Linear Groups

General Linear Group

Claim that the set of all linear operators together with operation of ordinary matrix multiplication is a group. We write it as , where denotes the dimension of the corresponding square matrix. That is:

Special Linear Group

Claim that the set of all linear operators such that determinant is is a group. Write it as where denotes the dimension of the corresponding square matrix.

Orthogonal Group

The orthogonal group, often denoted , is the group of all orthogonal transformations of -dimensional real space under the operation of composition.

Rotation Group

The rotation group, often denoted , is the group of all rotations about the origin of -dimensional Euclidean space under the operation of composition.

e.g. In , counterclockwise rotation about the positive axis by some angle is given by

Proposition

.

Projective Linear Group

Projective Linear Group

The projective linear group on a vector space is the quotient group , where is the subgroup of all nonzero scalar transformations of . These are quotiented out because they act trivially on the associated projective space .