10.12.2 Tensor Transformation Matrix Coordinate Relation

Tensor Transformation Matrix Coordinate Relation is the equation expressing how the coordinate values assigned to a fixed point or a fixed vector change when the underlying coordinate system is changed, using the same transformation matrix that relates the two bases but applied specifically to numerical coordinate labels rather than to abstract tensor components in general. It connects the purely algebraic role of the transformation matrix, as established through its basis relation, to the more concrete and computational setting in which specific numerical coordinates are assigned to points or vectors and must be recomputed after a change of coordinate system.

The Relation Itself

Old and New Coordinates of the Same Vector

Given a vector with known coordinates relative to an old coordinate system, the coordinate relation expresses the coordinates of that same vector relative to a new coordinate system as a function of the old coordinates, mediated entirely by the transformation matrix.

This equation mirrors the vector component change rule exactly, since coordinates assigned to a vector are simply a particular instance of contravariant components, expressed here using the letter conventionally reserved for coordinate labels rather than a generic component symbol.

Consistency With the Basis Relation

The coordinate relation is not an independent postulate but follows directly from combining the transformation matrix basis relation with the requirement that the point or vector being described remain fixed, exactly as the general component transformation rules do for arbitrary tensors.

Special Considerations for Coordinates

Coordinates as a Special Case of Contravariant Components

Because coordinates describe the position or displacement of a point relative to an origin and a basis, they behave under a change of basis in the same contravariant manner as vector components generally, making the coordinate relation a direct specialization rather than a separate rule.

Role of the Origin in Affine Settings

When coordinates describe points in an affine space rather than displacement vectors from a fixed origin, the coordinate relation may additionally require a translation term accounting for a shift in origin, alongside the linear transformation governed by the matrix, though the matrix itself continues to govern the linear part of the relation unchanged.

Consequences of the Relation

Enabling Practical Coordinate Computations

The coordinate relation is what allows a concrete numerical coordinate, obtained by measurement or calculation in one coordinate system, to be converted directly into the corresponding coordinate in another system, using only the transformation matrix without needing to re-derive the underlying tensor transformation theory each time.

Preserving Geometric or Physical Meaning

Because the coordinate relation is a direct instance of the general vector component change rule, the point or vector being coordinatized remains the same fixed geometric or physical object throughout, with only its numerical description changing according to the relation.

Schematic Representation

The diagram shows a single fixed point described by two different coordinate labels, related to one another by the transformation matrix through the coordinate relation, while the point itself remains unchanged.