Rotation and Transform

A transformation matrix describes the rotation of a coordinate system while an object remains fixed.

In contrast, a rotation matrix describes the rotation of an object in a fixed coordinate system. 

Much confusion arises from the amazing fact that each matrix (transform and rotation) is just the transpose of the other! 

An example of vector decomposition on coordinate system for the transform of coordinate system.

 

 

Coordinate Transformations, continuummechanics.org

Rotation Matrices, continuummechanics.org