📢The Orthogonality and Invariance of Cartesian Affine Transformations

The derivation shows that a Cartesian coordinate transformation, which is an affine transformation that preserves the form of the metric tensor, necessarily implies that the transformation matrix is orthogonal. This is mathematically expressed as the orthogonality condition. This requirement ensures that the transformation represents a rigid-body motion in Euclidean space. Furthermore, using this orthogonality condition, the inverse relationship can be derived and shown to have the same affine form.

Affine Transformations and the Orthogonality of Cartesian Rotations | Proof and Derivationviadean on Notion