10.12.1 Tensor Transformation Matrix Basis Relation

Tensor Transformation Matrix Basis Relation is the defining equation expressing the new basis vectors as linear combinations of the old basis vectors, with the entries of the change-of-basis matrix serving precisely as the coefficients of that linear combination, establishing the transformation matrix not as an abstract object chosen independently but as a direct encoding of how one basis is built out of another. It is the foundational relation from which the transformation matrix's role throughout tensor change of basis originates, since every other use of the matrix, for components of any tensor type, ultimately traces back to this single relation between the two bases.

The Relation Itself

Expressing New Basis Vectors as Combinations of Old Ones

The transformation matrix basis relation states that each new basis vector equals a specific linear combination of the old basis vectors, with the coefficients of that combination given by one row, or one column depending on convention, of the transformation matrix.

Reading the Matrix Entries Directly From the Relation

Each individual entry of the transformation matrix can be read directly from this relation as the coefficient multiplying a specific old basis vector in the expansion of a specific new basis vector, giving every entry of the matrix a concrete geometric or algebraic meaning rather than treating the matrix as a purely formal device.

Consequences of the Relation

Determining the Matrix From the Two Bases

Given any two bases of the same vector space, the transformation matrix basis relation can always be solved to produce the unique matrix relating them, since the coefficients expressing one basis in terms of the other are determined uniquely by the linear independence of the basis vectors.

Requiring Invertibility as a Consequence of the Relation

Because both the old and new basis vectors must each be expressible as linear combinations of the other, following the basis relation in both directions, the transformation matrix arising from this relation is automatically invertible whenever both sets of vectors genuinely form bases of the same space.

Deriving Further Roles From This Relation

Origin of the Component Transformation Rules

Every rule for transforming vector, covector, and higher-rank tensor components is derived by substituting this basis relation into the requirement that tensors remain invariant under a change of basis, meaning the entire body of component transformation rules is a direct logical consequence of this single relation between basis vectors.

Foundation for the Dual Basis Transformation

The transformation of the dual basis covectors, and consequently of covariant components, is likewise derived from this same basis relation, combined with the defining pairing between basis vectors and dual basis covectors that fixes how the dual basis must respond to any given change in the primal basis.

Schematic Representation

The diagram shows a new basis vector, drawn in blue, expressed as a specific combination of the old basis vectors, with the coefficients of that combination constituting one row of the transformation matrix as fixed by the basis relation.