10.4.2 Tensor Change Matrix Target Basis Coordinates

Learn how to transform tensor coordinates between bases using change matrices in tensor algebra.

Tensor Change Matrix Target Basis Coordinates is the structural convention in which the entries of the change-of-basis matrix are populated with the coordinates of the target basis vectors expressed in the source basis, forming the standard forward matrix (A) used throughout the ordinary component transformation law.

The Coordinate Structure Defined

Each Column Holds One Target Basis Vector's Old Coordinates

Under this convention, the (i)-th column of the matrix (A) contains the coordinates of the target basis vector (e'_i), expressed relative to the source basis ({e_j}).

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

Every entry (A^j_{\ i}) is therefore directly readable as one coordinate value: specifically, the amount of the source basis vector (e_j) contained in the target basis vector (e'_i).

Why This Is the Standard Convention

This target-basis-coordinates structure is the convention most commonly adopted because it aligns directly with the forward direction of a transformation, from a known source basis to a desired target basis, so that the matrix (A) itself directly encodes the destination basis being sought.

Structural Consequences of This Convention

The Contravariant Rule Requires the Inverse

Because (A) is built from target-basis-in-source-coordinates, and because contravariant components must transform oppositely to the basis vectors to preserve the underlying tensor, this convention is exactly what produces the requirement that contravariant components use (A^{-1}) rather than (A) itself.

v^{i}' = {(A^{- 1})}_{j}^{i} v^{j}

Covariant Components Match the Coordinate Structure Directly

Covariant components, by contrast, transform directly with (A), matching the same target-basis-coordinates structure used to build the matrix from the basis vectors themselves.

ω_{i}' = A_{i}^{j} ω_{j}

Reading Coordinates Directly From a Concrete Basis Change

A Worked Numerical Illustration

If a new basis vector is given explicitly as (e'_1 = 2e_1 - e_2), the target-basis-coordinates structure identifies the first column of (A) immediately: the entry pairing with (e_1) is (2), and the entry pairing with (e_2) is (-1), with no further computation needed beyond reading the given expansion.

A_{1}^{1} = 2, A_{1}^{2} = - 1

Relationship to the Source Basis Columns Convention

This target-basis-coordinates matrix (A) is exactly the matrix inverse of the source-basis-columns matrix built from the reverse expansion, confirming that the two structural conventions are mutually consistent descriptions of the same underlying pair of bases, related by inversion rather than by any independent computation.

A = B^{- 1}

Visual Illustration

Why This Coordinate Structure Is Foundational

Recognizing every entry of the standard change-of-basis matrix as literally the coordinate of a target basis vector in the source basis is what makes the matrix constructible directly from concrete geometric or algebraic data, rather than treated as an abstract symbol manipulated only formally. This structural understanding underlies both the practical construction of (A) from explicit basis vectors and the theoretical derivation of why contravariant and covariant components require opposite matrix factors during transformation.