10.4.1 Tensor Change Matrix Source Basis Columns

Tensor Change Matrix Source Basis Columns is the specific structural convention in which the columns of a change-of-basis matrix are populated with the coordinates of the source basis vectors expressed in the target basis, an alternative arrangement to the convention that places target basis vectors' coordinates in the columns, and one that governs an equally valid but oppositely directed matrix.

The Column Convention Defined

Each Column Holds One Source Basis Vector's New Coordinates

Under this convention, the matrix (B) is built so that its (i)-th column contains the coordinates of the source basis vector (e_i), expressed relative to the target basis ({e'_i}).

Concretely, this means expanding each source basis vector (e_i) in terms of the target basis vectors and reading off the resulting coefficients as one column of (B), which is the reverse expansion direction from the more commonly presented convention.

Relationship to the Standard Forward Matrix

This source-basis-columns matrix (B) is precisely the inverse of the standard forward matrix (A), since (A) is built from target basis vectors expanded in the source basis while (B) is built from source basis vectors expanded in the target basis.

Why This Alternative Convention Arises

Symmetry Between the Two Bases

Because the labeling of source and target is a relative choice, expanding the target basis in the source basis and expanding the source basis in the target basis are equally natural operations; the source-basis-columns matrix arises whenever a derivation or textbook happens to build its matrix starting from the basis being transformed away from, rather than the basis being transformed toward.

Consistency Requirement When Adopting This Convention

Once the source-basis-columns convention is adopted for a given calculation, the component transformation formulas must be adjusted accordingly, applying (B) wherever the standard exposition would apply (A^{-1}), and applying (B^{-1}) wherever it would apply (A); mixing the two conventions within a single calculation without this adjustment produces a numerically inconsistent result.

Reading Individual Columns

A Column as a Direct Coordinate Tuple

Just as with the standard convention, an individual column of (B) is directly readable as the coordinate tuple of one specific source basis vector, this time expressed in the target basis, giving each column a concrete geometric meaning rather than treating the matrix as an abstract array of numbers.

Practical Considerations When Using This Structure

Verifying Which Convention a Source Uses

Because both conventions produce valid, mutually inverse matrices, a practical first step when consulting an external derivation is to determine explicitly which convention, target-basis-columns or source-basis-columns, is in use, since applying formulas from one convention to a matrix built under the other produces systematically inverted results.

Building B Directly When Convenient

In situations where the source basis vectors' coordinates in the target basis are more directly available than the reverse, constructing (B) directly from this data, rather than constructing (A) first and inverting it, is often the more efficient practical route to the same final transformation.

Visual Illustration

Why Recognizing This Convention Matters

Recognizing the source-basis-columns structure as a legitimate, systematically inverse alternative to the more commonly presented convention prevents confusion when working across different texts or derivations that may adopt either arrangement. Understanding the relationship (B = A^{-1}) between the two conventions is what allows a practitioner to translate confidently between them without redoing the underlying basis change calculation from scratch.