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).