The rotation group is the set of all rotations of a coordinate frame. One can write a realization of this group as a set of 3d matrices that map to :
This is tedious to write out! We will compress this notationally to this expression for each (the th) of the vector coordinates:
In many physics books - especially at higher levels - it is pointless to even write the summation sign; people use the Einstein summation convention that repeated indices in an expression are to be summed:
(three equations, note well, one each for ).
One can easily prove that the transformations of this group leave the lengths (magnitudes) but not the directions of position vectors unchanged. Indeed, the ``correct'' way to derive a representation for the rotation matrix (which we will also write ) is to find the set of all infinitesimal transformations that leave the length of a vector unchanged - their composition forms the Lie (continuous) rotation group, SO(3).