed sum. Rewriting this relation to isolate the new components in terms of the old, and inverting the matrix relation, yields the component transformation formula:

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

which reproduces the standard covector component transformation rule, now derived from the requirement that the two expansions of φ agree rather than assumed outright.

Index Placement Conventions

Upper and Lower Index Roles

In the formula φ'_i = A^j_i φ_j, the index i on φ' is a lower index, matching the lower index convention for covector components. The matrix A carries one upper and one lower index, A^j_i, reflecting its role as an object that consumes a lower-indexed quantity, φ_j, and produces another lower-indexed quantity, φ'_i, after summing over the shared index j.

The Einstein Summation Convention

Under the Einstein summation convention, a repeated index appearing once as an upper index and once as a lower index in the same term is automatically summed over, allowing the transformation formula to be written without an explicit summation symbol:

φ_{i}^{'} = A_{i}^{j} φ_{j}

with the sum over j from 1 to n implied by its repetition.

Matrix Form of the Transformation

Writing Components as a Column

The components φ_1, ..., φ_n can be arranged into a column vector Φ, and similarly φ'_1, ..., φ'_n into a column vector Φ'. The transformation formula then takes the compact matrix form:

Φ' = A Φ

where A is treated as an ordinary n × n matrix acting on the column Φ by matrix multiplication.

Row Versus Column Convention

Some presentations instead arrange covector components as a row vector, in which case the transformation is written as Φ' = ΦA^T using the transpose of A, reflecting the fact that covectors are naturally dual, or row-vector-like, objects when vectors are treated as columns. Both conventions describe the same underlying transformation; the choice between them is a matter of notational preference rather than mathematical content.

Worked Example in Three Dimensions

The Setup

Let V be three-dimensional with old basis e_1, e_2, e_3 and a new basis given by the change-of-basis matrix:

A = (\begin{matrix} 2 & 0 & 0 \\ 0 & 1 & 1 \\ 0 & 0 & 1 \end{matrix})

Applying the Transformation

Given old covector components φ_1 = 5, φ_2 = 2, φ_3 = 1, the new components are computed row by row:

φ_{1}^{'} = 2 \cdot 5 = 10

φ_{2}^{'} = 1 \cdot 2 + 1 \cdot 1 = 3

φ_{3}^{'} = 1 \cdot 1 = 1

giving new components 10, 3, 1, illustrating how each new component is a specific linear combination of the old components determined by the corresponding row of A.

Diagrammatic Summary

The diagram represents the matrix form of the component transformation as an ordinary matrix-vector product, with the matrix A acting on the column Φ of old components to produce the column Φ' of new components.